From 764cbc8f9c47c64efdb9fd516735cb87d4a64e9d Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Wed, 15 May 2024 21:07:29 +0300 Subject: [PATCH 01/79] Update Wavelet.mdx --- content/6_Advanced/Wavelet.mdx | 170 ++++++++++++++++++++++++++++----- 1 file changed, 147 insertions(+), 23 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index a01b117038..d6ec142b8b 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -1,40 +1,164 @@ --- id: wavelet title: 'Wavelet Tree' -author: Benjamin Qi +author: Benjamin Qi, Crafticat prerequisites: - RURQ -description: "?" +description: Segment tree over values" frequency: 0 --- -## Wavelet Tree +# Wavelet Tree - -Like a segment tree on values rather than indices. - - - Introduces Wavelet Tree - - - Link in blog post is broken, check my comment. - + + Introduces Wavelet Tree + + + Link in blog post is broken, check my comment. + -### Solution - Range K-th Smallest +Suppose you want to support the following queries: + - Given a range l, r count number of occurrences of value x. + - Given a range l, r find the k smallest element + +With wavelet tree you can easily support those queries in log(n) time. + + It is possible to support with updates in log(n)^2 + + +## Solving the First Type of Query (Given a range l, r count number of occurrences of value x.) + +To calculate the number of occurrences from 𝑙 to 𝑟, we can use the following formula: +$$ +\begin{aligned} +occurrences(l, r) = occurrences(r) - occurrences(l) +\end{aligned} +$$ +This reduces the problem to counting the number of occurrences in a prefix. + +### Approaching the Solution + +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than 𝑥 +However, let's explore a different approach that can also be extended to the second type of query. + +### A different way to find the Index of 𝑟 in the List of Vertices +Instead of binary searching on the leaf, we update 𝑟 as we recurse down the tree. +If we are able to determine the position (index) of 𝑟 in the left and right children of a node +We can recurse down the tree and determine its position in the leaf node. + +### Determining the Position in Child Nodes +To find the position of 𝑟 in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede 𝑟. +This can be done using a prefix sum. + +Let's define: +c[i] = as 1 if index[i] is smaller than mid otherwise 0 +prefixB[i] as prefix sum of c[i] + +Formally +$$ +c[i] = index[i] < mid ? 1 : 0; +prefixB[i] = prefixB[i - 1] + c[i] +$$ + +### Updating 𝑟 as We Recurse +To know the value of 𝑟 if we recurse left, we use prefixB[r] +If we recurse right, we use 𝑟 - prefixB[r] + +## Solving the second type of query (Given a range l, r find the k smallest element) +We will determine whether the answer for a given node is in the left or the right segment. +We can calculate how many times the elements within the segments' ranges appear in our range (l, r) using our first type of query. +Note that this also works for non-leaf nodes using the following formula: +$$ +occurrences(l, r) = r - l +$$ + +Therefore, the occurrences of the left node is +$$ +left_occurrences = prefixB[r] - prefixB[l] +$$ + - If left_occurrences is less than k, it means the k-th smallest element is in the left subtree. Therefore, we update our range and recurse into the left child + - If left_occurrences left_occurrences is greater or equal to k it means the k-th smallest element is in the right subtree. We adjust k by subtracting left_occurrences from k, update our range, and recurse into the right child + + + Notice we still update l, r accordingly when we go left or right + + +the answer then will be the value of the node we end up on (leaf) + +## implemention + + + + ```cpp + struct Segment { + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r, vector> indices) :l(l), r(r), mid((r + l) / 2), indices(std::move(indices)) { + calculateB(); +} + + void update() { + if (children) return; + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + for (auto [ind, val] : indices) + if (val < mid) + leftIndices.emplace_back(ind,val); + else + rightIndices.emplace_back(ind,val); + + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); +} +} + + // Calculates the prefix B + void calculateB() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; +} +} + + int findKSmallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = lr - lb; + + if (k <= inLeft) return left->findKSmallest(lb,lr,k); + else return right->findKSmallest(a - lb,b - lr, k - inLeft); +} +}; + ``` + + - ### Problems From fd171f2f9c3d06af281a3d0e21ad2cfb66a7ac5d Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Wed, 15 May 2024 21:08:00 +0300 Subject: [PATCH 02/79] Update Wavelet.mdx --- content/6_Advanced/Wavelet.mdx | 193 ++++++++++++++++++--------------- 1 file changed, 108 insertions(+), 85 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index d6ec142b8b..f72140cdeb 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -12,85 +12,109 @@ frequency: 0 - - Introduces Wavelet Tree - - - Link in blog post is broken, check my comment. - + + Introduces Wavelet Tree + + + Link in blog post is broken, check my comment. + Suppose you want to support the following queries: - - Given a range l, r count number of occurrences of value x. - - Given a range l, r find the k smallest element + +- Given a range l, r count number of occurrences of value x. +- Given a range l, r find the k smallest element With wavelet tree you can easily support those queries in log(n) time. + It is possible to support with updates in log(n)^2 ## Solving the First Type of Query (Given a range l, r count number of occurrences of value x.) -To calculate the number of occurrences from 𝑙 to 𝑟, we can use the following formula: +To calculate the number of occurrences from 𝑙 to 𝑟, we can use the following +formula: + $$ \begin{aligned} occurrences(l, r) = occurrences(r) - occurrences(l) \end{aligned} $$ + This reduces the problem to counting the number of occurrences in a prefix. ### Approaching the Solution -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than 𝑥 -However, let's explore a different approach that can also be extended to the second type of query. +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than 𝑥 +However, let's explore a different approach that can also be extended to the +second type of query. ### A different way to find the Index of 𝑟 in the List of Vertices -Instead of binary searching on the leaf, we update 𝑟 as we recurse down the tree. -If we are able to determine the position (index) of 𝑟 in the left and right children of a node + +Instead of binary searching on the leaf, we update 𝑟 as we recurse down the +tree. +If we are able to determine the position (index) of 𝑟 in the left and right +children of a node We can recurse down the tree and determine its position in the leaf node. ### Determining the Position in Child Nodes -To find the position of 𝑟 in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede 𝑟. + +To find the position of 𝑟 in a node's left and right children, we need to +determine how many indices are smaller than the middle value (mid) and precede +𝑟. This can be done using a prefix sum. -Let's define: -c[i] = as 1 if index[i] is smaller than mid otherwise 0 +Let's define: +c[i] = as 1 if index[i] is smaller than mid otherwise 0 prefixB[i] as prefix sum of c[i] Formally + $$ c[i] = index[i] < mid ? 1 : 0; prefixB[i] = prefixB[i - 1] + c[i] $$ ### Updating 𝑟 as We Recurse -To know the value of 𝑟 if we recurse left, we use prefixB[r] + +To know the value of 𝑟 if we recurse left, we use prefixB[r] If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query (Given a range l, r find the k smallest element) -We will determine whether the answer for a given node is in the left or the right segment. -We can calculate how many times the elements within the segments' ranges appear in our range (l, r) using our first type of query. + +We will determine whether the answer for a given node is in the left or the +right segment. +We can calculate how many times the elements within the segments' ranges appear +in our range (l, r) using our first type of query. Note that this also works for non-leaf nodes using the following formula: + $$ occurrences(l, r) = r - l $$ Therefore, the occurrences of the left node is + $$ left_occurrences = prefixB[r] - prefixB[l] $$ - - If left_occurrences is less than k, it means the k-th smallest element is in the left subtree. Therefore, we update our range and recurse into the left child - - If left_occurrences left_occurrences is greater or equal to k it means the k-th smallest element is in the right subtree. We adjust k by subtracting left_occurrences from k, update our range, and recurse into the right child + +- If left_occurrences is less than k, it means the k-th smallest element is in + the left subtree. Therefore, we update our range and recurse into the left + child +- If left_occurrences left_occurrences is greater or equal to k it means the + k-th smallest element is in the right subtree. We adjust k by subtracting + left_occurrences from k, update our range, and recurse into the right child Notice we still update l, r accordingly when we go left or right @@ -102,64 +126,63 @@ the answer then will be the value of the node we end up on (leaf) - ```cpp - struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r, vector> indices) :l(l), r(r), mid((r + l) / 2), indices(std::move(indices)) { - calculateB(); -} - - void update() { - if (children) return; - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - for (auto [ind, val] : indices) - if (val < mid) - leftIndices.emplace_back(ind,val); - else - rightIndices.emplace_back(ind,val); - - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); -} -} - - // Calculates the prefix B - void calculateB() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; -} -} - - int findKSmallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = lr - lb; - - if (k <= inLeft) return left->findKSmallest(lb,lr,k); - else return right->findKSmallest(a - lb,b - lr, k - inLeft); -} +```cpp +struct Segment { + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r, vector> indices) :l(l), r(r), mid((r + l) / 2), indices(std::move(indices)) { + calculateB(); + } + + void update() { + if (children) return; + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + for (auto [ind, val] : indices) + if (val < mid) + leftIndices.emplace_back(ind,val); + else + rightIndices.emplace_back(ind,val); + + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + // Calculates the prefix B + void calculateB() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int findKSmallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = lr - lb; + + if (k <= inLeft) return left->findKSmallest(lb,lr,k); + else return right->findKSmallest(a - lb,b - lr, k - inLeft); + } }; - ``` +``` - ### Problems From 9103e4468fce42289e77f5b2c64d4b7331b03bce Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Thu, 16 May 2024 09:54:19 +0300 Subject: [PATCH 03/79] Update Wavelet.mdx --- content/6_Advanced/Wavelet.mdx | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index f72140cdeb..d9670cfe45 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -40,6 +40,10 @@ With wavelet tree you can easily support those queries in log(n) time. It is possible to support with updates in log(n)^2 +#The datastructor +We are going to create a segment tree but instead of on indices on values. +such that a segment between l and r denotes all indices who has a value between l and r. + ## Solving the First Type of Query (Given a range l, r count number of occurrences of value x.) To calculate the number of occurrences from 𝑙 to 𝑟, we can use the following From 0640ad12741dd6f4591d8bd318be69160e6701f5 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Wed, 19 Jun 2024 16:04:00 +0300 Subject: [PATCH 04/79] Update Wavelet.mdx --- content/6_Advanced/Wavelet.mdx | 65 +++++++++++++++++++++++++++++----- 1 file changed, 57 insertions(+), 8 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index d9670cfe45..9a0dcfddba 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -8,6 +8,7 @@ description: Segment tree over values" frequency: 0 --- + # Wavelet Tree @@ -37,12 +38,28 @@ Suppose you want to support the following queries: With wavelet tree you can easily support those queries in log(n) time. - It is possible to support with updates in log(n)^2 + It is possible to support with updates in $\mathcal{O}(\log n^2)$ -#The datastructor -We are going to create a segment tree but instead of on indices on values. -such that a segment between l and r denotes all indices who has a value between l and r. +## Wavelet Tree Structure + +A wavelet tree is a binary tree where each node represents a range of values. +The root node covers the entire range, and each subsequent level splits the +range into two halves. + +## First lets define our Data Structures + +We are going to maintain a segment tree over values instead of indices. +Each segment contains indices whose values lie within the segment's range. +Let's save those indicies in a vector. +Notice that a index can be in at most log(N) segments + + + +Lets say our array is: [3,5,3,1,2,2,3,4,5,5] + +![Wavelet tree Visualization](https://imgur.com/Z7fpTFy.png) + ## Solving the First Type of Query (Given a range l, r count number of occurrences of value x.) @@ -60,11 +77,11 @@ This reduces the problem to counting the number of occurrences in a prefix. ### Approaching the Solution One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than 𝑥 +and perform a binary search for the number of indices less than 𝑟 However, let's explore a different approach that can also be extended to the second type of query. -### A different way to find the Index of 𝑟 in the List of Vertices +### A different way to find the Index of 𝑟 in the list of Vertices Instead of binary searching on the leaf, we update 𝑟 as we recurse down the tree. @@ -106,6 +123,9 @@ Note that this also works for non-leaf nodes using the following formula: $$ occurrences(l, r) = r - l $$ + +This is simular to counting how many times a value appears up to index 𝑅 in our previous query. We did this by using the new 𝑅 value at the leaf node. But now, we consider the difference between the updated 𝑅 and 𝐿 + Therefore, the occurrences of the left node is @@ -113,10 +133,15 @@ $$ left_occurrences = prefixB[r] - prefixB[l] $$ -- If left_occurrences is less than k, it means the k-th smallest element is in + + Note that left_occurrences is the amount of indices between l and r that their value is less than mid + + + +- If left_occurrences left_occurrences is greater or equal to k, it means the k-th smallest element is in the left subtree. Therefore, we update our range and recurse into the left child -- If left_occurrences left_occurrences is greater or equal to k it means the +- If left_occurrences is less than k, it means the k-th smallest element is in the right subtree. We adjust k by subtracting left_occurrences from k, update our range, and recurse into the right child @@ -185,6 +210,30 @@ struct Segment { }; ``` +## Supporting updates + +Lets support updates of type: + - change value at index i to y + +What we can do is recurse down to your leaf in order to remove the old element. And also recurse down to add the new element. + +So what do the updates change? + - + Our indicies vector + Our prefix vector + +### Changing the indicies vector +What we can do is instead of storing a vector use a set. +Then erasing and adding values becomes easy. + +### Changing the prefix vector +Since each update could change our prefix vector a lot, we cant maintain just he normal vector. +What we could do use use a sparse segment tree. + - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index + - quering for a prefix can be done by quering the segment tree from 0 till i +This approach is not really memory effient and also requires the implemention of a segment tree. +A more friendly approach would be using an order statistics tree. + ### Problems From 70f47946fe472e020bd05ac00122fe0596e58a49 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Wed, 19 Jun 2024 16:05:01 +0300 Subject: [PATCH 05/79] Update Wavelet.problems.json --- content/6_Advanced/Wavelet.problems.json | 37 ++++++++++++++++++++++++ 1 file changed, 37 insertions(+) diff --git a/content/6_Advanced/Wavelet.problems.json b/content/6_Advanced/Wavelet.problems.json index f8b1fee726..98bce90ecc 100644 --- a/content/6_Advanced/Wavelet.problems.json +++ b/content/6_Advanced/Wavelet.problems.json @@ -16,6 +16,42 @@ } ], "wavelet": [ + { + "uniqueId": "codeforces_Destiny", + "name": "Destiny", + "url": "https://codeforces.com/contest/840/problem/D", + "source": "CF", + "difficulty": "Normal", + "isStarred": false, + "tags": ["Wavelet"], + "solutionMetadata": { + "kind": "none" + } + }, + { + "uniqueId": "ILKQUERY2", + "name": "ILKQUERY2", + "url": "https://www.spoj.com/problems/ILKQUERY2/", + "source": "SPOJ", + "difficulty": "Normal", + "isStarred": false, + "tags": ["Wavelet"], + "solutionMetadata": { + "kind": "none" + } + }, + { + "uniqueId": "coci-2020-index", + "name": "Index", + "url": "https://evaluator.hsin.hr/tasks/HONI202167index/", + "source": "COCI", + "difficulty": "Normal", + "isStarred": false, + "tags": ["Wavelet"], + "solutionMetadata": { + "kind": "none" + } + }, { "uniqueId": "kattis-easyquery", "name": "Easy Query", @@ -40,6 +76,7 @@ "kind": "autogen-label-from-site", "site": "DMOJ" } + } ] } From 8d77548a1b68452062073ac84d1cea6b0f3c35f3 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Wed, 19 Jun 2024 13:09:10 +0000 Subject: [PATCH 06/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 142 +++++++++++------------ content/6_Advanced/Wavelet.problems.json | 1 - 2 files changed, 70 insertions(+), 73 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 9a0dcfddba..602d9d7194 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -43,15 +43,15 @@ With wavelet tree you can easily support those queries in log(n) time. ## Wavelet Tree Structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. ## First lets define our Data Structures -We are going to maintain a segment tree over values instead of indices. -Each segment contains indices whose values lie within the segment's range. -Let's save those indicies in a vector. +We are going to maintain a segment tree over values instead of indices. +Each segment contains indices whose values lie within the segment's range. +Let's save those indicies in a vector. Notice that a index can be in at most log(N) segments @@ -76,28 +76,28 @@ This reduces the problem to counting the number of occurrences in a prefix. ### Approaching the Solution -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than 𝑟 +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than 𝑟 However, let's explore a different approach that can also be extended to the second type of query. ### A different way to find the Index of 𝑟 in the list of Vertices Instead of binary searching on the leaf, we update 𝑟 as we recurse down the -tree. +tree. If we are able to determine the position (index) of 𝑟 in the left and right -children of a node +children of a node We can recurse down the tree and determine its position in the leaf node. ### Determining the Position in Child Nodes To find the position of 𝑟 in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -𝑟. +𝑟. This can be done using a prefix sum. -Let's define: -c[i] = as 1 if index[i] is smaller than mid otherwise 0 +Let's define: +c[i] = as 1 if index[i] is smaller than mid otherwise 0 prefixB[i] as prefix sum of c[i] Formally @@ -109,15 +109,15 @@ $$ ### Updating 𝑟 as We Recurse -To know the value of 𝑟 if we recurse left, we use prefixB[r] +To know the value of 𝑟 if we recurse left, we use prefixB[r] If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query (Given a range l, r find the k smallest element) We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range (l, r) using our first type of query. +in our range (l, r) using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ @@ -134,7 +134,7 @@ left_occurrences = prefixB[r] - prefixB[l] $$ - Note that left_occurrences is the amount of indices between l and r that their value is less than mid + Note that left_occurrences is the amount of indices between l and r that their value is less than mid @@ -157,81 +157,79 @@ the answer then will be the value of the node we end up on (leaf) ```cpp struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r, vector> indices) :l(l), r(r), mid((r + l) / 2), indices(std::move(indices)) { - calculateB(); - } - - void update() { - if (children) return; - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - for (auto [ind, val] : indices) - if (val < mid) - leftIndices.emplace_back(ind,val); - else - rightIndices.emplace_back(ind,val); - - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - // Calculates the prefix B - void calculateB() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int findKSmallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = lr - lb; - - if (k <= inLeft) return left->findKSmallest(lb,lr,k); - else return right->findKSmallest(a - lb,b - lr, k - inLeft); - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r, vector> indices) + : l(l), r(r), mid((r + l) / 2), indices(std::move(indices)) { + calculateB(); + } + + void update() { + if (children) return; + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + for (auto [ind, val] : indices) + if (val < mid) leftIndices.emplace_back(ind, val); + else rightIndices.emplace_back(ind, val); + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + + // Calculates the prefix B + void calculateB() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int findKSmallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = lr - lb; + + if (k <= inLeft) return left->findKSmallest(lb, lr, k); + else return right->findKSmallest(a - lb, b - lr, k - inLeft); + } }; ``` ## Supporting updates Lets support updates of type: - - change value at index i to y + - change value at index i to y What we can do is recurse down to your leaf in order to remove the old element. And also recurse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indicies vector Our prefix vector ### Changing the indicies vector -What we can do is instead of storing a vector use a set. +What we can do is instead of storing a vector use a set. Then erasing and adding values becomes easy. ### Changing the prefix vector -Since each update could change our prefix vector a lot, we cant maintain just he normal vector. -What we could do use use a sparse segment tree. +Since each update could change our prefix vector a lot, we cant maintain just he normal vector. +What we could do use use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - quering for a prefix can be done by quering the segment tree from 0 till i -This approach is not really memory effient and also requires the implemention of a segment tree. +This approach is not really memory effient and also requires the implemention of a segment tree. A more friendly approach would be using an order statistics tree. diff --git a/content/6_Advanced/Wavelet.problems.json b/content/6_Advanced/Wavelet.problems.json index 98bce90ecc..3f218ef0fd 100644 --- a/content/6_Advanced/Wavelet.problems.json +++ b/content/6_Advanced/Wavelet.problems.json @@ -76,7 +76,6 @@ "kind": "autogen-label-from-site", "site": "DMOJ" } - } ] } From 3c29d6973d0d27e7e4898f0215f907d248cd21e2 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Sat, 22 Jun 2024 22:39:37 +0300 Subject: [PATCH 07/79] Requested changes --- content/6_Advanced/Wavelet.mdx | 222 +++++++++++++++++++-------------- 1 file changed, 129 insertions(+), 93 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 602d9d7194..bcf2b3a212 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -8,8 +8,8 @@ description: Segment tree over values" frequency: 0 --- - # Wavelet Tree +Segment trees over indices. @@ -32,10 +32,10 @@ frequency: 0 Suppose you want to support the following queries: -- Given a range l, r count number of occurrences of value x. -- Given a range l, r find the k smallest element +- Given a range $l$, $r$ count number of occurrences of value $x$. +- Given a range $l$, $r$ find the $k$ smallest element -With wavelet tree you can easily support those queries in log(n) time. +With wavelet tree you can easily support those queries in $log(n)$ time. It is possible to support with updates in $\mathcal{O}(\log n^2)$ @@ -43,27 +43,28 @@ With wavelet tree you can easily support those queries in log(n) time. ## Wavelet Tree Structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. -## First lets define our Data Structures +## Defining the structure -We are going to maintain a segment tree over values instead of indices. -Each segment contains indices whose values lie within the segment's range. -Let's save those indicies in a vector. -Notice that a index can be in at most log(N) segments +We are going to maintain a segment tree over values instead of indices. +Each segment contains indices whose values lie within the segment's range. +Let's save those indicies in a vector. +Notice that a index can be in at most $log(N)$ segments -Lets say our array is: [3,5,3,1,2,2,3,4,5,5] +Lets say our array is: $[3,5,3,1,2,2,3,4,5,5]$ -![Wavelet tree Visualization](https://imgur.com/Z7fpTFy.png) +![Wavelet Tree Visualization](./assets/diagram.png) -## Solving the First Type of Query (Given a range l, r count number of occurrences of value x.) +## Solving the First Type of query +**Given a range l, r count number of occurrences of value x.** -To calculate the number of occurrences from 𝑙 to 𝑟, we can use the following +To calculate the number of occurrences from $𝑙$ to $𝑟$, we can use the following formula: $$ @@ -76,29 +77,30 @@ This reduces the problem to counting the number of occurrences in a prefix. ### Approaching the Solution -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than 𝑟 +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. -### A different way to find the Index of 𝑟 in the list of Vertices +### A different way +**A different way to find the Index of $𝑟$ in the list of vertices** -Instead of binary searching on the leaf, we update 𝑟 as we recurse down the -tree. -If we are able to determine the position (index) of 𝑟 in the left and right -children of a node +Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the +tree. +If we are able to determine the position (index) of $𝑟$ in the left and right +children of a node We can recurse down the tree and determine its position in the leaf node. ### Determining the Position in Child Nodes -To find the position of 𝑟 in a node's left and right children, we need to +To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -𝑟. +$𝑟$. This can be done using a prefix sum. -Let's define: -c[i] = as 1 if index[i] is smaller than mid otherwise 0 -prefixB[i] as prefix sum of c[i] +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +- $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -107,17 +109,18 @@ c[i] = index[i] < mid ? 1 : 0; prefixB[i] = prefixB[i - 1] + c[i] $$ -### Updating 𝑟 as We Recurse +### Updating $𝑟$ as We Recurse -To know the value of 𝑟 if we recurse left, we use prefixB[r] +To know the value of 𝑟 if we recurse left, we use prefixB[r] If we recurse right, we use 𝑟 - prefixB[r] -## Solving the second type of query (Given a range l, r find the k smallest element) +## Solving the second type of query +**Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range (l, r) using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ @@ -134,102 +137,135 @@ left_occurrences = prefixB[r] - prefixB[l] $$ - Note that left_occurrences is the amount of indices between l and r that their value is less than mid + Note that left_occurrences is the amount of indices between l and r that their value is less than mid -- If left_occurrences left_occurrences is greater or equal to k, it means the k-th smallest element is in +- If left_occurrences left_occurrences is greater or equal to $k$, it means the $k$-th smallest element is in the left subtree. Therefore, we update our range and recurse into the left child -- If left_occurrences is less than k, it means the - k-th smallest element is in the right subtree. We adjust k by subtracting - left_occurrences from k, update our range, and recurse into the right child +- If left_occurrences is less than $k$, it means the + $k$-th smallest element is in the right subtree. We adjust k by subtracting + left_occurrences from $k$, update our range, and recurse into the right child - Notice we still update l, r accordingly when we go left or right + Notice we still update $l, r$ accordingly when we go left or right the answer then will be the value of the node we end up on (leaf) +## Time complexity +For each query, we traverse left or right, meaning the number of nodes visited is proportional to the height of the tree. +For each node, we compute the $l$ or $r$ using prefix sums, which is $\mathcal{O}(1)$ for each query. +Hence, the overall time complexity is $\mathcal{O}(\log n)$ + ## implemention ```cpp +#include + +using namespace std; + struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r, vector> indices) - : l(l), r(r), mid((r + l) / 2), indices(std::move(indices)) { - calculateB(); - } - - void update() { - if (children) return; - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - for (auto [ind, val] : indices) - if (val < mid) leftIndices.emplace_back(ind, val); - else rightIndices.emplace_back(ind, val); - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - // Calculates the prefix B - void calculateB() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int findKSmallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = lr - lb; - - if (k <= inLeft) return left->findKSmallest(lb, lr, k); - else return right->findKSmallest(a - lb, b - lr, k - inLeft); - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) return; + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), + [this](const pair& elem) { + return elem.second < mid; + }); + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) + return left->find_k_smallest(lb,lr,k); // Appears in left + else + return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right + } }; + + +int main() { + int n, q; cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; cin >> v; + indices.emplace_back(i,v); + } + Segment seg(0,1e9 + 2,indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } + + return 0; +} + ``` ## Supporting updates Lets support updates of type: - - change value at index i to y + - change value at index $i$ to $y$ What we can do is recurse down to your leaf in order to remove the old element. And also recurse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indicies vector Our prefix vector ### Changing the indicies vector -What we can do is instead of storing a vector use a set. +What we can do is instead of storing a vector use a set. Then erasing and adding values becomes easy. ### Changing the prefix vector -Since each update could change our prefix vector a lot, we cant maintain just he normal vector. -What we could do use use a sparse segment tree. +Since each update could change our prefix vector a lot, we cant maintain just he normal vector. +What we could do use use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - - quering for a prefix can be done by quering the segment tree from 0 till i -This approach is not really memory effient and also requires the implemention of a segment tree. + - quering for a prefix can be done by quering the segment tree from 0 till $i$ +This approach is not really memory effient and also requires the implemention of a segment tree. A more friendly approach would be using an order statistics tree. From 54abaaf4312aa99d07f439fec3a65820925df286 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Sat, 22 Jun 2024 19:40:50 +0000 Subject: [PATCH 08/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 191 +++++++++++++++++---------------- 1 file changed, 98 insertions(+), 93 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index bcf2b3a212..c8c4c1c523 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -43,15 +43,15 @@ With wavelet tree you can easily support those queries in $log(n)$ time. ## Wavelet Tree Structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. ## Defining the structure -We are going to maintain a segment tree over values instead of indices. -Each segment contains indices whose values lie within the segment's range. -Let's save those indicies in a vector. +We are going to maintain a segment tree over values instead of indices. +Each segment contains indices whose values lie within the segment's range. +Let's save those indicies in a vector. Notice that a index can be in at most $log(N)$ segments @@ -77,8 +77,8 @@ This reduces the problem to counting the number of occurrences in a prefix. ### Approaching the Solution -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. @@ -86,20 +86,20 @@ second type of query. **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. +tree. If we are able to determine the position (index) of $𝑟$ in the left and right -children of a node +children of a node We can recurse down the tree and determine its position in the leaf node. ### Determining the Position in Child Nodes To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -111,16 +111,16 @@ $$ ### Updating $𝑟$ as We Recurse -To know the value of 𝑟 if we recurse left, we use prefixB[r] +To know the value of 𝑟 if we recurse left, we use prefixB[r] If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ @@ -137,7 +137,7 @@ left_occurrences = prefixB[r] - prefixB[l] $$ - Note that left_occurrences is the amount of indices between l and r that their value is less than mid + Note that left_occurrences is the amount of indices between l and r that their value is less than mid @@ -155,8 +155,8 @@ $$ the answer then will be the value of the node we end up on (leaf) ## Time complexity -For each query, we traverse left or right, meaning the number of nodes visited is proportional to the height of the tree. -For each node, we compute the $l$ or $r$ using prefix sums, which is $\mathcal{O}(1)$ for each query. +For each query, we traverse left or right, meaning the number of nodes visited is proportional to the height of the tree. +For each node, we compute the $l$ or $r$ using prefix sums, which is $\mathcal{O}(1)$ for each query. Hence, the overall time complexity is $\mathcal{O}(\log n)$ ## implemention @@ -169,103 +169,108 @@ Hence, the overall time complexity is $\mathcal{O}(\log n)$ using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) return; - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), - [this](const pair& elem) { - return elem.second < mid; - }); - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) - return left->find_k_smallest(lb,lr,k); // Appears in left - else - return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r, const vector> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) return; + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(), + rightIndices.begin(), + [this](const pair &elem) { + return elem.second < mid; + }); + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = + lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) + return left->find_k_smallest(lb, lr, k); // Appears in left + else + return right->find_k_smallest(a - lb, b - lr, + k - inLeft); // Appears in right + } }; - int main() { - int n, q; cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; cin >> v; - indices.emplace_back(i,v); - } - Segment seg(0,1e9 + 2,indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } - - return 0; + int n, q; + cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + indices.emplace_back(i, v); + } + Segment seg(0, 1e9 + 2, indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } + + return 0; } - ``` ## Supporting updates Lets support updates of type: - - change value at index $i$ to $y$ + - change value at index $i$ to $y$ What we can do is recurse down to your leaf in order to remove the old element. And also recurse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indicies vector Our prefix vector ### Changing the indicies vector -What we can do is instead of storing a vector use a set. +What we can do is instead of storing a vector use a set. Then erasing and adding values becomes easy. ### Changing the prefix vector -Since each update could change our prefix vector a lot, we cant maintain just he normal vector. -What we could do use use a sparse segment tree. +Since each update could change our prefix vector a lot, we cant maintain just he normal vector. +What we could do use use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - quering for a prefix can be done by quering the segment tree from 0 till $i$ -This approach is not really memory effient and also requires the implemention of a segment tree. +This approach is not really memory effient and also requires the implemention of a segment tree. A more friendly approach would be using an order statistics tree. 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zNOWB45GA=XcX=4btG^YLxMz_f_*ps&R5q^^)zQSt0;?bwsM6sCi{6W2c18>+v4x*_ zIjR(p{LmJ`j~%8JTV8Mn94oh1e{M_JdBa-OQ>{XPEDkwm2Wm~01z=Wm{6jYQ*J9Ca z76qMy&vlB5E&j#+JVbH3q@Ve+fFX(e0mo8D&Pq`cg$4Qk Date: Mon, 24 Jun 2024 19:11:50 +0300 Subject: [PATCH 10/79] Req changes v2.mdx --- content/6_Advanced/Wavelet.mdx | 219 ++++++++++++++++----------------- 1 file changed, 107 insertions(+), 112 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index c8c4c1c523..37e1444da0 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -1,15 +1,15 @@ --- id: wavelet title: 'Wavelet Tree' -author: Benjamin Qi, Crafticat +author: Benjamin Qi, Omri Jerbi prerequisites: - RURQ -description: Segment tree over values" +description: Wavelet trees are data structures that support efficient queries for the k-th minimum element in a range by maintaining a segment tree over values instead of indices" frequency: 0 --- # Wavelet Tree -Segment trees over indices. +Wavelet trees are data structures that support efficient queries for the k-th minimum element in a range by maintaining a segment tree over values instead of indices. @@ -37,21 +37,17 @@ Suppose you want to support the following queries: With wavelet tree you can easily support those queries in $log(n)$ time. - - It is possible to support with updates in $\mathcal{O}(\log n^2)$ - - ## Wavelet Tree Structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. ## Defining the structure -We are going to maintain a segment tree over values instead of indices. -Each segment contains indices whose values lie within the segment's range. -Let's save those indicies in a vector. +We are going to maintain a segment tree over values instead of indices. +Each segment contains indices whose values lie within the segment's range. +Let's save those indicies in a vector. Notice that a index can be in at most $log(N)$ segments @@ -69,7 +65,7 @@ formula: $$ \begin{aligned} -occurrences(l, r) = occurrences(r) - occurrences(l) +\texttt{occurrences}(l, r) = \texttt{occurrences}(r) - \texttt{occurrences}(l) \end{aligned} $$ @@ -77,8 +73,8 @@ This reduces the problem to counting the number of occurrences in a prefix. ### Approaching the Solution -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. @@ -86,20 +82,20 @@ second type of query. **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. +tree. If we are able to determine the position (index) of $𝑟$ in the left and right -children of a node +children of a node We can recurse down the tree and determine its position in the leaf node. ### Determining the Position in Child Nodes To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -111,20 +107,20 @@ $$ ### Updating $𝑟$ as We Recurse -To know the value of 𝑟 if we recurse left, we use prefixB[r] +To know the value of 𝑟 if we recurse left, we use prefixB[r] If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ -occurrences(l, r) = r - l +\texttt{occurrences}(l, r) = r - l $$ This is simular to counting how many times a value appears up to index 𝑅 in our previous query. We did this by using the new 𝑅 value at the leaf node. But now, we consider the difference between the updated 𝑅 and 𝐿 @@ -133,20 +129,20 @@ This is simular to counting how many times a value appears up to index 𝑅 in o Therefore, the occurrences of the left node is $$ -left_occurrences = prefixB[r] - prefixB[l] +\texttt{left\_occurrences} = prefixB[r] - prefixB[l] $$ - Note that left_occurrences is the amount of indices between l and r that their value is less than mid + Note that $\texttt{left\_occurrences}$ is the amount of indices between l and r that their value is less than mid -- If left_occurrences left_occurrences is greater or equal to $k$, it means the $k$-th smallest element is in +- If $\texttt{left\_occurrences}$ is greater or equal to $k$, it means the $k$-th smallest element is in the left subtree. Therefore, we update our range and recurse into the left child -- If left_occurrences is less than $k$, it means the +- If $\texttt{left\_occurrences}$ is less than $k$, it means the $k$-th smallest element is in the right subtree. We adjust k by subtracting - left_occurrences from $k$, update our range, and recurse into the right child + $\texttt{left\_occurrences}$ from $k$, update our range, and recurse into the right child Notice we still update $l, r$ accordingly when we go left or right @@ -155,12 +151,13 @@ $$ the answer then will be the value of the node we end up on (leaf) ## Time complexity -For each query, we traverse left or right, meaning the number of nodes visited is proportional to the height of the tree. -For each node, we compute the $l$ or $r$ using prefix sums, which is $\mathcal{O}(1)$ for each query. +For each query, we traverse left or right, meaning the number of nodes visited is proportional to the height of the tree. +For each node, we compute the $l$ or $r$ using prefix sums, which is $\mathcal{O}(1)$ for each query. Hence, the overall time complexity is $\mathcal{O}(\log n)$ ## implemention + ```cpp @@ -169,111 +166,109 @@ Hence, the overall time complexity is $\mathcal{O}(\log n)$ using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r, const vector> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) return; - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(), - rightIndices.begin(), - [this](const pair &elem) { - return elem.second < mid; - }); - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = - lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) - return left->find_k_smallest(lb, lr, k); // Appears in left - else - return right->find_k_smallest(a - lb, b - lr, - k - inLeft); // Appears in right - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) return; + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), + [this](const pair& elem) { + return elem.second < mid; + }); + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) + return left->find_k_smallest(lb,lr,k); // Appears in left + else + return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right + } }; + int main() { - int n, q; - cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - indices.emplace_back(i, v); - } - Segment seg(0, 1e9 + 2, indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } - - return 0; + int n, q; cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; cin >> v; + indices.emplace_back(i,v); + } + Segment seg(0,1e9 + 2,indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } + + return 0; } + ``` + + + + ## Supporting updates Lets support updates of type: - - change value at index $i$ to $y$ + - change value at index $i$ to $y$ What we can do is recurse down to your leaf in order to remove the old element. And also recurse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indicies vector Our prefix vector ### Changing the indicies vector -What we can do is instead of storing a vector use a set. +What we can do is instead of storing a vector use a set. Then erasing and adding values becomes easy. ### Changing the prefix vector -Since each update could change our prefix vector a lot, we cant maintain just he normal vector. -What we could do use use a sparse segment tree. +Since each update could change our prefix vector a lot, we cant maintain just he normal vector. +What we could do use use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - quering for a prefix can be done by quering the segment tree from 0 till $i$ -This approach is not really memory effient and also requires the implemention of a segment tree. +This approach is not really memory effient and also requires the implemention of a segment tree. A more friendly approach would be using an order statistics tree. - ### Problems From 2978c677e18483d5fc0533b57d91f6d8335c4ceb Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Mon, 24 Jun 2024 16:13:17 +0000 Subject: [PATCH 11/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 191 +++++++++++++++++---------------- 1 file changed, 98 insertions(+), 93 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 37e1444da0..014c37d81e 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -39,15 +39,15 @@ With wavelet tree you can easily support those queries in $log(n)$ time. ## Wavelet Tree Structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. ## Defining the structure -We are going to maintain a segment tree over values instead of indices. -Each segment contains indices whose values lie within the segment's range. -Let's save those indicies in a vector. +We are going to maintain a segment tree over values instead of indices. +Each segment contains indices whose values lie within the segment's range. +Let's save those indicies in a vector. Notice that a index can be in at most $log(N)$ segments @@ -73,8 +73,8 @@ This reduces the problem to counting the number of occurrences in a prefix. ### Approaching the Solution -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. @@ -82,20 +82,20 @@ second type of query. **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. +tree. If we are able to determine the position (index) of $𝑟$ in the left and right -children of a node +children of a node We can recurse down the tree and determine its position in the leaf node. ### Determining the Position in Child Nodes To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -107,16 +107,16 @@ $$ ### Updating $𝑟$ as We Recurse -To know the value of 𝑟 if we recurse left, we use prefixB[r] +To know the value of 𝑟 if we recurse left, we use prefixB[r] If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ @@ -133,7 +133,7 @@ $$ $$ - Note that $\texttt{left\_occurrences}$ is the amount of indices between l and r that their value is less than mid + Note that $\texttt{left\_occurrences}$ is the amount of indices between l and r that their value is less than mid @@ -151,8 +151,8 @@ $$ the answer then will be the value of the node we end up on (leaf) ## Time complexity -For each query, we traverse left or right, meaning the number of nodes visited is proportional to the height of the tree. -For each node, we compute the $l$ or $r$ using prefix sums, which is $\mathcal{O}(1)$ for each query. +For each query, we traverse left or right, meaning the number of nodes visited is proportional to the height of the tree. +For each node, we compute the $l$ or $r$ using prefix sums, which is $\mathcal{O}(1)$ for each query. Hence, the overall time complexity is $\mathcal{O}(\log n)$ ## implemention @@ -166,79 +166,84 @@ Hence, the overall time complexity is $\mathcal{O}(\log n)$ using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) return; - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), - [this](const pair& elem) { - return elem.second < mid; - }); - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) - return left->find_k_smallest(lb,lr,k); // Appears in left - else - return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r, const vector> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) return; + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(), + rightIndices.begin(), + [this](const pair &elem) { + return elem.second < mid; + }); + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = + lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) + return left->find_k_smallest(lb, lr, k); // Appears in left + else + return right->find_k_smallest(a - lb, b - lr, + k - inLeft); // Appears in right + } }; - int main() { - int n, q; cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; cin >> v; - indices.emplace_back(i,v); - } - Segment seg(0,1e9 + 2,indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } - - return 0; + int n, q; + cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + indices.emplace_back(i, v); + } + Segment seg(0, 1e9 + 2, indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } + + return 0; } - ``` @@ -248,25 +253,25 @@ int main() { ## Supporting updates Lets support updates of type: - - change value at index $i$ to $y$ + - change value at index $i$ to $y$ What we can do is recurse down to your leaf in order to remove the old element. And also recurse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indicies vector Our prefix vector ### Changing the indicies vector -What we can do is instead of storing a vector use a set. +What we can do is instead of storing a vector use a set. Then erasing and adding values becomes easy. ### Changing the prefix vector -Since each update could change our prefix vector a lot, we cant maintain just he normal vector. -What we could do use use a sparse segment tree. +Since each update could change our prefix vector a lot, we cant maintain just he normal vector. +What we could do use use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - quering for a prefix can be done by quering the segment tree from 0 till $i$ -This approach is not really memory effient and also requires the implemention of a segment tree. +This approach is not really memory effient and also requires the implemention of a segment tree. A more friendly approach would be using an order statistics tree. From a49cd775b2fcd43784f02f86083e8816503691e1 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Sun, 30 Jun 2024 14:39:26 +0300 Subject: [PATCH 12/79] Update Wavelet.mdx Changed time complexity Added braces --- content/6_Advanced/Wavelet.mdx | 203 ++++++++++++++++----------------- 1 file changed, 100 insertions(+), 103 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 014c37d81e..388e1ae5a8 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -35,20 +35,21 @@ Suppose you want to support the following queries: - Given a range $l$, $r$ count number of occurrences of value $x$. - Given a range $l$, $r$ find the $k$ smallest element -With wavelet tree you can easily support those queries in $log(n)$ time. +With wavelet tree you can easily support those queries in $log(M)$ time. +when M is the maximum value in the array. ## Wavelet Tree Structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. ## Defining the structure -We are going to maintain a segment tree over values instead of indices. -Each segment contains indices whose values lie within the segment's range. -Let's save those indicies in a vector. -Notice that a index can be in at most $log(N)$ segments +We are going to maintain a segment tree over values instead of indices. +Each segment contains indices whose values lie within the segment's range. +Let's save those indicies in a vector. +Notice that a index can be in at most $log(M)$ segments @@ -73,8 +74,8 @@ This reduces the problem to counting the number of occurrences in a prefix. ### Approaching the Solution -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. @@ -82,20 +83,20 @@ second type of query. **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. +tree. If we are able to determine the position (index) of $𝑟$ in the left and right -children of a node +children of a node We can recurse down the tree and determine its position in the leaf node. ### Determining the Position in Child Nodes To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -107,16 +108,16 @@ $$ ### Updating $𝑟$ as We Recurse -To know the value of 𝑟 if we recurse left, we use prefixB[r] +To know the value of 𝑟 if we recurse left, we use prefixB[r] If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ @@ -133,7 +134,7 @@ $$ $$ - Note that $\texttt{left\_occurrences}$ is the amount of indices between l and r that their value is less than mid + Note that $\texttt{left\_occurrences}$ is the amount of indices between l and r that their value is less than mid @@ -150,12 +151,8 @@ $$ the answer then will be the value of the node we end up on (leaf) -## Time complexity -For each query, we traverse left or right, meaning the number of nodes visited is proportional to the height of the tree. -For each node, we compute the $l$ or $r$ using prefix sums, which is $\mathcal{O}(1)$ for each query. -Hence, the overall time complexity is $\mathcal{O}(\log n)$ - ## implemention +**Time Complexity:** $\mathcal{O}(Q * log(M))$ @@ -166,84 +163,84 @@ Hence, the overall time complexity is $\mathcal{O}(\log n)$ using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r, const vector> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) return; - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(), - rightIndices.begin(), - [this](const pair &elem) { - return elem.second < mid; - }); - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = - lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) - return left->find_k_smallest(lb, lr, k); // Appears in left - else - return right->find_k_smallest(a - lb, b - lr, - k - inLeft); // Appears in right - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) { + return; + } + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), + [this](const pair& elem) { + return elem.second < mid; + }); + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) { + return l; + } + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) { + return left->find_k_smallest(lb,lr,k); // Appears in left + } else { + return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right + } + } }; + int main() { - int n, q; - cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - indices.emplace_back(i, v); - } - Segment seg(0, 1e9 + 2, indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } - - return 0; + int n, q; cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; cin >> v; + indices.emplace_back(i,v); + } + Segment seg(0,1e9 + 2,indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } + + return 0; } + ``` @@ -253,25 +250,25 @@ int main() { ## Supporting updates Lets support updates of type: - - change value at index $i$ to $y$ + - change value at index $i$ to $y$ What we can do is recurse down to your leaf in order to remove the old element. And also recurse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indicies vector Our prefix vector ### Changing the indicies vector -What we can do is instead of storing a vector use a set. +What we can do is instead of storing a vector use a set. Then erasing and adding values becomes easy. ### Changing the prefix vector -Since each update could change our prefix vector a lot, we cant maintain just he normal vector. -What we could do use use a sparse segment tree. +Since each update could change our prefix vector a lot, we cant maintain just he normal vector. +What we could do use use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - quering for a prefix can be done by quering the segment tree from 0 till $i$ -This approach is not really memory effient and also requires the implemention of a segment tree. +This approach is not really memory effient and also requires the implemention of a segment tree. A more friendly approach would be using an order statistics tree. From 6df8c0bc466edbdaea232a29225a8c7531ef25d4 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Sun, 30 Jun 2024 14:40:09 +0300 Subject: [PATCH 13/79] Update Wavelet.problems.json Changed the uniqueID --- content/6_Advanced/Wavelet.problems.json | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/content/6_Advanced/Wavelet.problems.json b/content/6_Advanced/Wavelet.problems.json index 3f218ef0fd..4ef913b01c 100644 --- a/content/6_Advanced/Wavelet.problems.json +++ b/content/6_Advanced/Wavelet.problems.json @@ -17,7 +17,7 @@ ], "wavelet": [ { - "uniqueId": "codeforces_Destiny", + "uniqueId": "cf_840D", "name": "Destiny", "url": "https://codeforces.com/contest/840/problem/D", "source": "CF", @@ -29,7 +29,7 @@ } }, { - "uniqueId": "ILKQUERY2", + "uniqueId": "spoj-ILKQUERY2", "name": "ILKQUERY2", "url": "https://www.spoj.com/problems/ILKQUERY2/", "source": "SPOJ", @@ -76,6 +76,7 @@ "kind": "autogen-label-from-site", "site": "DMOJ" } + } ] } From 0379489c4e4f9793055300307d76e9596875b00a Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Sun, 30 Jun 2024 11:40:38 +0000 Subject: [PATCH 14/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 195 +++++++++++++++++---------------- 1 file changed, 98 insertions(+), 97 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 388e1ae5a8..16d02369ae 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -35,20 +35,20 @@ Suppose you want to support the following queries: - Given a range $l$, $r$ count number of occurrences of value $x$. - Given a range $l$, $r$ find the $k$ smallest element -With wavelet tree you can easily support those queries in $log(M)$ time. +With wavelet tree you can easily support those queries in $log(M)$ time. when M is the maximum value in the array. ## Wavelet Tree Structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. ## Defining the structure -We are going to maintain a segment tree over values instead of indices. -Each segment contains indices whose values lie within the segment's range. -Let's save those indicies in a vector. +We are going to maintain a segment tree over values instead of indices. +Each segment contains indices whose values lie within the segment's range. +Let's save those indicies in a vector. Notice that a index can be in at most $log(M)$ segments @@ -74,8 +74,8 @@ This reduces the problem to counting the number of occurrences in a prefix. ### Approaching the Solution -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. @@ -83,20 +83,20 @@ second type of query. **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. +tree. If we are able to determine the position (index) of $𝑟$ in the left and right -children of a node +children of a node We can recurse down the tree and determine its position in the leaf node. ### Determining the Position in Child Nodes To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -108,16 +108,16 @@ $$ ### Updating $𝑟$ as We Recurse -To know the value of 𝑟 if we recurse left, we use prefixB[r] +To know the value of 𝑟 if we recurse left, we use prefixB[r] If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ @@ -134,7 +134,7 @@ $$ $$ - Note that $\texttt{left\_occurrences}$ is the amount of indices between l and r that their value is less than mid + Note that $\texttt{left\_occurrences}$ is the amount of indices between l and r that their value is less than mid @@ -163,84 +163,85 @@ the answer then will be the value of the node we end up on (leaf) using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) { - return; - } - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), - [this](const pair& elem) { - return elem.second < mid; - }); - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) { - return l; - } - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) { - return left->find_k_smallest(lb,lr,k); // Appears in left - } else { - return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right - } - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r, const vector> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(), + rightIndices.begin(), + [this](const pair &elem) { + return elem.second < mid; + }); + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) { return l; } + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = + lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) { + return left->find_k_smallest(lb, lr, k); // Appears in left + } else { + return right->find_k_smallest(a - lb, b - lr, + k - inLeft); // Appears in right + } + } }; - int main() { - int n, q; cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; cin >> v; - indices.emplace_back(i,v); - } - Segment seg(0,1e9 + 2,indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } - - return 0; + int n, q; + cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + indices.emplace_back(i, v); + } + Segment seg(0, 1e9 + 2, indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } + + return 0; } - ``` @@ -250,25 +251,25 @@ int main() { ## Supporting updates Lets support updates of type: - - change value at index $i$ to $y$ + - change value at index $i$ to $y$ What we can do is recurse down to your leaf in order to remove the old element. And also recurse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indicies vector Our prefix vector ### Changing the indicies vector -What we can do is instead of storing a vector use a set. +What we can do is instead of storing a vector use a set. Then erasing and adding values becomes easy. ### Changing the prefix vector -Since each update could change our prefix vector a lot, we cant maintain just he normal vector. -What we could do use use a sparse segment tree. +Since each update could change our prefix vector a lot, we cant maintain just he normal vector. +What we could do use use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - quering for a prefix can be done by quering the segment tree from 0 till $i$ -This approach is not really memory effient and also requires the implemention of a segment tree. +This approach is not really memory effient and also requires the implemention of a segment tree. A more friendly approach would be using an order statistics tree. From a2ae4c1d4bedbd757a1a0d4f2d5b3f34bd222331 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Sun, 7 Jul 2024 22:09:20 +0300 Subject: [PATCH 15/79] Update Wavelet.mdx --- content/6_Advanced/Wavelet.mdx | 214 ++++++++++++++++----------------- 1 file changed, 105 insertions(+), 109 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 16d02369ae..5845016b3b 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -32,34 +32,31 @@ Wavelet trees are data structures that support efficient queries for the k-th mi Suppose you want to support the following queries: -- Given a range $l$, $r$ count number of occurrences of value $x$. +- Given a range $l$, $r$ count the number of occurrences of value $x$. - Given a range $l$, $r$ find the $k$ smallest element -With wavelet tree you can easily support those queries in $log(M)$ time. +With wavelet tree, you can easily support those queries in $log(M)$ time. when M is the maximum value in the array. ## Wavelet Tree Structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. ## Defining the structure -We are going to maintain a segment tree over values instead of indices. -Each segment contains indices whose values lie within the segment's range. -Let's save those indicies in a vector. -Notice that a index can be in at most $log(M)$ segments +We are going to maintain a segment tree over values instead of indices. Each segment will contain indices whose values lie within the segment's range. We'll save those indices in a vector. Notice that an index can be in at most $log(M)$ segments -Lets say our array is: $[3,5,3,1,2,2,3,4,5,5]$ +Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ ![Wavelet Tree Visualization](./assets/diagram.png) ## Solving the First Type of query -**Given a range l, r count number of occurrences of value x.** +**Given a range l, r count the number of occurrences of value x.** To calculate the number of occurrences from $𝑙$ to $𝑟$, we can use the following formula: @@ -74,8 +71,8 @@ This reduces the problem to counting the number of occurrences in a prefix. ### Approaching the Solution -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. @@ -83,20 +80,20 @@ second type of query. **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. -If we are able to determine the position (index) of $𝑟$ in the left and right -children of a node +tree. +If we can determine the position (index) of $𝑟$ in the left and right +children of a node. We can recurse down the tree and determine its position in the leaf node. ### Determining the Position in Child Nodes To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -108,23 +105,23 @@ $$ ### Updating $𝑟$ as We Recurse -To know the value of 𝑟 if we recurse left, we use prefixB[r] +To know the value of 𝑟 if we recurse left, we use prefixB[r] If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ \texttt{occurrences}(l, r) = r - l $$ -This is simular to counting how many times a value appears up to index 𝑅 in our previous query. We did this by using the new 𝑅 value at the leaf node. But now, we consider the difference between the updated 𝑅 and 𝐿 +This is similar to counting how many times a value appears up to index 𝑅 in our previous query. We did this by using the new 𝑅 value at the leaf node. But now, we consider the difference between the updated 𝑅 and 𝐿 Therefore, the occurrences of the left node is @@ -134,7 +131,7 @@ $$ $$ - Note that $\texttt{left\_occurrences}$ is the amount of indices between l and r that their value is less than mid + Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid @@ -163,85 +160,84 @@ the answer then will be the value of the node we end up on (leaf) using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r, const vector> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(), - rightIndices.begin(), - [this](const pair &elem) { - return elem.second < mid; - }); - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) { return l; } - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = - lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) { - return left->find_k_smallest(lb, lr, k); // Appears in left - } else { - return right->find_k_smallest(a - lb, b - lr, - k - inLeft); // Appears in right - } - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) { + return; + } + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), + [this](const pair& elem) { + return elem.second < mid; + }); + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) { + return l; + } + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) { + return left->find_k_smallest(lb,lr,k); // Appears in left + } else { + return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right + } + } }; + int main() { - int n, q; - cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - indices.emplace_back(i, v); - } - Segment seg(0, 1e9 + 2, indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } - - return 0; + int n, q; cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; cin >> v; + indices.emplace_back(i,v); + } + Segment seg(0,1e9 + 2,indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } + + return 0; } + ``` @@ -250,26 +246,26 @@ int main() { ## Supporting updates -Lets support updates of type: - - change value at index $i$ to $y$ +Let's support updates of type: + - change value at index $i$ to $y$ -What we can do is recurse down to your leaf in order to remove the old element. And also recurse down to add the new element. +We can traverse down to the leaf to remove the old element and also traverse down to add the new element. -So what do the updates change? - - - Our indicies vector +So what do the updates change? + - + Our indices vector Our prefix vector -### Changing the indicies vector -What we can do is instead of storing a vector use a set. +### Changing the indices vector +What we can do is instead of storing a vector use a set. Then erasing and adding values becomes easy. ### Changing the prefix vector -Since each update could change our prefix vector a lot, we cant maintain just he normal vector. -What we could do use use a sparse segment tree. +Since each update could change our prefix vector a lot, we can't maintain just the normal vector. +What we could do is use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - - quering for a prefix can be done by quering the segment tree from 0 till $i$ -This approach is not really memory effient and also requires the implemention of a segment tree. + - querying for a prefix can be done by querying the segment tree from 0 to $i$ +This approach is not memory efficient and requires a segment tree's implementation. A more friendly approach would be using an order statistics tree. From 76bd73487d0a2c01b40db6009a0a0d0cbb0c06a7 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Sun, 7 Jul 2024 22:10:19 +0300 Subject: [PATCH 16/79] Update Wavelet.problems.json --- content/6_Advanced/Wavelet.problems.json | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/6_Advanced/Wavelet.problems.json b/content/6_Advanced/Wavelet.problems.json index 4ef913b01c..cdc0ab38ef 100644 --- a/content/6_Advanced/Wavelet.problems.json +++ b/content/6_Advanced/Wavelet.problems.json @@ -17,7 +17,7 @@ ], "wavelet": [ { - "uniqueId": "cf_840D", + "uniqueId": "cf-840D", "name": "Destiny", "url": "https://codeforces.com/contest/840/problem/D", "source": "CF", From 22d3b843e17158a1c2fcd5d43651f4edc304cc47 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Sun, 7 Jul 2024 19:10:31 +0000 Subject: [PATCH 17/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 189 ++++++++++++----------- content/6_Advanced/Wavelet.problems.json | 1 - 2 files changed, 95 insertions(+), 95 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 5845016b3b..d4acc70707 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -35,12 +35,12 @@ Suppose you want to support the following queries: - Given a range $l$, $r$ count the number of occurrences of value $x$. - Given a range $l$, $r$ find the $k$ smallest element -With wavelet tree, you can easily support those queries in $log(M)$ time. +With wavelet tree, you can easily support those queries in $log(M)$ time. when M is the maximum value in the array. ## Wavelet Tree Structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. @@ -71,8 +71,8 @@ This reduces the problem to counting the number of occurrences in a prefix. ### Approaching the Solution -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. @@ -80,20 +80,20 @@ second type of query. **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. +tree. If we can determine the position (index) of $𝑟$ in the left and right -children of a node. +children of a node. We can recurse down the tree and determine its position in the leaf node. ### Determining the Position in Child Nodes To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -105,16 +105,16 @@ $$ ### Updating $𝑟$ as We Recurse -To know the value of 𝑟 if we recurse left, we use prefixB[r] +To know the value of 𝑟 if we recurse left, we use prefixB[r] If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ @@ -131,7 +131,7 @@ $$ $$ - Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid + Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid @@ -160,84 +160,85 @@ the answer then will be the value of the node we end up on (leaf) using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) { - return; - } - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), - [this](const pair& elem) { - return elem.second < mid; - }); - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) { - return l; - } - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) { - return left->find_k_smallest(lb,lr,k); // Appears in left - } else { - return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right - } - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r, const vector> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(), + rightIndices.begin(), + [this](const pair &elem) { + return elem.second < mid; + }); + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) { return l; } + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = + lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) { + return left->find_k_smallest(lb, lr, k); // Appears in left + } else { + return right->find_k_smallest(a - lb, b - lr, + k - inLeft); // Appears in right + } + } }; - int main() { - int n, q; cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; cin >> v; - indices.emplace_back(i,v); - } - Segment seg(0,1e9 + 2,indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } - - return 0; + int n, q; + cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + indices.emplace_back(i, v); + } + Segment seg(0, 1e9 + 2, indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } + + return 0; } - ``` @@ -247,25 +248,25 @@ int main() { ## Supporting updates Let's support updates of type: - - change value at index $i$ to $y$ + - change value at index $i$ to $y$ We can traverse down to the leaf to remove the old element and also traverse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indices vector Our prefix vector ### Changing the indices vector -What we can do is instead of storing a vector use a set. +What we can do is instead of storing a vector use a set. Then erasing and adding values becomes easy. ### Changing the prefix vector -Since each update could change our prefix vector a lot, we can't maintain just the normal vector. -What we could do is use a sparse segment tree. +Since each update could change our prefix vector a lot, we can't maintain just the normal vector. +What we could do is use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - querying for a prefix can be done by querying the segment tree from 0 to $i$ -This approach is not memory efficient and requires a segment tree's implementation. +This approach is not memory efficient and requires a segment tree's implementation. A more friendly approach would be using an order statistics tree. diff --git a/content/6_Advanced/Wavelet.problems.json b/content/6_Advanced/Wavelet.problems.json index cdc0ab38ef..45ac5b8786 100644 --- a/content/6_Advanced/Wavelet.problems.json +++ b/content/6_Advanced/Wavelet.problems.json @@ -76,7 +76,6 @@ "kind": "autogen-label-from-site", "site": "DMOJ" } - } ] } From dc752bdd994bb2187d39263c62c182b04e74bbdd Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Tue, 9 Jul 2024 12:52:11 +0300 Subject: [PATCH 18/79] Update Wavelet.mdx --- content/6_Advanced/Wavelet.mdx | 203 ++++++++++++++++----------------- 1 file changed, 96 insertions(+), 107 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index d4acc70707..7c0b861a8a 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -35,17 +35,15 @@ Suppose you want to support the following queries: - Given a range $l$, $r$ count the number of occurrences of value $x$. - Given a range $l$, $r$ find the $k$ smallest element -With wavelet tree, you can easily support those queries in $log(M)$ time. +With wavelet tree, you can easily support those queries in $log(M)$ time. when M is the maximum value in the array. ## Wavelet Tree Structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. -## Defining the structure - We are going to maintain a segment tree over values instead of indices. Each segment will contain indices whose values lie within the segment's range. We'll save those indices in a vector. Notice that an index can be in at most $log(M)$ segments @@ -69,10 +67,8 @@ $$ This reduces the problem to counting the number of occurrences in a prefix. -### Approaching the Solution - -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. @@ -80,20 +76,18 @@ second type of query. **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. +tree. If we can determine the position (index) of $𝑟$ in the left and right -children of a node. +children of a node. We can recurse down the tree and determine its position in the leaf node. -### Determining the Position in Child Nodes - To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -103,18 +97,18 @@ c[i] = index[i] < mid ? 1 : 0; prefixB[i] = prefixB[i - 1] + c[i] $$ -### Updating $𝑟$ as We Recurse -To know the value of 𝑟 if we recurse left, we use prefixB[r] -If we recurse right, we use 𝑟 - prefixB[r] +To update $r$ as we recurse down, we do the following: +- To know the value of 𝑟 if we recurse left, we use prefixB[r] +- If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ @@ -131,7 +125,7 @@ $$ $$ - Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid + Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid @@ -160,85 +154,84 @@ the answer then will be the value of the node we end up on (leaf) using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r, const vector> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(), - rightIndices.begin(), - [this](const pair &elem) { - return elem.second < mid; - }); - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) { return l; } - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = - lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) { - return left->find_k_smallest(lb, lr, k); // Appears in left - } else { - return right->find_k_smallest(a - lb, b - lr, - k - inLeft); // Appears in right - } - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) { + return; + } + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), + [this](const pair& elem) { + return elem.second < mid; + }); + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) { + return l; + } + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) { + return left->find_k_smallest(lb,lr,k); // Appears in left + } else { + return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right + } + } }; + int main() { - int n, q; - cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - indices.emplace_back(i, v); - } - Segment seg(0, 1e9 + 2, indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } - - return 0; + int n, q; cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; cin >> v; + indices.emplace_back(i,v); + } + Segment seg(0,1e9 + 2,indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } + + return 0; } + ``` @@ -248,28 +241,24 @@ int main() { ## Supporting updates Let's support updates of type: - - change value at index $i$ to $y$ + - change value at index $i$ to $y$ We can traverse down to the leaf to remove the old element and also traverse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indices vector Our prefix vector - -### Changing the indices vector -What we can do is instead of storing a vector use a set. + +To change the indices vector, what we can do is, instead of storing a vector, use a set. Then erasing and adding values becomes easy. -### Changing the prefix vector -Since each update could change our prefix vector a lot, we can't maintain just the normal vector. -What we could do is use a sparse segment tree. +On the other hand, To change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. What we could do is use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - querying for a prefix can be done by querying the segment tree from 0 to $i$ -This approach is not memory efficient and requires a segment tree's implementation. +This approach is not memory efficient and requires a segment tree's implementation. A more friendly approach would be using an order statistics tree. - ### Problems From 8d1c959a132b08a03387eb217f6ff4eb76bb0a57 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Tue, 9 Jul 2024 09:53:28 +0000 Subject: [PATCH 19/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 189 +++++++++++++++++---------------- 1 file changed, 95 insertions(+), 94 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 7c0b861a8a..26cbcf0da5 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -35,12 +35,12 @@ Suppose you want to support the following queries: - Given a range $l$, $r$ count the number of occurrences of value $x$. - Given a range $l$, $r$ find the $k$ smallest element -With wavelet tree, you can easily support those queries in $log(M)$ time. +With wavelet tree, you can easily support those queries in $log(M)$ time. when M is the maximum value in the array. ## Wavelet Tree Structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. @@ -67,8 +67,8 @@ $$ This reduces the problem to counting the number of occurrences in a prefix. -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. @@ -76,18 +76,18 @@ second type of query. **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. +tree. If we can determine the position (index) of $𝑟$ in the left and right -children of a node. +children of a node. We can recurse down the tree and determine its position in the leaf node. To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -98,17 +98,17 @@ prefixB[i] = prefixB[i - 1] + c[i] $$ -To update $r$ as we recurse down, we do the following: -- To know the value of 𝑟 if we recurse left, we use prefixB[r] +To update $r$ as we recurse down, we do the following: +- To know the value of 𝑟 if we recurse left, we use prefixB[r] - If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ @@ -125,7 +125,7 @@ $$ $$ - Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid + Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid @@ -154,84 +154,85 @@ the answer then will be the value of the node we end up on (leaf) using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) { - return; - } - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), - [this](const pair& elem) { - return elem.second < mid; - }); - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) { - return l; - } - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) { - return left->find_k_smallest(lb,lr,k); // Appears in left - } else { - return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right - } - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r, const vector> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(), + rightIndices.begin(), + [this](const pair &elem) { + return elem.second < mid; + }); + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) { return l; } + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = + lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) { + return left->find_k_smallest(lb, lr, k); // Appears in left + } else { + return right->find_k_smallest(a - lb, b - lr, + k - inLeft); // Appears in right + } + } }; - int main() { - int n, q; cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; cin >> v; - indices.emplace_back(i,v); - } - Segment seg(0,1e9 + 2,indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } - - return 0; + int n, q; + cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + indices.emplace_back(i, v); + } + Segment seg(0, 1e9 + 2, indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } + + return 0; } - ``` @@ -241,22 +242,22 @@ int main() { ## Supporting updates Let's support updates of type: - - change value at index $i$ to $y$ + - change value at index $i$ to $y$ We can traverse down to the leaf to remove the old element and also traverse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indices vector Our prefix vector - -To change the indices vector, what we can do is, instead of storing a vector, use a set. + +To change the indices vector, what we can do is, instead of storing a vector, use a set. Then erasing and adding values becomes easy. On the other hand, To change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. What we could do is use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - querying for a prefix can be done by querying the segment tree from 0 to $i$ -This approach is not memory efficient and requires a segment tree's implementation. +This approach is not memory efficient and requires a segment tree's implementation. A more friendly approach would be using an order statistics tree. ### Problems From f4ad194bb837ce364329582d38d6a9f0eb6aee05 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Tue, 9 Jul 2024 12:54:01 +0300 Subject: [PATCH 20/79] Update Wavelet.mdx --- content/6_Advanced/Wavelet.mdx | 191 ++++++++++++++++----------------- 1 file changed, 95 insertions(+), 96 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 26cbcf0da5..61813cf7f8 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -35,12 +35,12 @@ Suppose you want to support the following queries: - Given a range $l$, $r$ count the number of occurrences of value $x$. - Given a range $l$, $r$ find the $k$ smallest element -With wavelet tree, you can easily support those queries in $log(M)$ time. +With wavelet tree, you can easily support those queries in $log(M)$ time. when M is the maximum value in the array. ## Wavelet Tree Structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. @@ -67,8 +67,8 @@ $$ This reduces the problem to counting the number of occurrences in a prefix. -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. @@ -76,18 +76,18 @@ second type of query. **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. +tree. If we can determine the position (index) of $𝑟$ in the left and right -children of a node. +children of a node. We can recurse down the tree and determine its position in the leaf node. To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -98,17 +98,17 @@ prefixB[i] = prefixB[i - 1] + c[i] $$ -To update $r$ as we recurse down, we do the following: -- To know the value of 𝑟 if we recurse left, we use prefixB[r] +To update $r$ as we recurse down, we do the following: +- To know the value of 𝑟 if we recurse left, we use prefixB[r] - If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ @@ -125,7 +125,7 @@ $$ $$ - Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid + Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid @@ -142,7 +142,7 @@ $$ the answer then will be the value of the node we end up on (leaf) -## implemention +## Implemention **Time Complexity:** $\mathcal{O}(Q * log(M))$ @@ -154,85 +154,84 @@ the answer then will be the value of the node we end up on (leaf) using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r, const vector> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(), - rightIndices.begin(), - [this](const pair &elem) { - return elem.second < mid; - }); - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) { return l; } - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = - lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) { - return left->find_k_smallest(lb, lr, k); // Appears in left - } else { - return right->find_k_smallest(a - lb, b - lr, - k - inLeft); // Appears in right - } - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) { + return; + } + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), + [this](const pair& elem) { + return elem.second < mid; + }); + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) { + return l; + } + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) { + return left->find_k_smallest(lb,lr,k); // Appears in left + } else { + return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right + } + } }; + int main() { - int n, q; - cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - indices.emplace_back(i, v); - } - Segment seg(0, 1e9 + 2, indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } - - return 0; + int n, q; cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; cin >> v; + indices.emplace_back(i,v); + } + Segment seg(0,1e9 + 2,indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } + + return 0; } + ``` @@ -242,22 +241,22 @@ int main() { ## Supporting updates Let's support updates of type: - - change value at index $i$ to $y$ + - change value at index $i$ to $y$ We can traverse down to the leaf to remove the old element and also traverse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indices vector Our prefix vector - -To change the indices vector, what we can do is, instead of storing a vector, use a set. + +To change the indices vector, what we can do is, instead of storing a vector, use a set. Then erasing and adding values becomes easy. On the other hand, To change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. What we could do is use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - querying for a prefix can be done by querying the segment tree from 0 to $i$ -This approach is not memory efficient and requires a segment tree's implementation. +This approach is not memory efficient and requires a segment tree's implementation. A more friendly approach would be using an order statistics tree. ### Problems From a5df3388f86d3a2aa0e8d252d8674b38706c1767 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Tue, 9 Jul 2024 09:56:02 +0000 Subject: [PATCH 21/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 189 +++++++++++++++++---------------- 1 file changed, 95 insertions(+), 94 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 61813cf7f8..b04c9404e2 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -35,12 +35,12 @@ Suppose you want to support the following queries: - Given a range $l$, $r$ count the number of occurrences of value $x$. - Given a range $l$, $r$ find the $k$ smallest element -With wavelet tree, you can easily support those queries in $log(M)$ time. +With wavelet tree, you can easily support those queries in $log(M)$ time. when M is the maximum value in the array. ## Wavelet Tree Structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. @@ -67,8 +67,8 @@ $$ This reduces the problem to counting the number of occurrences in a prefix. -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. @@ -76,18 +76,18 @@ second type of query. **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. +tree. If we can determine the position (index) of $𝑟$ in the left and right -children of a node. +children of a node. We can recurse down the tree and determine its position in the leaf node. To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -98,17 +98,17 @@ prefixB[i] = prefixB[i - 1] + c[i] $$ -To update $r$ as we recurse down, we do the following: -- To know the value of 𝑟 if we recurse left, we use prefixB[r] +To update $r$ as we recurse down, we do the following: +- To know the value of 𝑟 if we recurse left, we use prefixB[r] - If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ @@ -125,7 +125,7 @@ $$ $$ - Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid + Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid @@ -154,84 +154,85 @@ the answer then will be the value of the node we end up on (leaf) using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) { - return; - } - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), - [this](const pair& elem) { - return elem.second < mid; - }); - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) { - return l; - } - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) { - return left->find_k_smallest(lb,lr,k); // Appears in left - } else { - return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right - } - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r, const vector> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(), + rightIndices.begin(), + [this](const pair &elem) { + return elem.second < mid; + }); + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) { return l; } + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = + lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) { + return left->find_k_smallest(lb, lr, k); // Appears in left + } else { + return right->find_k_smallest(a - lb, b - lr, + k - inLeft); // Appears in right + } + } }; - int main() { - int n, q; cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; cin >> v; - indices.emplace_back(i,v); - } - Segment seg(0,1e9 + 2,indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } - - return 0; + int n, q; + cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + indices.emplace_back(i, v); + } + Segment seg(0, 1e9 + 2, indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } + + return 0; } - ``` @@ -241,22 +242,22 @@ int main() { ## Supporting updates Let's support updates of type: - - change value at index $i$ to $y$ + - change value at index $i$ to $y$ We can traverse down to the leaf to remove the old element and also traverse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indices vector Our prefix vector - -To change the indices vector, what we can do is, instead of storing a vector, use a set. + +To change the indices vector, what we can do is, instead of storing a vector, use a set. Then erasing and adding values becomes easy. On the other hand, To change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. What we could do is use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - querying for a prefix can be done by querying the segment tree from 0 to $i$ -This approach is not memory efficient and requires a segment tree's implementation. +This approach is not memory efficient and requires a segment tree's implementation. A more friendly approach would be using an order statistics tree. ### Problems From fd9509e4999be944b5c35f1a92d0a98c216e73b4 Mon Sep 17 00:00:00 2001 From: Ryan Chou <81596991+ryanchou-dev@users.noreply.github.com> Date: Thu, 11 Jul 2024 00:21:24 -0700 Subject: [PATCH 22/79] fix build --- content/5_Plat/DC-SRQ.problems.json | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/5_Plat/DC-SRQ.problems.json b/content/5_Plat/DC-SRQ.problems.json index d2fbe4581a..afe5fb2528 100644 --- a/content/5_Plat/DC-SRQ.problems.json +++ b/content/5_Plat/DC-SRQ.problems.json @@ -93,11 +93,11 @@ { "uniqueId": "cf-840D", "name": "Destiny", - "url": "https://codeforces.com/problemset/problem/840/D", + "url": "https://codeforces.com/contest/840/problem/D", "source": "CF", "difficulty": "Hard", "isStarred": false, - "tags": [], + "tags": ["Wavelet"], "solutionMetadata": { "kind": "autogen-label-from-site", "site": "CF" From c5394c414af04ceb4b30ee900f7bb7b11cd0e6b9 Mon Sep 17 00:00:00 2001 From: Ryan Chou <81596991+ryanchou-dev@users.noreply.github.com> Date: Thu, 11 Jul 2024 00:21:52 -0700 Subject: [PATCH 23/79] solution metadata --- content/6_Advanced/Wavelet.problems.json | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/content/6_Advanced/Wavelet.problems.json b/content/6_Advanced/Wavelet.problems.json index 45ac5b8786..5e3b7e11ec 100644 --- a/content/6_Advanced/Wavelet.problems.json +++ b/content/6_Advanced/Wavelet.problems.json @@ -25,7 +25,8 @@ "isStarred": false, "tags": ["Wavelet"], "solutionMetadata": { - "kind": "none" + "kind": "autogen-label-from-site", + "site": "CF" } }, { From b5864e22b4ca89ddab8dae735f368a864a51b62c Mon Sep 17 00:00:00 2001 From: Ryan Chou <81596991+ryanchou-dev@users.noreply.github.com> Date: Thu, 11 Jul 2024 00:28:17 -0700 Subject: [PATCH 24/79] format --- content/6_Advanced/Wavelet.mdx | 25 ++++++++++--------------- 1 file changed, 10 insertions(+), 15 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index b04c9404e2..ec9ed028f7 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -35,7 +35,7 @@ Suppose you want to support the following queries: - Given a range $l$, $r$ count the number of occurrences of value $x$. - Given a range $l$, $r$ find the $k$ smallest element -With wavelet tree, you can easily support those queries in $log(M)$ time. +With a wavelet tree, you can easily support those queries in $log(M)$ time. when M is the maximum value in the array. ## Wavelet Tree Structure @@ -65,7 +65,7 @@ $$ \end{aligned} $$ -This reduces the problem to counting the number of occurrences in a prefix. +This reduces the problem of counting the number of occurrences in a prefix. One way to solve the problem is to go to the leaf node and perform a binary search for the number of indices less than $𝑟$ @@ -87,7 +87,7 @@ $𝑟$. This can be done using a prefix sum. Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +- $c[i]$ as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -114,7 +114,7 @@ Note that this also works for non-leaf nodes using the following formula: $$ \texttt{occurrences}(l, r) = r - l $$ - + This is similar to counting how many times a value appears up to index 𝑅 in our previous query. We did this by using the new 𝑅 value at the leaf node. But now, we consider the difference between the updated 𝑅 and 𝐿 @@ -230,8 +230,6 @@ int main() { k++; cout << seg.find_k_smallest(a, b, k) << " "; } - - return 0; } ``` @@ -244,20 +242,17 @@ int main() { Let's support updates of type: - change value at index $i$ to $y$ -We can traverse down to the leaf to remove the old element and also traverse down to add the new element. +We can traverse down to the leaf to remove the old element and traverse again to add the new element. These updates only change our indices vector and prefix vector. -So what do the updates change? - - - Our indices vector - Our prefix vector +To make erasing and adding values easier, we can have the indices vector store sets instead of vectors. -To change the indices vector, what we can do is, instead of storing a vector, use a set. -Then erasing and adding values becomes easy. +On the other hand, to change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. -On the other hand, To change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. What we could do is use a sparse segment tree. +Instead, we can use a sparse segment tree. These support the operations: - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - querying for a prefix can be done by querying the segment tree from 0 to $i$ -This approach is not memory efficient and requires a segment tree's implementation. + +This approach is not memory efficient and requires a segment tree implementation. A more friendly approach would be using an order statistics tree. ### Problems From 56a111c3a85458a99ae2304bd3841fa6f0557167 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Thu, 25 Jul 2024 07:30:37 +0300 Subject: [PATCH 25/79] Update Wavelet.mdx --- content/6_Advanced/Wavelet.mdx | 204 +++++++++++++++++---------------- 1 file changed, 104 insertions(+), 100 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index ec9ed028f7..361739877d 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -35,12 +35,12 @@ Suppose you want to support the following queries: - Given a range $l$, $r$ count the number of occurrences of value $x$. - Given a range $l$, $r$ find the $k$ smallest element -With a wavelet tree, you can easily support those queries in $log(M)$ time. +With wavelet tree, you can easily support those queries in $log(M)$ time. when M is the maximum value in the array. ## Wavelet Tree Structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. @@ -48,12 +48,13 @@ We are going to maintain a segment tree over values instead of indices. Each seg -Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ +Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ +Each node has an array representing the indecies of every number between l and r ![Wavelet Tree Visualization](./assets/diagram.png) -## Solving the First Type of query +## Solving the First type of query **Given a range l, r count the number of occurrences of value x.** To calculate the number of occurrences from $𝑙$ to $𝑟$, we can use the following @@ -65,29 +66,28 @@ $$ \end{aligned} $$ -This reduces the problem of counting the number of occurrences in a prefix. +This reduces the problem to counting the number of occurrences in a prefix. -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. -### A different way **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. +tree. If we can determine the position (index) of $𝑟$ in the left and right -children of a node. +children of a node. We can recurse down the tree and determine its position in the leaf node. To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -98,23 +98,23 @@ prefixB[i] = prefixB[i - 1] + c[i] $$ -To update $r$ as we recurse down, we do the following: -- To know the value of 𝑟 if we recurse left, we use prefixB[r] +To update $r$ as we recurse down, we do the following: +- To know the value of 𝑟 if we recurse left, we use prefixB[r] - If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ \texttt{occurrences}(l, r) = r - l $$ - + This is similar to counting how many times a value appears up to index 𝑅 in our previous query. We did this by using the new 𝑅 value at the leaf node. But now, we consider the difference between the updated 𝑅 and 𝐿 @@ -125,7 +125,7 @@ $$ $$ - Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid + Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid @@ -154,83 +154,84 @@ the answer then will be the value of the node we end up on (leaf) using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r, const vector> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(), - rightIndices.begin(), - [this](const pair &elem) { - return elem.second < mid; - }); - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) { return l; } - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = - lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) { - return left->find_k_smallest(lb, lr, k); // Appears in left - } else { - return right->find_k_smallest(a - lb, b - lr, - k - inLeft); // Appears in right - } - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) { + return; + } + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), + [this](const pair& elem) { + return elem.second < mid; + }); + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) { + return l; + } + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) { + return left->find_k_smallest(lb,lr,k); // Appears in left + } else { + return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right + } + } }; + int main() { - int n, q; - cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - indices.emplace_back(i, v); - } - Segment seg(0, 1e9 + 2, indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } + int n, q; cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; cin >> v; + indices.emplace_back(i,v); + } + Segment seg(0,1e9 + 2,indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } + + return 0; } + ``` @@ -240,19 +241,22 @@ int main() { ## Supporting updates Let's support updates of type: - - change value at index $i$ to $y$ + - change value at index $i$ to $y$ -We can traverse down to the leaf to remove the old element and traverse again to add the new element. These updates only change our indices vector and prefix vector. +We can traverse down to the leaf to remove the old element and also traverse down to add the new element. -To make erasing and adding values easier, we can have the indices vector store sets instead of vectors. +So what do the updates change? + - + Our indices vector + Our prefix vector + +To change the indices vector, what we can do is, instead of storing a vector, use a set. +Then erasing and adding values becomes easy. -On the other hand, to change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. - -Instead, we can use a sparse segment tree. These support the operations: +On the other hand, To change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. What we could do is use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - querying for a prefix can be done by querying the segment tree from 0 to $i$ - -This approach is not memory efficient and requires a segment tree implementation. +This approach is not memory efficient and requires a segment tree's implementation. A more friendly approach would be using an order statistics tree. ### Problems From 10585a2c62fbb55cb7ab19809ee95e1f8cc6c884 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Thu, 25 Jul 2024 07:31:52 +0300 Subject: [PATCH 26/79] Update Wavelet.mdx --- content/6_Advanced/Wavelet.mdx | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 361739877d..9ec71825bc 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -38,7 +38,7 @@ Suppose you want to support the following queries: With wavelet tree, you can easily support those queries in $log(M)$ time. when M is the maximum value in the array. -## Wavelet Tree Structure +## Wavelet tree structure A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the @@ -54,7 +54,7 @@ Each node has an array representing the indecies of every number between l and r ![Wavelet Tree Visualization](./assets/diagram.png) -## Solving the First type of query +## Solving the first type of query **Given a range l, r count the number of occurrences of value x.** To calculate the number of occurrences from $𝑙$ to $𝑟$, we can use the following From 8eb4b89f1504bb6e593fd0bcb36f8e5cb28f599a Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Thu, 25 Jul 2024 04:31:51 +0000 Subject: [PATCH 27/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 191 +++++++++++++++++---------------- 1 file changed, 96 insertions(+), 95 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 9ec71825bc..f443163f5a 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -35,12 +35,12 @@ Suppose you want to support the following queries: - Given a range $l$, $r$ count the number of occurrences of value $x$. - Given a range $l$, $r$ find the $k$ smallest element -With wavelet tree, you can easily support those queries in $log(M)$ time. +With wavelet tree, you can easily support those queries in $log(M)$ time. when M is the maximum value in the array. ## Wavelet tree structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. @@ -48,7 +48,7 @@ We are going to maintain a segment tree over values instead of indices. Each seg -Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ +Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ Each node has an array representing the indecies of every number between l and r ![Wavelet Tree Visualization](./assets/diagram.png) @@ -68,26 +68,26 @@ $$ This reduces the problem to counting the number of occurrences in a prefix. -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. +tree. If we can determine the position (index) of $𝑟$ in the left and right -children of a node. +children of a node. We can recurse down the tree and determine its position in the leaf node. To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -98,17 +98,17 @@ prefixB[i] = prefixB[i - 1] + c[i] $$ -To update $r$ as we recurse down, we do the following: -- To know the value of 𝑟 if we recurse left, we use prefixB[r] +To update $r$ as we recurse down, we do the following: +- To know the value of 𝑟 if we recurse left, we use prefixB[r] - If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ @@ -125,7 +125,7 @@ $$ $$ - Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid + Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid @@ -154,84 +154,85 @@ the answer then will be the value of the node we end up on (leaf) using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) { - return; - } - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), - [this](const pair& elem) { - return elem.second < mid; - }); - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) { - return l; - } - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) { - return left->find_k_smallest(lb,lr,k); // Appears in left - } else { - return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right - } - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r, const vector> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(), + rightIndices.begin(), + [this](const pair &elem) { + return elem.second < mid; + }); + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) { return l; } + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = + lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) { + return left->find_k_smallest(lb, lr, k); // Appears in left + } else { + return right->find_k_smallest(a - lb, b - lr, + k - inLeft); // Appears in right + } + } }; - int main() { - int n, q; cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; cin >> v; - indices.emplace_back(i,v); - } - Segment seg(0,1e9 + 2,indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } - - return 0; + int n, q; + cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + indices.emplace_back(i, v); + } + Segment seg(0, 1e9 + 2, indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } + + return 0; } - ``` @@ -241,22 +242,22 @@ int main() { ## Supporting updates Let's support updates of type: - - change value at index $i$ to $y$ + - change value at index $i$ to $y$ We can traverse down to the leaf to remove the old element and also traverse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indices vector Our prefix vector - -To change the indices vector, what we can do is, instead of storing a vector, use a set. + +To change the indices vector, what we can do is, instead of storing a vector, use a set. Then erasing and adding values becomes easy. On the other hand, To change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. What we could do is use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - querying for a prefix can be done by querying the segment tree from 0 to $i$ -This approach is not memory efficient and requires a segment tree's implementation. +This approach is not memory efficient and requires a segment tree's implementation. A more friendly approach would be using an order statistics tree. ### Problems From f964801adaefa1cd1328526ab910881c5ac61c06 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Fri, 26 Jul 2024 09:31:22 +0300 Subject: [PATCH 28/79] Update Wavelet.mdx --- content/6_Advanced/Wavelet.mdx | 192 ++++++++++++++++----------------- 1 file changed, 96 insertions(+), 96 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index f443163f5a..1def84226b 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -35,12 +35,12 @@ Suppose you want to support the following queries: - Given a range $l$, $r$ count the number of occurrences of value $x$. - Given a range $l$, $r$ find the $k$ smallest element -With wavelet tree, you can easily support those queries in $log(M)$ time. +With wavelet tree, you can easily support those queries in $log(M)$ time. when M is the maximum value in the array. ## Wavelet tree structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. @@ -48,7 +48,7 @@ We are going to maintain a segment tree over values instead of indices. Each seg -Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ +Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ Each node has an array representing the indecies of every number between l and r ![Wavelet Tree Visualization](./assets/diagram.png) @@ -68,26 +68,26 @@ $$ This reduces the problem to counting the number of occurrences in a prefix. -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. +tree. If we can determine the position (index) of $𝑟$ in the left and right -children of a node. +children of a node. We can recurse down the tree and determine its position in the leaf node. To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -98,17 +98,17 @@ prefixB[i] = prefixB[i - 1] + c[i] $$ -To update $r$ as we recurse down, we do the following: -- To know the value of 𝑟 if we recurse left, we use prefixB[r] +To update $r$ as we recurse down, we do the following: +- To know the value of 𝑟 if we recurse left, we use prefixB[r] - If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ @@ -125,7 +125,7 @@ $$ $$ - Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid + Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid @@ -154,85 +154,84 @@ the answer then will be the value of the node we end up on (leaf) using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r, const vector> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(), - rightIndices.begin(), - [this](const pair &elem) { - return elem.second < mid; - }); - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) { return l; } - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = - lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) { - return left->find_k_smallest(lb, lr, k); // Appears in left - } else { - return right->find_k_smallest(a - lb, b - lr, - k - inLeft); // Appears in right - } - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) { + return; + } + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), + [this](const pair& elem) { + return elem.second < mid; + }); + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) { + return l; + } + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) { + return left->find_k_smallest(lb,lr,k); // Appears in left + } else { + return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right + } + } }; + int main() { - int n, q; - cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - indices.emplace_back(i, v); - } - Segment seg(0, 1e9 + 2, indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } - - return 0; + int n, q; cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; cin >> v; + indices.emplace_back(i,v); + } + Segment seg(0,1e9 + 2,indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } + + return 0; } + ``` @@ -242,23 +241,24 @@ int main() { ## Supporting updates Let's support updates of type: - - change value at index $i$ to $y$ + - change value at index $i$ to $y$ We can traverse down to the leaf to remove the old element and also traverse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indices vector Our prefix vector - -To change the indices vector, what we can do is, instead of storing a vector, use a set. + +To change the indices vector, what we can do is, instead of storing a vector, use a set. Then erasing and adding values becomes easy. On the other hand, To change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. What we could do is use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - querying for a prefix can be done by querying the segment tree from 0 to $i$ -This approach is not memory efficient and requires a segment tree's implementation. +This approach is not memory efficient and requires a segment tree's implementation. A more friendly approach would be using an order statistics tree. +Such that querying for a prefix would be equivalent to order_of_key($i$) ### Problems From f4590841abfdb57c493415c8a0d9505c576a8091 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Fri, 26 Jul 2024 06:32:33 +0000 Subject: [PATCH 29/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 191 +++++++++++++++++---------------- 1 file changed, 96 insertions(+), 95 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 1def84226b..9a8148c433 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -35,12 +35,12 @@ Suppose you want to support the following queries: - Given a range $l$, $r$ count the number of occurrences of value $x$. - Given a range $l$, $r$ find the $k$ smallest element -With wavelet tree, you can easily support those queries in $log(M)$ time. +With wavelet tree, you can easily support those queries in $log(M)$ time. when M is the maximum value in the array. ## Wavelet tree structure -A wavelet tree is a binary tree where each node represents a range of values. +A wavelet tree is a binary tree where each node represents a range of values. The root node covers the entire range, and each subsequent level splits the range into two halves. @@ -48,7 +48,7 @@ We are going to maintain a segment tree over values instead of indices. Each seg -Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ +Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ Each node has an array representing the indecies of every number between l and r ![Wavelet Tree Visualization](./assets/diagram.png) @@ -68,26 +68,26 @@ $$ This reduces the problem to counting the number of occurrences in a prefix. -One way to solve the problem is to go to the leaf node -and perform a binary search for the number of indices less than $𝑟$ +One way to solve the problem is to go to the leaf node +and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. **A different way to find the Index of $𝑟$ in the list of vertices** Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the -tree. +tree. If we can determine the position (index) of $𝑟$ in the left and right -children of a node. +children of a node. We can recurse down the tree and determine its position in the leaf node. To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede -$𝑟$. +$𝑟$. This can be done using a prefix sum. -Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +Let's define: +- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 - $prefixB[i]$ as prefix sum of $c[i]$ Formally @@ -98,17 +98,17 @@ prefixB[i] = prefixB[i - 1] + c[i] $$ -To update $r$ as we recurse down, we do the following: -- To know the value of 𝑟 if we recurse left, we use prefixB[r] +To update $r$ as we recurse down, we do the following: +- To know the value of 𝑟 if we recurse left, we use prefixB[r] - If we recurse right, we use 𝑟 - prefixB[r] ## Solving the second type of query **Given a range l, r find the k smallest element** We will determine whether the answer for a given node is in the left or the -right segment. +right segment. We can calculate how many times the elements within the segments' ranges appear -in our range $(l, r)$ using our first type of query. +in our range $(l, r)$ using our first type of query. Note that this also works for non-leaf nodes using the following formula: $$ @@ -125,7 +125,7 @@ $$ $$ - Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid + Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid @@ -154,84 +154,85 @@ the answer then will be the value of the node we end up on (leaf) using namespace std; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - vector> indices; // Index, Value - vector prefixB; - - Segment(int l, int r,const vector> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) { - return; - } - children = true; - if (r - l > 1) { - // Split the indices for left and right child - vector> leftIndices, rightIndices; - partition_copy(indices.begin(), indices.end(), leftIndices.begin(),rightIndices.begin(), - [this](const pair& elem) { - return elem.second < mid; - }); - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefixB.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefixB[i++] = j; - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) { - return l; - } - - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid - - if (k <= inLeft) { - return left->find_k_smallest(lb,lr,k); // Appears in left - } else { - return right->find_k_smallest(a - lb,b - lr, k - inLeft); // Appears in right - } - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + vector> indices; // Index, Value + vector prefixB; + + Segment(int l, int r, const vector> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) { + calculate_b(); + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + if (r - l > 1) { + // Split the indices for left and right child + vector> leftIndices, rightIndices; + partition_copy(indices.begin(), indices.end(), leftIndices.begin(), + rightIndices.begin(), + [this](const pair &elem) { + return elem.second < mid; + }); + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + + // Calculates the prefix B + void calculate_b() { + int i = 1; + int j = 0; + prefixB.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefixB[i++] = j; + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) { return l; } + + int lb = prefixB[a]; + int lr = prefixB[b]; + int inLeft = + lr - lb; // Amount of values in range (a,b) that are less the mid + + if (k <= inLeft) { + return left->find_k_smallest(lb, lr, k); // Appears in left + } else { + return right->find_k_smallest(a - lb, b - lr, + k - inLeft); // Appears in right + } + } }; - int main() { - int n, q; cin >> n >> q; - - vector> indices; - for (int i = 0; i < n; ++i) { - int v; cin >> v; - indices.emplace_back(i,v); - } - Segment seg(0,1e9 + 2,indices); - - for (int i = 0; i < q; ++i) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } - - return 0; + int n, q; + cin >> n >> q; + + vector> indices; + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + indices.emplace_back(i, v); + } + Segment seg(0, 1e9 + 2, indices); + + for (int i = 0; i < q; ++i) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } + + return 0; } - ``` @@ -241,22 +242,22 @@ int main() { ## Supporting updates Let's support updates of type: - - change value at index $i$ to $y$ + - change value at index $i$ to $y$ We can traverse down to the leaf to remove the old element and also traverse down to add the new element. -So what do the updates change? - - +So what do the updates change? + - Our indices vector Our prefix vector - -To change the indices vector, what we can do is, instead of storing a vector, use a set. + +To change the indices vector, what we can do is, instead of storing a vector, use a set. Then erasing and adding values becomes easy. On the other hand, To change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. What we could do is use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - querying for a prefix can be done by querying the segment tree from 0 to $i$ -This approach is not memory efficient and requires a segment tree's implementation. +This approach is not memory efficient and requires a segment tree's implementation. A more friendly approach would be using an order statistics tree. Such that querying for a prefix would be equivalent to order_of_key($i$) From 5738143044254ecc4a1385e8e16c982ab6fce366 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Thu, 15 Aug 2024 15:10:35 +0300 Subject: [PATCH 30/79] Update Wavelet.problems.json --- content/6_Advanced/Wavelet.problems.json | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/content/6_Advanced/Wavelet.problems.json b/content/6_Advanced/Wavelet.problems.json index 5e3b7e11ec..7a9e06f9e2 100644 --- a/content/6_Advanced/Wavelet.problems.json +++ b/content/6_Advanced/Wavelet.problems.json @@ -25,8 +25,7 @@ "isStarred": false, "tags": ["Wavelet"], "solutionMetadata": { - "kind": "autogen-label-from-site", - "site": "CF" + "kind": "none" } }, { @@ -42,13 +41,13 @@ } }, { - "uniqueId": "coci-2020-index", + "uniqueId": "coci-20-index", "name": "Index", "url": "https://evaluator.hsin.hr/tasks/HONI202167index/", "source": "COCI", "difficulty": "Normal", "isStarred": false, - "tags": ["Wavelet"], + "tags": ["Wavelet, Persistent Segtree"], "solutionMetadata": { "kind": "none" } @@ -77,6 +76,7 @@ "kind": "autogen-label-from-site", "site": "DMOJ" } + } ] } From 89c01e03d364624d962b6d0e85e7229e8904d811 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Thu, 15 Aug 2024 12:11:53 +0000 Subject: [PATCH 31/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.problems.json | 1 - 1 file changed, 1 deletion(-) diff --git a/content/6_Advanced/Wavelet.problems.json b/content/6_Advanced/Wavelet.problems.json index 7a9e06f9e2..4335c86ad8 100644 --- a/content/6_Advanced/Wavelet.problems.json +++ b/content/6_Advanced/Wavelet.problems.json @@ -76,7 +76,6 @@ "kind": "autogen-label-from-site", "site": "DMOJ" } - } ] } From 27ca815faaf546e9f9256ade248c40e918e225ff Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Thu, 15 Aug 2024 16:01:28 +0300 Subject: [PATCH 32/79] Update Wavelet.problems.json --- content/6_Advanced/Wavelet.problems.json | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/content/6_Advanced/Wavelet.problems.json b/content/6_Advanced/Wavelet.problems.json index 4335c86ad8..42bbfc6dd4 100644 --- a/content/6_Advanced/Wavelet.problems.json +++ b/content/6_Advanced/Wavelet.problems.json @@ -42,7 +42,7 @@ }, { "uniqueId": "coci-20-index", - "name": "Index", + "name": "2021 - Index", "url": "https://evaluator.hsin.hr/tasks/HONI202167index/", "source": "COCI", "difficulty": "Normal", @@ -76,6 +76,7 @@ "kind": "autogen-label-from-site", "site": "DMOJ" } + } ] } From 0dbb8e1842d3de36c836ad3847b4fb7baf7f3acd Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Thu, 15 Aug 2024 13:03:05 +0000 Subject: [PATCH 33/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.problems.json | 1 - 1 file changed, 1 deletion(-) diff --git a/content/6_Advanced/Wavelet.problems.json b/content/6_Advanced/Wavelet.problems.json index 42bbfc6dd4..9b7ffd82a1 100644 --- a/content/6_Advanced/Wavelet.problems.json +++ b/content/6_Advanced/Wavelet.problems.json @@ -76,7 +76,6 @@ "kind": "autogen-label-from-site", "site": "DMOJ" } - } ] } From 3466558799bcea99307f61b7f0e29df38bf9f731 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Wed, 21 Aug 2024 22:10:49 +0300 Subject: [PATCH 34/79] Update content/6_Advanced/Wavelet.mdx Co-authored-by: Ryan Chou <81596991+ryanchou-dev@users.noreply.github.com> --- content/6_Advanced/Wavelet.mdx | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 9a8148c433..65834cdea9 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -143,7 +143,7 @@ $$ the answer then will be the value of the node we end up on (leaf) ## Implemention -**Time Complexity:** $\mathcal{O}(Q * log(M))$ +**Time Complexity:** $\mathcal{O}(Q \cdot \log(M))$ From cb82a0c631c20d534fdf4b96d258aea2caa8b1ad Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Wed, 21 Aug 2024 22:11:16 +0300 Subject: [PATCH 35/79] Update content/6_Advanced/Wavelet.mdx Co-authored-by: Ryan Chou <81596991+ryanchou-dev@users.noreply.github.com> --- content/6_Advanced/Wavelet.mdx | 1 - 1 file changed, 1 deletion(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 65834cdea9..e0cb79589d 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -231,7 +231,6 @@ int main() { cout << seg.find_k_smallest(a, b, k) << " "; } - return 0; } ``` From 163120501a0fc6ea2eeef10d7b09e76a5d1bc952 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Wed, 21 Aug 2024 19:15:22 +0000 Subject: [PATCH 36/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 1 - 1 file changed, 1 deletion(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index e0cb79589d..292d955421 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -230,7 +230,6 @@ int main() { k++; cout << seg.find_k_smallest(a, b, k) << " "; } - } ``` From 75ebdca9cc35f0cea9609a4c45cf3b71183e0a8c Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Wed, 2 Oct 2024 17:00:07 +0300 Subject: [PATCH 37/79] Changed problem name.json --- content/6_Advanced/Wavelet.problems.json | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/content/6_Advanced/Wavelet.problems.json b/content/6_Advanced/Wavelet.problems.json index 9b7ffd82a1..553fab27de 100644 --- a/content/6_Advanced/Wavelet.problems.json +++ b/content/6_Advanced/Wavelet.problems.json @@ -30,7 +30,7 @@ }, { "uniqueId": "spoj-ILKQUERY2", - "name": "ILKQUERY2", + "name": "I Love Kd-Trees", "url": "https://www.spoj.com/problems/ILKQUERY2/", "source": "SPOJ", "difficulty": "Normal", @@ -76,6 +76,7 @@ "kind": "autogen-label-from-site", "site": "DMOJ" } + } ] } From c49efd4c8ad05349ecf35e0dd6a0e07e0bb01fbf Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Wed, 2 Oct 2024 18:48:35 +0000 Subject: [PATCH 38/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 6 ++---- content/6_Advanced/Wavelet.problems.json | 1 - 2 files changed, 2 insertions(+), 5 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 292d955421..17b9bfdbcd 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -173,8 +173,7 @@ struct Segment { // Split the indices for left and right child vector> leftIndices, rightIndices; partition_copy(indices.begin(), indices.end(), leftIndices.begin(), - rightIndices.begin(), - [this](const pair &elem) { + rightIndices.begin(), [this](const pair &elem) { return elem.second < mid; }); @@ -200,8 +199,7 @@ struct Segment { int lb = prefixB[a]; int lr = prefixB[b]; - int inLeft = - lr - lb; // Amount of values in range (a,b) that are less the mid + int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid if (k <= inLeft) { return left->find_k_smallest(lb, lr, k); // Appears in left diff --git a/content/6_Advanced/Wavelet.problems.json b/content/6_Advanced/Wavelet.problems.json index 553fab27de..c691111031 100644 --- a/content/6_Advanced/Wavelet.problems.json +++ b/content/6_Advanced/Wavelet.problems.json @@ -76,7 +76,6 @@ "kind": "autogen-label-from-site", "site": "DMOJ" } - } ] } From fd77752ef2432f258dd94863f46fc6e42807a704 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Thu, 3 Oct 2024 10:11:47 +0300 Subject: [PATCH 39/79] Changed description.mdx --- content/6_Advanced/Wavelet.mdx | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 17b9bfdbcd..573dc36cb7 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -4,7 +4,7 @@ title: 'Wavelet Tree' author: Benjamin Qi, Omri Jerbi prerequisites: - RURQ -description: Wavelet trees are data structures that support efficient queries for the k-th minimum element in a range by maintaining a segment tree over values instead of indices" +description: Wavelet trees support efficient queries for the kth minimum element in a range" frequency: 0 --- @@ -173,7 +173,8 @@ struct Segment { // Split the indices for left and right child vector> leftIndices, rightIndices; partition_copy(indices.begin(), indices.end(), leftIndices.begin(), - rightIndices.begin(), [this](const pair &elem) { + rightIndices.begin(), + [this](const pair &elem) { return elem.second < mid; }); @@ -199,7 +200,8 @@ struct Segment { int lb = prefixB[a]; int lr = prefixB[b]; - int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid + int inLeft = + lr - lb; // Amount of values in range (a,b) that are less the mid if (k <= inLeft) { return left->find_k_smallest(lb, lr, k); // Appears in left From 03ddc495277ae5b781f8bee951815c799ab5f406 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Thu, 3 Oct 2024 07:13:26 +0000 Subject: [PATCH 40/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 6 ++---- 1 file changed, 2 insertions(+), 4 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 573dc36cb7..47838c5972 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -173,8 +173,7 @@ struct Segment { // Split the indices for left and right child vector> leftIndices, rightIndices; partition_copy(indices.begin(), indices.end(), leftIndices.begin(), - rightIndices.begin(), - [this](const pair &elem) { + rightIndices.begin(), [this](const pair &elem) { return elem.second < mid; }); @@ -200,8 +199,7 @@ struct Segment { int lb = prefixB[a]; int lr = prefixB[b]; - int inLeft = - lr - lb; // Amount of values in range (a,b) that are less the mid + int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid if (k <= inLeft) { return left->find_k_smallest(lb, lr, k); // Appears in left From 5013297c10eecf866107719599031dab45f5d4ce Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Thu, 3 Oct 2024 10:15:05 +0300 Subject: [PATCH 41/79] Some minor changes.mdx --- content/6_Advanced/Wavelet.mdx | 10 ++++++---- 1 file changed, 6 insertions(+), 4 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 47838c5972..60b0632004 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -49,7 +49,7 @@ We are going to maintain a segment tree over values instead of indices. Each seg Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ -Each node has an array representing the indecies of every number between l and r +Each node has an array representing the indices of every number between l and r ![Wavelet Tree Visualization](./assets/diagram.png) @@ -105,7 +105,7 @@ To update $r$ as we recurse down, we do the following: ## Solving the second type of query **Given a range l, r find the k smallest element** -We will determine whether the answer for a given node is in the left or the +We will determine whether the answer for a given node is on the left or the right segment. We can calculate how many times the elements within the segments' ranges appear in our range $(l, r)$ using our first type of query. @@ -173,7 +173,8 @@ struct Segment { // Split the indices for left and right child vector> leftIndices, rightIndices; partition_copy(indices.begin(), indices.end(), leftIndices.begin(), - rightIndices.begin(), [this](const pair &elem) { + rightIndices.begin(), + [this](const pair &elem) { return elem.second < mid; }); @@ -199,7 +200,8 @@ struct Segment { int lb = prefixB[a]; int lr = prefixB[b]; - int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid + int inLeft = + lr - lb; // Amount of values in range (a,b) that are less the mid if (k <= inLeft) { return left->find_k_smallest(lb, lr, k); // Appears in left From a3f89ebcc3766fa3f093aa90228c2f91175f09b1 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Thu, 3 Oct 2024 07:16:17 +0000 Subject: [PATCH 42/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 6 ++---- 1 file changed, 2 insertions(+), 4 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 60b0632004..c7492cdf0a 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -173,8 +173,7 @@ struct Segment { // Split the indices for left and right child vector> leftIndices, rightIndices; partition_copy(indices.begin(), indices.end(), leftIndices.begin(), - rightIndices.begin(), - [this](const pair &elem) { + rightIndices.begin(), [this](const pair &elem) { return elem.second < mid; }); @@ -200,8 +199,7 @@ struct Segment { int lb = prefixB[a]; int lr = prefixB[b]; - int inLeft = - lr - lb; // Amount of values in range (a,b) that are less the mid + int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid if (k <= inLeft) { return left->find_k_smallest(lb, lr, k); // Appears in left From 39519df9415520ec8e75062f2fef1af66248a5e7 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Sun, 6 Oct 2024 13:25:16 +0300 Subject: [PATCH 43/79] Changed variable name styles (to snake cases).mdx --- content/6_Advanced/Wavelet.mdx | 20 +++++++++++--------- 1 file changed, 11 insertions(+), 9 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index c7492cdf0a..7cde9b66de 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -49,7 +49,7 @@ We are going to maintain a segment tree over values instead of indices. Each seg Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ -Each node has an array representing the indices of every number between l and r +Each node has an array representing the indecies of every number between l and r ![Wavelet Tree Visualization](./assets/diagram.png) @@ -105,7 +105,7 @@ To update $r$ as we recurse down, we do the following: ## Solving the second type of query **Given a range l, r find the k smallest element** -We will determine whether the answer for a given node is on the left or the +We will determine whether the answer for a given node is in the left or the right segment. We can calculate how many times the elements within the segments' ranges appear in our range $(l, r)$ using our first type of query. @@ -158,7 +158,7 @@ struct Segment { int l, r, mid; bool children = false; vector> indices; // Index, Value - vector prefixB; + vector prefix_b; Segment(int l, int r, const vector> &indices) : l(l), r(r), mid((r + l) / 2), indices(indices) { @@ -173,7 +173,8 @@ struct Segment { // Split the indices for left and right child vector> leftIndices, rightIndices; partition_copy(indices.begin(), indices.end(), leftIndices.begin(), - rightIndices.begin(), [this](const pair &elem) { + rightIndices.begin(), + [this](const pair &elem) { return elem.second < mid; }); @@ -186,10 +187,10 @@ struct Segment { void calculate_b() { int i = 1; int j = 0; - prefixB.resize(indices.size() + 1); + prefix_b.resize(indices.size() + 1); for (auto [ind, val] : indices) { if (val < mid) j++; - prefixB[i++] = j; + prefix_b[i++] = j; } } @@ -197,9 +198,10 @@ struct Segment { update(); if (r - l <= 1) { return l; } - int lb = prefixB[a]; - int lr = prefixB[b]; - int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid + int lb = prefix_b[a]; + int lr = prefix_b[b]; + int inLeft = + lr - lb; // Amount of values in range (a,b) that are less the mid if (k <= inLeft) { return left->find_k_smallest(lb, lr, k); // Appears in left From 3a1c37a173529bcfb1e4c0deed23f03ef5dd2379 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Sun, 6 Oct 2024 10:26:31 +0000 Subject: [PATCH 44/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 6 ++---- 1 file changed, 2 insertions(+), 4 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 7cde9b66de..34ec809d1f 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -173,8 +173,7 @@ struct Segment { // Split the indices for left and right child vector> leftIndices, rightIndices; partition_copy(indices.begin(), indices.end(), leftIndices.begin(), - rightIndices.begin(), - [this](const pair &elem) { + rightIndices.begin(), [this](const pair &elem) { return elem.second < mid; }); @@ -200,8 +199,7 @@ struct Segment { int lb = prefix_b[a]; int lr = prefix_b[b]; - int inLeft = - lr - lb; // Amount of values in range (a,b) that are less the mid + int inLeft = lr - lb; // Amount of values in range (a,b) that are less the mid if (k <= inLeft) { return left->find_k_smallest(lb, lr, k); // Appears in left From 0cd5cfa60dab46b896bb7b5d0403db0400451895 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Tue, 8 Oct 2024 19:33:36 +0300 Subject: [PATCH 45/79] Update content/6_Advanced/Wavelet.mdx Removed unnecessary details Co-authored-by: Ryan Chou <81596991+ryanchou-dev@users.noreply.github.com> --- content/6_Advanced/Wavelet.mdx | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 34ec809d1f..bb6f59da7a 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -32,8 +32,8 @@ Wavelet trees are data structures that support efficient queries for the k-th mi Suppose you want to support the following queries: -- Given a range $l$, $r$ count the number of occurrences of value $x$. -- Given a range $l$, $r$ find the $k$ smallest element +- Given a range, count the number of occurrences of value $x$. +- Given a range, find the $k$ smallest element With wavelet tree, you can easily support those queries in $log(M)$ time. when M is the maximum value in the array. From 4ee7ff8320c611e0d1456d8fa764abce1bc6d9da Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Tue, 8 Oct 2024 19:34:26 +0300 Subject: [PATCH 46/79] Update content/6_Advanced/Wavelet.mdx Latex consistency changes Co-authored-by: Ryan Chou <81596991+ryanchou-dev@users.noreply.github.com> --- content/6_Advanced/Wavelet.mdx | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index bb6f59da7a..7027cc99bd 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -35,8 +35,8 @@ Suppose you want to support the following queries: - Given a range, count the number of occurrences of value $x$. - Given a range, find the $k$ smallest element -With wavelet tree, you can easily support those queries in $log(M)$ time. -when M is the maximum value in the array. +With a wavelet tree, you can easily support those queries in $\log(M)$ time, +where $M$ is the maximum value in the array. ## Wavelet tree structure From 43abe7116010ae2be5d639bbcc5801324060811a Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Tue, 8 Oct 2024 19:35:02 +0300 Subject: [PATCH 47/79] Update content/6_Advanced/Wavelet.mdx Latex consistency Co-authored-by: Ryan Chou <81596991+ryanchou-dev@users.noreply.github.com> From 8586c51447e1d4bebeab7a8dfdd3d73c5da10f82 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Tue, 8 Oct 2024 19:57:14 +0300 Subject: [PATCH 48/79] Some grammer and latex changes + added motivation --- content/6_Advanced/Wavelet.mdx | 15 +++++++++------ 1 file changed, 9 insertions(+), 6 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 7027cc99bd..bd7540e4ec 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -46,10 +46,15 @@ range into two halves. We are going to maintain a segment tree over values instead of indices. Each segment will contain indices whose values lie within the segment's range. We'll save those indices in a vector. Notice that an index can be in at most $log(M)$ segments + + + The motivation for maintaining a segment tree over values lies in the types of queries we want to answer. While traditional segment trees handle range-based queries like sums effectively, they fall short for value-based queries. By maintaining segments over values, we can efficiently address these value-specific queries. + + Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ -Each node has an array representing the indecies of every number between l and r +Each node has an array representing the indices of every number between l and r ![Wavelet Tree Visualization](./assets/diagram.png) @@ -73,8 +78,6 @@ and perform a binary search for the number of indices less than $𝑟$ However, let's explore a different approach that can also be extended to the second type of query. -**A different way to find the Index of $𝑟$ in the list of vertices** - Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the tree. If we can determine the position (index) of $𝑟$ in the left and right @@ -99,11 +102,11 @@ $$ To update $r$ as we recurse down, we do the following: -- To know the value of 𝑟 if we recurse left, we use prefixB[r] -- If we recurse right, we use 𝑟 - prefixB[r] +- To know the value of $r$ if we recurse left, we use prefixB[r] +- If we recurse right, we use $r$ - prefixB[r] ## Solving the second type of query -**Given a range l, r find the k smallest element** +**Given a range $l$, $r$ find the k smallest element** We will determine whether the answer for a given node is in the left or the right segment. From 3774454a5a334a225b7134665dc152365aba2833 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Mon, 21 Oct 2024 10:53:26 +0300 Subject: [PATCH 49/79] Structure changes + latex errors --- content/6_Advanced/Wavelet.mdx | 79 +++++++++++++++++----------------- 1 file changed, 39 insertions(+), 40 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index bd7540e4ec..b7e0b3dc35 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -4,7 +4,7 @@ title: 'Wavelet Tree' author: Benjamin Qi, Omri Jerbi prerequisites: - RURQ -description: Wavelet trees support efficient queries for the kth minimum element in a range" +description: Wavelet trees support efficient queries for the kth minimum element in a range frequency: 0 --- @@ -30,28 +30,20 @@ Wavelet trees are data structures that support efficient queries for the k-th mi +## Introduction Suppose you want to support the following queries: - Given a range, count the number of occurrences of value $x$. - Given a range, find the $k$ smallest element -With a wavelet tree, you can easily support those queries in $\log(M)$ time, +With a wavelet tree, you can easily support those queries in $\mathcal{O}(\log M)$ time, where $M$ is the maximum value in the array. -## Wavelet tree structure +### Wavelet tree structure -A wavelet tree is a binary tree where each node represents a range of values. -The root node covers the entire range, and each subsequent level splits the -range into two halves. +To answer value-based queries efficiently, we'll create a segment tree where each node represents a range of values, instead of indices. Just like a normal segment tree, each subsequent level splits the range into two halves. Note that an index can appear in at most $\log(M)$ nodes. -We are going to maintain a segment tree over values instead of indices. Each segment will contain indices whose values lie within the segment's range. We'll save those indices in a vector. Notice that an index can be in at most $log(M)$ segments - - - - The motivation for maintaining a segment tree over values lies in the types of queries we want to answer. While traditional segment trees handle range-based queries like sums effectively, they fall short for value-based queries. By maintaining segments over values, we can efficiently address these value-specific queries. - - - + Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ Each node has an array representing the indices of every number between l and r @@ -59,8 +51,11 @@ Each node has an array representing the indices of every number between l and r ![Wavelet Tree Visualization](./assets/diagram.png) +### Solution - Range K-th Smallest +Before we solve this problem, let's consider a simpler version where we are asked, given a range, to count the number of occurrences of value $x$. + ## Solving the first type of query -**Given a range l, r count the number of occurrences of value x.** +**Given a range $l$, $r$, count the number of occurrences of value x.** To calculate the number of occurrences from $𝑙$ to $𝑟$, we can use the following formula: @@ -80,9 +75,7 @@ second type of query. Instead of binary searching on the leaf, we update $𝑟$ as we recurse down the tree. -If we can determine the position (index) of $𝑟$ in the left and right -children of a node. -We can recurse down the tree and determine its position in the leaf node. +If we can determine the position (index) of $r$ in the left and right children of a node, we can recurse down the tree and determine its position in the leaf node. To find the position of $𝑟$ in a node's left and right children, we need to determine how many indices are smaller than the middle value (mid) and precede @@ -91,20 +84,26 @@ This can be done using a prefix sum. Let's define: - $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 -- $prefixB[i]$ as prefix sum of $c[i]$ +- $prefix_b[i]$ as prefix sum of $c[i]$ Formally $$ -c[i] = index[i] < mid ? 1 : 0; -prefixB[i] = prefixB[i - 1] + c[i] +c[i] = \begin{cases} + 1, & \text{if } \text{index}[i] < \text{mid} \\ + 0, & \text{otherwise} +\end{cases} +$$ +$$ +\text{prefixB}[i] = \text{prefixB}[i - 1] + c[i] $$ To update $r$ as we recurse down, we do the following: -- To know the value of $r$ if we recurse left, we use prefixB[r] -- If we recurse right, we use $r$ - prefixB[r] +- To know the value of $r$ if we recurse left, we use prefix_b[r] +- If we recurse right, we use $r$ - prefix_b[r] +Now let's try to solve our main problem. ## Solving the second type of query **Given a range $l$, $r$ find the k smallest element** @@ -117,14 +116,14 @@ Note that this also works for non-leaf nodes using the following formula: $$ \texttt{occurrences}(l, r) = r - l $$ - -This is similar to counting how many times a value appears up to index 𝑅 in our previous query. We did this by using the new 𝑅 value at the leaf node. But now, we consider the difference between the updated 𝑅 and 𝐿 + +This is similar to counting how many times a value appears up to index $r$ in our previous query. We did this by using the new $r$ value at the leaf node. But now, we consider the difference between the updated $r$ and $l$ Therefore, the occurrences of the left node is $$ -\texttt{left\_occurrences} = prefixB[r] - prefixB[l] +\texttt{left\_occurrences} = prefix_b[r] - prefix_b[l] $$ @@ -145,6 +144,8 @@ $$ the answer then will be the value of the node we end up on (leaf) +In conclusion we maintain our ranges l and r as we recurse down to our child, and when we reach the child node we can return $R$ - $L$. + ## Implemention **Time Complexity:** $\mathcal{O}(Q \cdot \log(M))$ @@ -155,6 +156,7 @@ the answer then will be the value of the node we end up on (leaf) #include using namespace std; +constexpr int MAX_VAL = 1e9 + 2; struct Segment { Segment *left = nullptr, *right = nullptr; @@ -165,7 +167,15 @@ struct Segment { Segment(int l, int r, const vector> &indices) : l(l), r(r), mid((r + l) / 2), indices(indices) { - calculate_b(); + + // Calcultes the prefix B + int i = 1; + int j = 0; + prefix_b.resize(indices.size() + 1); + for (auto [ind, val] : indices) { + if (val < mid) j++; + prefix_b[i++] = j; + } } // Sparse since values can go up to 1e9 @@ -185,17 +195,6 @@ struct Segment { } } - // Calculates the prefix B - void calculate_b() { - int i = 1; - int j = 0; - prefix_b.resize(indices.size() + 1); - for (auto [ind, val] : indices) { - if (val < mid) j++; - prefix_b[i++] = j; - } - } - int find_k_smallest(int a, int b, int k) { update(); if (r - l <= 1) { return l; } @@ -223,7 +222,7 @@ int main() { cin >> v; indices.emplace_back(i, v); } - Segment seg(0, 1e9 + 2, indices); + Segment seg(0, MAX_VAL, indices); for (int i = 0; i < q; ++i) { int a, b, k; @@ -253,7 +252,7 @@ So what do the updates change? To change the indices vector, what we can do is, instead of storing a vector, use a set. Then erasing and adding values becomes easy. -On the other hand, To change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. What we could do is use a sparse segment tree. +On the other hand, to change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. What we could do is use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - querying for a prefix can be done by querying the segment tree from 0 to $i$ This approach is not memory efficient and requires a segment tree's implementation. From d36e36fb7d007a87fdcf851d105c26ab66011965 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Mon, 21 Oct 2024 07:54:40 +0000 Subject: [PATCH 50/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index b7e0b3dc35..768e319695 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -30,7 +30,7 @@ Wavelet trees are data structures that support efficient queries for the k-th mi -## Introduction +## Introduction Suppose you want to support the following queries: - Given a range, count the number of occurrences of value $x$. @@ -89,7 +89,7 @@ Let's define: Formally $$ -c[i] = \begin{cases} +c[i] = \begin{cases} 1, & \text{if } \text{index}[i] < \text{mid} \\ 0, & \text{otherwise} \end{cases} From 920e150191f284d2b7f0444c251e008ca9673228 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Fri, 25 Oct 2024 11:14:39 +0300 Subject: [PATCH 51/79] Update content/6_Advanced/Wavelet.mdx Co-authored-by: Ryan Chou <81596991+ryanchou-dev@users.noreply.github.com> --- content/6_Advanced/Wavelet.mdx | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 768e319695..ab5f49ee66 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -168,7 +168,7 @@ struct Segment { Segment(int l, int r, const vector> &indices) : l(l), r(r), mid((r + l) / 2), indices(indices) { - // Calcultes the prefix B + // Calculate the prefix B int i = 1; int j = 0; prefix_b.resize(indices.size() + 1); From e688cb5b5acf5224ccfeacac7d2de857e606a244 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Fri, 25 Oct 2024 11:15:15 +0300 Subject: [PATCH 52/79] Update content/6_Advanced/Wavelet.mdx Co-authored-by: Ryan Chou <81596991+ryanchou-dev@users.noreply.github.com> --- content/6_Advanced/Wavelet.mdx | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index ab5f49ee66..ee9028e02e 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -249,8 +249,7 @@ So what do the updates change? Our indices vector Our prefix vector -To change the indices vector, what we can do is, instead of storing a vector, use a set. -Then erasing and adding values becomes easy. +To support erasing/adding values to the indices vector quickly, we can choose to use a set instead of a vector. On the other hand, to change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. What we could do is use a sparse segment tree. - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index From af86360e720daf9aec2adf9114ebb69f249ea86e Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Fri, 25 Oct 2024 11:15:21 +0300 Subject: [PATCH 53/79] Update content/6_Advanced/Wavelet.mdx Co-authored-by: Ryan Chou <81596991+ryanchou-dev@users.noreply.github.com> --- content/6_Advanced/Wavelet.mdx | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index ee9028e02e..1e676c2aa9 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -83,7 +83,7 @@ $𝑟$. This can be done using a prefix sum. Let's define: -- $c[i]$ = as 1 if $index[i]$ is smaller than mid otherwise 0 +- $c[i]$ = as $1$ if $index[i]$ is smaller than mid otherwise $0$ - $prefix_b[i]$ as prefix sum of $c[i]$ Formally From f5d7c2ab920c72c560b6e33862c311f6994eb3ee Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Tue, 5 Nov 2024 21:10:46 +0200 Subject: [PATCH 54/79] Added code for updates.mdx --- content/6_Advanced/Wavelet.mdx | 129 +++++++++++++++++++++++++++++++-- 1 file changed, 122 insertions(+), 7 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 1e676c2aa9..08b5bac584 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -90,12 +90,12 @@ Formally $$ c[i] = \begin{cases} - 1, & \text{if } \text{index}[i] < \text{mid} \\ - 0, & \text{otherwise} + 1, & \texttt{if } \texttt{index}[i] < \texttt{mid} \\ + 0, & \texttt{otherwise} \end{cases} $$ $$ -\text{prefixB}[i] = \text{prefixB}[i - 1] + c[i] +\texttt{prefixB}[i] = \texttt{prefixB}[i - 1] + c[i] $$ @@ -126,10 +126,7 @@ $$ \texttt{left\_occurrences} = prefix_b[r] - prefix_b[l] $$ - - Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid - - +Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid - If $\texttt{left\_occurrences}$ is greater or equal to $k$, it means the $k$-th smallest element is in the left subtree. Therefore, we update our range and recurse into the left @@ -258,6 +255,124 @@ This approach is not memory efficient and requires a segment tree's implementati A more friendly approach would be using an order statistics tree. Such that querying for a prefix would be equivalent to order_of_key($i$) +## Implemention +**Time Complexity:** $\mathcal{O}(Q \cdot \log(M) \cdot \log(N))$ + + + +```cpp +#include + +using namespace std; + +#include +using namespace __gnu_pbds; +template +using Tree = + tree, rb_tree_tag, tree_order_statistics_node_update>; + +struct Segment { + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // Index, Value + + Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) {return;} + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) + leftIndices.insert({index, value}); + else + rightIndices.insert({index, value}); + } + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) + return left->find_k_smallest(a,b,k); // Appears in left + else + return right->find_k_smallest(a,b, k - inLeft); // Appears in right + } + + void update(int index, int v, int oldV) { + update(oldV, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insertOperation) { + if (insertOperation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) + left->update(v, index, insertOperation); + else + right->update(v, index, insertOperation); + } +}; + + +int main() { + int n, q; cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { + indices.insert({i,0}); + } + + Segment seg(0,1e9 + 2,indices); + vector arr(n); + + for (int i = 0; i < n; ++i) { + int v; cin >> v; + seg.update(i, v, arr[i]); + arr[i] =v; + } + + for (int i = 0; i < q; ++i) { + int t; cin >> t; + if (t == 1) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } else { + int a, b; cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; +} + +``` + + + + ### Problems From 6f4a74a300e40cc784bf9c140d9a1fd2ecd751e9 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Tue, 5 Nov 2024 19:11:56 +0000 Subject: [PATCH 55/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 180 ++++++++++++++++----------------- 1 file changed, 87 insertions(+), 93 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 08b5bac584..edac005bcc 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -269,105 +269,99 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // Index, Value - - Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) {return;} - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) - leftIndices.insert({index, value}); - else - rightIndices.insert({index, value}); - } - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) - return left->find_k_smallest(a,b,k); // Appears in left - else - return right->find_k_smallest(a,b, k - inLeft); // Appears in right - } - - void update(int index, int v, int oldV) { - update(oldV, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insertOperation) { - if (insertOperation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) - left->update(v, index, insertOperation); - else - right->update(v, index, insertOperation); - } -}; + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // Index, Value + + Segment(int l, int r, const Tree> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) {} + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) leftIndices.insert({index, value}); + else rightIndices.insert({index, value}); + } + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left + else return right->find_k_smallest(a, b, k - inLeft); // Appears in right + } + + void update(int index, int v, int oldV) { + update(oldV, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insertOperation) { + if (insertOperation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) left->update(v, index, insertOperation); + else right->update(v, index, insertOperation); + } +}; int main() { - int n, q; cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { - indices.insert({i,0}); - } - - Segment seg(0,1e9 + 2,indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; cin >> v; - seg.update(i, v, arr[i]); - arr[i] =v; - } - - for (int i = 0; i < q; ++i) { - int t; cin >> t; - if (t == 1) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } else { - int a, b; cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; -} + int n, q; + cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } + + Segment seg(0, 1e9 + 2, indices); + vector arr(n); + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + seg.update(i, v, arr[i]); + arr[i] = v; + } + + for (int i = 0; i < q; ++i) { + int t; + cin >> t; + if (t == 1) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } else { + int a, b; + cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; +} ``` From 0c64b49d710b5d30f6a000a93ce3a093857a1377 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Tue, 5 Nov 2024 21:12:47 +0200 Subject: [PATCH 56/79] var names changed.mdx --- content/6_Advanced/Wavelet.mdx | 179 +++++++++++++++++---------------- 1 file changed, 93 insertions(+), 86 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index edac005bcc..f834e7420b 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -269,99 +269,106 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // Index, Value - - Segment(int l, int r, const Tree> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) {} - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) leftIndices.insert({index, value}); - else rightIndices.insert({index, value}); - } - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left - else return right->find_k_smallest(a, b, k - inLeft); // Appears in right - } - - void update(int index, int v, int oldV) { - update(oldV, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insertOperation) { - if (insertOperation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) left->update(v, index, insertOperation); - else right->update(v, index, insertOperation); - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // Index, Value + + Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) {return;} + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) + leftIndices.insert({index, value}); + else + rightIndices.insert({index, value}); + } + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) + return left->find_k_smallest(a,b,k); // Appears in left + else + return right->find_k_smallest(a,b, k - inLeft); // Appears in right + } + + void update(int index, int v, int oldV) { + insert_erase_operation(oldV, index, false); + insert_erase_operation(v, index, true); + } + + void insert_erase_operation(int v, int index, bool insertOperation) { + if (insertOperation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) + left->update(v, index, insertOperation); + else + right->update(v, index, insertOperation); + } }; -int main() { - int n, q; - cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } - - Segment seg(0, 1e9 + 2, indices); - vector arr(n); - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - seg.update(i, v, arr[i]); - arr[i] = v; - } +int main() { + int n, q; cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { + indices.insert({i,0}); + } + + Segment seg(0,1e9 + 2,indices); + vector arr(n); + + for (int i = 0; i < n; ++i) { + int v; cin >> v; + seg.update(i, v, arr[i]); + arr[i] =v; + } + + for (int i = 0; i < q; ++i) { + int t; cin >> t; + if (t == 1) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } else { + int a, b; cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; +} - for (int i = 0; i < q; ++i) { - int t; - cin >> t; - if (t == 1) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } else { - int a, b; - cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - return 0; -} ``` From ee1f44034288e1156eaf5a7da884061dcf11dd72 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Tue, 5 Nov 2024 19:14:24 +0000 Subject: [PATCH 57/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 179 ++++++++++++++++----------------- 1 file changed, 86 insertions(+), 93 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index f834e7420b..e0c013bbd5 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -269,106 +269,99 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // Index, Value - - Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) {return;} - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) - leftIndices.insert({index, value}); - else - rightIndices.insert({index, value}); - } - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) - return left->find_k_smallest(a,b,k); // Appears in left - else - return right->find_k_smallest(a,b, k - inLeft); // Appears in right - } - - void update(int index, int v, int oldV) { - insert_erase_operation(oldV, index, false); - insert_erase_operation(v, index, true); - } - - void insert_erase_operation(int v, int index, bool insertOperation) { - if (insertOperation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) - left->update(v, index, insertOperation); - else - right->update(v, index, insertOperation); - } -}; + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // Index, Value + + Segment(int l, int r, const Tree> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) {} + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) leftIndices.insert({index, value}); + else rightIndices.insert({index, value}); + } + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left + else return right->find_k_smallest(a, b, k - inLeft); // Appears in right + } + + void update(int index, int v, int oldV) { + insert_erase_operation(oldV, index, false); + insert_erase_operation(v, index, true); + } + + void insert_erase_operation(int v, int index, bool insertOperation) { + if (insertOperation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + if (v < mid) left->update(v, index, insertOperation); + else right->update(v, index, insertOperation); + } +}; int main() { - int n, q; cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { - indices.insert({i,0}); - } - - Segment seg(0,1e9 + 2,indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; cin >> v; - seg.update(i, v, arr[i]); - arr[i] =v; - } - - for (int i = 0; i < q; ++i) { - int t; cin >> t; - if (t == 1) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } else { - int a, b; cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; -} + int n, q; + cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } + Segment seg(0, 1e9 + 2, indices); + vector arr(n); + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + seg.update(i, v, arr[i]); + arr[i] = v; + } + + for (int i = 0; i < q; ++i) { + int t; + cin >> t; + if (t == 1) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } else { + int a, b; + cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; +} ``` From 967677ce4fb67de1ebf6cb6883d0caff5a762930 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Wed, 27 Nov 2024 22:26:11 +0200 Subject: [PATCH 58/79] Formatting errors and grammar changes .mdx --- content/6_Advanced/Wavelet.mdx | 200 +++++++++++++++++---------------- 1 file changed, 103 insertions(+), 97 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index e0c013bbd5..7c1b1d257c 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -36,17 +36,17 @@ Suppose you want to support the following queries: - Given a range, count the number of occurrences of value $x$. - Given a range, find the $k$ smallest element -With a wavelet tree, you can easily support those queries in $\mathcal{O}(\log M)$ time, +With a wavelet tree, you can easily support these queries in $\mathcal{O}(\log M)$ time, where $M$ is the maximum value in the array. -### Wavelet tree structure +### Wavelet Tree Structure To answer value-based queries efficiently, we'll create a segment tree where each node represents a range of values, instead of indices. Just like a normal segment tree, each subsequent level splits the range into two halves. Note that an index can appear in at most $\log(M)$ nodes. Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ -Each node has an array representing the indices of every number between l and r +Each node has an array representing the indices of every number between $l$ and $r$ ![Wavelet Tree Visualization](./assets/diagram.png) @@ -95,7 +95,7 @@ c[i] = \begin{cases} \end{cases} $$ $$ -\texttt{prefixB}[i] = \texttt{prefixB}[i - 1] + c[i] +\texttt{prefix_b}[i] = \texttt{prefix_b}[i - 1] + c[i] $$ @@ -105,7 +105,7 @@ To update $r$ as we recurse down, we do the following: Now let's try to solve our main problem. ## Solving the second type of query -**Given a range $l$, $r$ find the k smallest element** +**Given a range $l$, $r$ find the k-th smallest element** We will determine whether the answer for a given node is in the left or the right segment. @@ -120,13 +120,13 @@ $$ This is similar to counting how many times a value appears up to index $r$ in our previous query. We did this by using the new $r$ value at the leaf node. But now, we consider the difference between the updated $r$ and $l$ -Therefore, the occurrences of the left node is +Therefore, the occurrences of the left node are $$ \texttt{left\_occurrences} = prefix_b[r] - prefix_b[l] $$ -Note that $\texttt{left\_occurrences}$ is the number of indices between l and r whose value is less than mid +Note that $\texttt{left\_occurrences}$ is the number of indices between $l$ and $r$ whose value is less than $\texttt{mid}$ - If $\texttt{left\_occurrences}$ is greater or equal to $k$, it means the $k$-th smallest element is in the left subtree. Therefore, we update our range and recurse into the left @@ -139,11 +139,11 @@ Note that $\texttt{left\_occurrences}$ is the number of indices between l and r Notice we still update $l, r$ accordingly when we go left or right -the answer then will be the value of the node we end up on (leaf) +The answer then will be the value of the node we end up on (leaf). -In conclusion we maintain our ranges l and r as we recurse down to our child, and when we reach the child node we can return $R$ - $L$. +In conclusion we maintain our ranges l and r as we recurse down to our child, and when we reach the child node we can return $r$ - $l$. -## Implemention +## Implementation **Time Complexity:** $\mathcal{O}(Q \cdot \log(M))$ @@ -269,99 +269,105 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // Index, Value - - Segment(int l, int r, const Tree> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) {} - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) leftIndices.insert({index, value}); - else rightIndices.insert({index, value}); - } - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left - else return right->find_k_smallest(a, b, k - inLeft); // Appears in right - } - - void update(int index, int v, int oldV) { - insert_erase_operation(oldV, index, false); - insert_erase_operation(v, index, true); - } - - void insert_erase_operation(int v, int index, bool insertOperation) { - if (insertOperation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) left->update(v, index, insertOperation); - else right->update(v, index, insertOperation); - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // Index, Value + + Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) {return;} + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) + leftIndices.insert({index, value}); + else + rightIndices.insert({index, value}); + } + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) + return left->find_k_smallest(a,b,k); // Appears in left + else + return right->find_k_smallest(a,b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) + left->update(v, index, insert_operation); + else + right->update(v, index, insert_operation); + } }; -int main() { - int n, q; - cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } - - Segment seg(0, 1e9 + 2, indices); - vector arr(n); - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - seg.update(i, v, arr[i]); - arr[i] = v; - } - - for (int i = 0; i < q; ++i) { - int t; - cin >> t; - if (t == 1) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } else { - int a, b; - cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; +int main() { + int n, q; cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { + indices.insert({i,0}); + } + + Segment seg(0,1e9 + 2,indices); + vector arr(n); + + for (int i = 0; i < n; ++i) { + int v; cin >> v; + seg.update(i, v, arr[i]); + arr[i] =v; + } + + for (int i = 0; i < q; ++i) { + int t; cin >> t; + if (t == 1) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } else { + int a, b; cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; } + ``` From acb53cf781968fea735832c7889ebd3d3c2d5b48 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Wed, 27 Nov 2024 20:27:23 +0000 Subject: [PATCH 59/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 180 ++++++++++++++++----------------- 1 file changed, 87 insertions(+), 93 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 7c1b1d257c..d88664fb40 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -269,105 +269,99 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // Index, Value - - Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) {return;} - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) - leftIndices.insert({index, value}); - else - rightIndices.insert({index, value}); - } - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) - return left->find_k_smallest(a,b,k); // Appears in left - else - return right->find_k_smallest(a,b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) - left->update(v, index, insert_operation); - else - right->update(v, index, insert_operation); - } -}; + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // Index, Value + + Segment(int l, int r, const Tree> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) {} + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) leftIndices.insert({index, value}); + else rightIndices.insert({index, value}); + } + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left + else return right->find_k_smallest(a, b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) left->update(v, index, insert_operation); + else right->update(v, index, insert_operation); + } +}; int main() { - int n, q; cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { - indices.insert({i,0}); - } - - Segment seg(0,1e9 + 2,indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; cin >> v; - seg.update(i, v, arr[i]); - arr[i] =v; - } - - for (int i = 0; i < q; ++i) { - int t; cin >> t; - if (t == 1) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } else { - int a, b; cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; -} + int n, q; + cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } + + Segment seg(0, 1e9 + 2, indices); + vector arr(n); + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + seg.update(i, v, arr[i]); + arr[i] = v; + } + + for (int i = 0; i < q; ++i) { + int t; + cin >> t; + if (t == 1) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } else { + int a, b; + cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; +} ``` From 1983f249a74a2e5e0c94b18677ffd37308e873a8 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Wed, 27 Nov 2024 22:27:52 +0200 Subject: [PATCH 60/79] Update Wavelet.mdx --- content/6_Advanced/Wavelet.mdx | 182 +++++++++++++++++---------------- 1 file changed, 94 insertions(+), 88 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index d88664fb40..5fa7425234 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -95,7 +95,7 @@ c[i] = \begin{cases} \end{cases} $$ $$ -\texttt{prefix_b}[i] = \texttt{prefix_b}[i - 1] + c[i] +\texttt{prefixB}[i] = \texttt{prefixB}[i - 1] + c[i] $$ @@ -269,99 +269,105 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // Index, Value - - Segment(int l, int r, const Tree> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) {} - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) leftIndices.insert({index, value}); - else rightIndices.insert({index, value}); - } - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left - else return right->find_k_smallest(a, b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) left->update(v, index, insert_operation); - else right->update(v, index, insert_operation); - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // Index, Value + + Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) {return;} + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) + leftIndices.insert({index, value}); + else + rightIndices.insert({index, value}); + } + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) + return left->find_k_smallest(a,b,k); // Appears in left + else + return right->find_k_smallest(a,b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) + left->update(v, index, insert_operation); + else + right->update(v, index, insert_operation); + } }; -int main() { - int n, q; - cin >> n >> q; - Tree> indices; - for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } - - Segment seg(0, 1e9 + 2, indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - seg.update(i, v, arr[i]); - arr[i] = v; - } - - for (int i = 0; i < q; ++i) { - int t; - cin >> t; - if (t == 1) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } else { - int a, b; - cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; +int main() { + int n, q; cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { + indices.insert({i,0}); + } + + Segment seg(0,1e9 + 2,indices); + vector arr(n); + + for (int i = 0; i < n; ++i) { + int v; cin >> v; + seg.update(i, v, arr[i]); + arr[i] =v; + } + + for (int i = 0; i < q; ++i) { + int t; cin >> t; + if (t == 1) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } else { + int a, b; cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; } + ``` From 970230680b0df2a171f38688a8370afe3985bee1 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Wed, 27 Nov 2024 20:30:04 +0000 Subject: [PATCH 61/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 180 ++++++++++++++++----------------- 1 file changed, 87 insertions(+), 93 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 5fa7425234..66720a40fd 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -269,105 +269,99 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // Index, Value - - Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) {return;} - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) - leftIndices.insert({index, value}); - else - rightIndices.insert({index, value}); - } - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) - return left->find_k_smallest(a,b,k); // Appears in left - else - return right->find_k_smallest(a,b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) - left->update(v, index, insert_operation); - else - right->update(v, index, insert_operation); - } -}; + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // Index, Value + + Segment(int l, int r, const Tree> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) {} + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) leftIndices.insert({index, value}); + else rightIndices.insert({index, value}); + } + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left + else return right->find_k_smallest(a, b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) left->update(v, index, insert_operation); + else right->update(v, index, insert_operation); + } +}; int main() { - int n, q; cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { - indices.insert({i,0}); - } - - Segment seg(0,1e9 + 2,indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; cin >> v; - seg.update(i, v, arr[i]); - arr[i] =v; - } - - for (int i = 0; i < q; ++i) { - int t; cin >> t; - if (t == 1) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } else { - int a, b; cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; -} + int n, q; + cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } + + Segment seg(0, 1e9 + 2, indices); + vector arr(n); + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + seg.update(i, v, arr[i]); + arr[i] = v; + } + + for (int i = 0; i < q; ++i) { + int t; + cin >> t; + if (t == 1) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } else { + int a, b; + cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; +} ``` From b4c4fbf4983cb1d35249128d913e995e36be3357 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Sat, 30 Nov 2024 15:43:38 +0200 Subject: [PATCH 62/79] Latex changes + desc for updates.mdx --- content/6_Advanced/Wavelet.mdx | 213 +++++++++++++++++---------------- 1 file changed, 108 insertions(+), 105 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 66720a40fd..90d32438f0 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -83,8 +83,8 @@ $𝑟$. This can be done using a prefix sum. Let's define: -- $c[i]$ = as $1$ if $index[i]$ is smaller than mid otherwise $0$ -- $prefix_b[i]$ as prefix sum of $c[i]$ +- $c[i]$ = as $1$ if $\texttt{index}$[$i$] is smaller than mid otherwise $0$ +- $\texttt{prefix\_b}$[$i$] as prefix sum of $c[i]$ Formally @@ -95,13 +95,13 @@ c[i] = \begin{cases} \end{cases} $$ $$ -\texttt{prefixB}[i] = \texttt{prefixB}[i - 1] + c[i] +\texttt{prefix\_b}[i] = \texttt{prefix\_b}[i - 1] + c[i] $$ To update $r$ as we recurse down, we do the following: -- To know the value of $r$ if we recurse left, we use prefix_b[r] -- If we recurse right, we use $r$ - prefix_b[r] +- To know the value of $r$ if we recurse left, we use $\texttt{prefix\_b}$[$r$] +- If we recurse right, we use $r$ - $\texttt{prefix\_b}$[$r$] Now let's try to solve our main problem. ## Solving the second type of query @@ -123,7 +123,7 @@ This is similar to counting how many times a value appears up to index $r$ in ou Therefore, the occurrences of the left node are $$ -\texttt{left\_occurrences} = prefix_b[r] - prefix_b[l] +\texttt{left\_occurrences} = \texttt{prefix\_b}[r] - \texttt{prefix\_b}[l] $$ Note that $\texttt{left\_occurrences}$ is the number of indices between $l$ and $r$ whose value is less than $\texttt{mid}$ @@ -235,26 +235,23 @@ int main() { ## Supporting updates - -Let's support updates of type: - - change value at index $i$ to $y$ +Let's support updates that change the value at index $i$ to $y$. We can traverse down to the leaf to remove the old element and also traverse down to add the new element. +Let's first solve for adding a new element; erasing is similar but the opposite. + So what do the updates change? - - - Our indices vector - Our prefix vector -To support erasing/adding values to the indices vector quickly, we can choose to use a set instead of a vector. +- Our indices vector +- Our prefix vector +At each step of our recursion, the indices vector will need to be modified; we need to insert the new index. Since we can no longer maintain a sorted vector of the indices, we will switch to a set. On the other hand, to change the prefix vector, since each update could change our prefix vector a lot, we can't maintain just the normal vector. What we could do is use a sparse segment tree. - - erasing and inserting can be done by just setting the value to 0 or 1 at the specific index - - querying for a prefix can be done by querying the segment tree from 0 to $i$ + - erasing and inserting can be done by just setting the value to $0$ or $1$ at the specific index + - querying for a prefix can be done by querying the segment tree from $0$ to $i$ This approach is not memory efficient and requires a segment tree's implementation. -A more friendly approach would be using an order statistics tree. -Such that querying for a prefix would be equivalent to order_of_key($i$) - +A more friendly approach would be using an order statistics tree, which is a binary search tree implementation in C++ that allows efficient queries for the rank of elements in a set. Querying for a prefix would then be equivalent to $\texttt{order\_of\_key}$($i$). ## Implemention **Time Complexity:** $\mathcal{O}(Q \cdot \log(M) \cdot \log(N))$ @@ -269,99 +266,105 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // Index, Value - - Segment(int l, int r, const Tree> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) {} - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) leftIndices.insert({index, value}); - else rightIndices.insert({index, value}); - } - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left - else return right->find_k_smallest(a, b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) left->update(v, index, insert_operation); - else right->update(v, index, insert_operation); - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // Index, Value + + Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) {return;} + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) + leftIndices.insert({index, value}); + else + rightIndices.insert({index, value}); + } + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) + return left->find_k_smallest(a,b,k); // Appears in left + else + return right->find_k_smallest(a,b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) + left->update(v, index, insert_operation); + else + right->update(v, index, insert_operation); + } }; -int main() { - int n, q; - cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } - - Segment seg(0, 1e9 + 2, indices); - vector arr(n); - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - seg.update(i, v, arr[i]); - arr[i] = v; - } - - for (int i = 0; i < q; ++i) { - int t; - cin >> t; - if (t == 1) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } else { - int a, b; - cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; +int main() { + int n, q; cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { + indices.insert({i,0}); + } + + Segment seg(0,1e9 + 2,indices); + vector arr(n); + + for (int i = 0; i < n; ++i) { + int v; cin >> v; + seg.update(i, v, arr[i]); + arr[i] =v; + } + + for (int i = 0; i < q; ++i) { + int t; cin >> t; + if (t == 1) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } else { + int a, b; cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; } + ``` From 46be5f5c1feaf9f642e04a366bca4fc9964682ef Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Sat, 30 Nov 2024 13:44:54 +0000 Subject: [PATCH 63/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 180 ++++++++++++++++----------------- 1 file changed, 87 insertions(+), 93 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 90d32438f0..ccecfbda5c 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -266,105 +266,99 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // Index, Value - - Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) {return;} - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) - leftIndices.insert({index, value}); - else - rightIndices.insert({index, value}); - } - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) - return left->find_k_smallest(a,b,k); // Appears in left - else - return right->find_k_smallest(a,b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) - left->update(v, index, insert_operation); - else - right->update(v, index, insert_operation); - } -}; + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // Index, Value + + Segment(int l, int r, const Tree> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) {} + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) leftIndices.insert({index, value}); + else rightIndices.insert({index, value}); + } + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left + else return right->find_k_smallest(a, b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) left->update(v, index, insert_operation); + else right->update(v, index, insert_operation); + } +}; int main() { - int n, q; cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { - indices.insert({i,0}); - } - - Segment seg(0,1e9 + 2,indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; cin >> v; - seg.update(i, v, arr[i]); - arr[i] =v; - } - - for (int i = 0; i < q; ++i) { - int t; cin >> t; - if (t == 1) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } else { - int a, b; cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; -} + int n, q; + cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } + + Segment seg(0, 1e9 + 2, indices); + vector arr(n); + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + seg.update(i, v, arr[i]); + arr[i] = v; + } + + for (int i = 0; i < q; ++i) { + int t; + cin >> t; + if (t == 1) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } else { + int a, b; + cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; +} ``` From e70879beb6cd2d013c7f4e912d2a8e45f0fb966c Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Mon, 16 Dec 2024 14:08:56 +0200 Subject: [PATCH 64/79] Removed introduction .mdx --- content/6_Advanced/Wavelet.mdx | 191 ++++++++++++++++----------------- 1 file changed, 94 insertions(+), 97 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index ccecfbda5c..16764b606b 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -30,15 +30,6 @@ Wavelet trees are data structures that support efficient queries for the k-th mi -## Introduction -Suppose you want to support the following queries: - -- Given a range, count the number of occurrences of value $x$. -- Given a range, find the $k$ smallest element - -With a wavelet tree, you can easily support these queries in $\mathcal{O}(\log M)$ time, -where $M$ is the maximum value in the array. - ### Wavelet Tree Structure To answer value-based queries efficiently, we'll create a segment tree where each node represents a range of values, instead of indices. Just like a normal segment tree, each subsequent level splits the range into two halves. Note that an index can appear in at most $\log(M)$ nodes. @@ -159,7 +150,7 @@ struct Segment { Segment *left = nullptr, *right = nullptr; int l, r, mid; bool children = false; - vector> indices; // Index, Value + vector> indices; // index, value vector prefix_b; Segment(int l, int r, const vector> &indices) @@ -266,99 +257,105 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // Index, Value - - Segment(int l, int r, const Tree> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) {} - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) leftIndices.insert({index, value}); - else rightIndices.insert({index, value}); - } - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left - else return right->find_k_smallest(a, b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) left->update(v, index, insert_operation); - else right->update(v, index, insert_operation); - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // index, value + + Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) {return;} + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) + leftIndices.insert({index, value}); + else + rightIndices.insert({index, value}); + } + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) + return left->find_k_smallest(a,b,k); // Appears in left + else + return right->find_k_smallest(a,b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) + left->update(v, index, insert_operation); + else + right->update(v, index, insert_operation); + } }; -int main() { - int n, q; - cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } - - Segment seg(0, 1e9 + 2, indices); - vector arr(n); - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - seg.update(i, v, arr[i]); - arr[i] = v; - } - - for (int i = 0; i < q; ++i) { - int t; - cin >> t; - if (t == 1) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } else { - int a, b; - cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; +int main() { + int n, q; cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { + indices.insert({i,0}); + } + + Segment seg(0,1e9 + 2,indices); + vector arr(n); + + for (int i = 0; i < n; ++i) { + int v; cin >> v; + seg.update(i, v, arr[i]); + arr[i] =v; + } + + for (int i = 0; i < q; ++i) { + int t; cin >> t; + if (t == 1) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } else { + int a, b; cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; } + ``` From da9947e68c4748ec92059ed2650206fea1e9594c Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Mon, 16 Dec 2024 12:10:15 +0000 Subject: [PATCH 65/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 180 ++++++++++++++++----------------- 1 file changed, 87 insertions(+), 93 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 16764b606b..e578e6c1eb 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -257,105 +257,99 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // index, value - - Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) {return;} - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) - leftIndices.insert({index, value}); - else - rightIndices.insert({index, value}); - } - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) - return left->find_k_smallest(a,b,k); // Appears in left - else - return right->find_k_smallest(a,b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) - left->update(v, index, insert_operation); - else - right->update(v, index, insert_operation); - } -}; + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // index, value + + Segment(int l, int r, const Tree> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) {} + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) leftIndices.insert({index, value}); + else rightIndices.insert({index, value}); + } + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left + else return right->find_k_smallest(a, b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) left->update(v, index, insert_operation); + else right->update(v, index, insert_operation); + } +}; int main() { - int n, q; cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { - indices.insert({i,0}); - } - - Segment seg(0,1e9 + 2,indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; cin >> v; - seg.update(i, v, arr[i]); - arr[i] =v; - } - - for (int i = 0; i < q; ++i) { - int t; cin >> t; - if (t == 1) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } else { - int a, b; cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; -} + int n, q; + cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } + + Segment seg(0, 1e9 + 2, indices); + vector arr(n); + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + seg.update(i, v, arr[i]); + arr[i] = v; + } + + for (int i = 0; i < q; ++i) { + int t; + cin >> t; + if (t == 1) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } else { + int a, b; + cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; +} ``` From e9fcb707d56c1826aff1fac33fa673c57ef83675 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Mon, 16 Dec 2024 15:53:54 +0200 Subject: [PATCH 66/79] Changed headers.mdx --- content/6_Advanced/Wavelet.mdx | 186 +++++++++++++++++---------------- 1 file changed, 95 insertions(+), 91 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index e578e6c1eb..ff9b788fde 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -45,8 +45,7 @@ Each node has an array representing the indices of every number between $l$ and ### Solution - Range K-th Smallest Before we solve this problem, let's consider a simpler version where we are asked, given a range, to count the number of occurrences of value $x$. -## Solving the first type of query -**Given a range $l$, $r$, count the number of occurrences of value x.** +## Given a range $l$, $r$, count the number of occurrences of value x. To calculate the number of occurrences from $𝑙$ to $𝑟$, we can use the following formula: @@ -95,8 +94,7 @@ To update $r$ as we recurse down, we do the following: - If we recurse right, we use $r$ - $\texttt{prefix\_b}$[$r$] Now let's try to solve our main problem. -## Solving the second type of query -**Given a range $l$, $r$ find the k-th smallest element** +## Given a range $l$, $r$ find the k-th smallest element We will determine whether the answer for a given node is in the left or the right segment. @@ -257,99 +255,105 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // index, value - - Segment(int l, int r, const Tree> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) {} - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) leftIndices.insert({index, value}); - else rightIndices.insert({index, value}); - } - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left - else return right->find_k_smallest(a, b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) left->update(v, index, insert_operation); - else right->update(v, index, insert_operation); - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // index, value + + Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) {return;} + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) + leftIndices.insert({index, value}); + else + rightIndices.insert({index, value}); + } + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) + return left->find_k_smallest(a,b,k); // Appears in left + else + return right->find_k_smallest(a,b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) + left->update(v, index, insert_operation); + else + right->update(v, index, insert_operation); + } }; -int main() { - int n, q; - cin >> n >> q; - Tree> indices; - for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } - - Segment seg(0, 1e9 + 2, indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - seg.update(i, v, arr[i]); - arr[i] = v; - } - - for (int i = 0; i < q; ++i) { - int t; - cin >> t; - if (t == 1) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } else { - int a, b; - cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; +int main() { + int n, q; cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { + indices.insert({i,0}); + } + + Segment seg(0,1e9 + 2,indices); + vector arr(n); + + for (int i = 0; i < n; ++i) { + int v; cin >> v; + seg.update(i, v, arr[i]); + arr[i] =v; + } + + for (int i = 0; i < q; ++i) { + int t; cin >> t; + if (t == 1) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } else { + int a, b; cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; } + ``` From 3f8622277d0aac0184d74c04621e6e856601c45e Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Mon, 16 Dec 2024 13:55:08 +0000 Subject: [PATCH 67/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 180 ++++++++++++++++----------------- 1 file changed, 87 insertions(+), 93 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index ff9b788fde..f920cb3939 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -255,105 +255,99 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // index, value - - Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) {return;} - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) - leftIndices.insert({index, value}); - else - rightIndices.insert({index, value}); - } - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) - return left->find_k_smallest(a,b,k); // Appears in left - else - return right->find_k_smallest(a,b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) - left->update(v, index, insert_operation); - else - right->update(v, index, insert_operation); - } -}; + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // index, value + + Segment(int l, int r, const Tree> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) {} + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) leftIndices.insert({index, value}); + else rightIndices.insert({index, value}); + } + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left + else return right->find_k_smallest(a, b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) left->update(v, index, insert_operation); + else right->update(v, index, insert_operation); + } +}; int main() { - int n, q; cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { - indices.insert({i,0}); - } - - Segment seg(0,1e9 + 2,indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; cin >> v; - seg.update(i, v, arr[i]); - arr[i] =v; - } - - for (int i = 0; i < q; ++i) { - int t; cin >> t; - if (t == 1) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } else { - int a, b; cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; -} + int n, q; + cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } + + Segment seg(0, 1e9 + 2, indices); + vector arr(n); + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + seg.update(i, v, arr[i]); + arr[i] = v; + } + + for (int i = 0; i < q; ++i) { + int t; + cin >> t; + if (t == 1) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } else { + int a, b; + cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; +} ``` From 72f93e7850c8fae06c5606af55f5c94db913b1c7 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Mon, 16 Dec 2024 16:32:00 +0200 Subject: [PATCH 68/79] Removed spoiler.mdx --- content/6_Advanced/Wavelet.mdx | 183 +++++++++++++++++---------------- 1 file changed, 94 insertions(+), 89 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index f920cb3939..dea4ab1fdb 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -34,13 +34,12 @@ Wavelet trees are data structures that support efficient queries for the k-th mi To answer value-based queries efficiently, we'll create a segment tree where each node represents a range of values, instead of indices. Just like a normal segment tree, each subsequent level splits the range into two halves. Note that an index can appear in at most $\log(M)$ nodes. - +#### Wavelet Tree Visualization Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ Each node has an array representing the indices of every number between $l$ and $r$ ![Wavelet Tree Visualization](./assets/diagram.png) - ### Solution - Range K-th Smallest Before we solve this problem, let's consider a simpler version where we are asked, given a range, to count the number of occurrences of value $x$. @@ -255,99 +254,105 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // index, value - - Segment(int l, int r, const Tree> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) {} - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) leftIndices.insert({index, value}); - else rightIndices.insert({index, value}); - } - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left - else return right->find_k_smallest(a, b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) left->update(v, index, insert_operation); - else right->update(v, index, insert_operation); - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // index, value + + Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) {return;} + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) + leftIndices.insert({index, value}); + else + rightIndices.insert({index, value}); + } + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) + return left->find_k_smallest(a,b,k); // Appears in left + else + return right->find_k_smallest(a,b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) + left->update(v, index, insert_operation); + else + right->update(v, index, insert_operation); + } }; -int main() { - int n, q; - cin >> n >> q; - Tree> indices; - for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } - - Segment seg(0, 1e9 + 2, indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - seg.update(i, v, arr[i]); - arr[i] = v; - } - - for (int i = 0; i < q; ++i) { - int t; - cin >> t; - if (t == 1) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } else { - int a, b; - cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; +int main() { + int n, q; cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { + indices.insert({i,0}); + } + + Segment seg(0,1e9 + 2,indices); + vector arr(n); + + for (int i = 0; i < n; ++i) { + int v; cin >> v; + seg.update(i, v, arr[i]); + arr[i] =v; + } + + for (int i = 0; i < q; ++i) { + int t; cin >> t; + if (t == 1) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } else { + int a, b; cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; } + ``` From 07c3afc057fcb046c9e092b1193e61785ff6d54d Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Mon, 16 Dec 2024 14:33:21 +0000 Subject: [PATCH 69/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 180 ++++++++++++++++----------------- 1 file changed, 87 insertions(+), 93 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index dea4ab1fdb..077d9417ce 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -254,105 +254,99 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // index, value - - Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) {return;} - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) - leftIndices.insert({index, value}); - else - rightIndices.insert({index, value}); - } - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) - return left->find_k_smallest(a,b,k); // Appears in left - else - return right->find_k_smallest(a,b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) - left->update(v, index, insert_operation); - else - right->update(v, index, insert_operation); - } -}; + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // index, value + + Segment(int l, int r, const Tree> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) {} + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) leftIndices.insert({index, value}); + else rightIndices.insert({index, value}); + } + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left + else return right->find_k_smallest(a, b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) left->update(v, index, insert_operation); + else right->update(v, index, insert_operation); + } +}; int main() { - int n, q; cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { - indices.insert({i,0}); - } - - Segment seg(0,1e9 + 2,indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; cin >> v; - seg.update(i, v, arr[i]); - arr[i] =v; - } - - for (int i = 0; i < q; ++i) { - int t; cin >> t; - if (t == 1) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } else { - int a, b; cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; -} + int n, q; + cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } + + Segment seg(0, 1e9 + 2, indices); + vector arr(n); + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + seg.update(i, v, arr[i]); + arr[i] = v; + } + + for (int i = 0; i < q; ++i) { + int t; + cin >> t; + if (t == 1) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } else { + int a, b; + cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; +} ``` From c2e62965abf1cbadbd6e3dd6a132f738caeb23de Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Mon, 16 Dec 2024 16:37:38 +0200 Subject: [PATCH 70/79] Changed # levels.mdx --- content/6_Advanced/Wavelet.mdx | 182 +++++++++++++++++---------------- 1 file changed, 94 insertions(+), 88 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 077d9417ce..ce0f402097 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -30,7 +30,7 @@ Wavelet trees are data structures that support efficient queries for the k-th mi -### Wavelet Tree Structure +## Wavelet Tree Structure To answer value-based queries efficiently, we'll create a segment tree where each node represents a range of values, instead of indices. Just like a normal segment tree, each subsequent level splits the range into two halves. Note that an index can appear in at most $\log(M)$ nodes. @@ -254,99 +254,105 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // index, value - - Segment(int l, int r, const Tree> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) {} - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) leftIndices.insert({index, value}); - else rightIndices.insert({index, value}); - } - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left - else return right->find_k_smallest(a, b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) left->update(v, index, insert_operation); - else right->update(v, index, insert_operation); - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // index, value + + Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) {return;} + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) + leftIndices.insert({index, value}); + else + rightIndices.insert({index, value}); + } + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) + return left->find_k_smallest(a,b,k); // Appears in left + else + return right->find_k_smallest(a,b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) + left->update(v, index, insert_operation); + else + right->update(v, index, insert_operation); + } }; -int main() { - int n, q; - cin >> n >> q; - Tree> indices; - for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } - - Segment seg(0, 1e9 + 2, indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - seg.update(i, v, arr[i]); - arr[i] = v; - } - - for (int i = 0; i < q; ++i) { - int t; - cin >> t; - if (t == 1) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } else { - int a, b; - cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; +int main() { + int n, q; cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { + indices.insert({i,0}); + } + + Segment seg(0,1e9 + 2,indices); + vector arr(n); + + for (int i = 0; i < n; ++i) { + int v; cin >> v; + seg.update(i, v, arr[i]); + arr[i] =v; + } + + for (int i = 0; i < q; ++i) { + int t; cin >> t; + if (t == 1) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } else { + int a, b; cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; } + ``` From 657e3adc3ddb3a608c84a04d30a7fb7241e5d0ec Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Mon, 16 Dec 2024 14:38:52 +0000 Subject: [PATCH 71/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 180 ++++++++++++++++----------------- 1 file changed, 87 insertions(+), 93 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index ce0f402097..8ed75acf5c 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -254,105 +254,99 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // index, value - - Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) {return;} - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) - leftIndices.insert({index, value}); - else - rightIndices.insert({index, value}); - } - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) - return left->find_k_smallest(a,b,k); // Appears in left - else - return right->find_k_smallest(a,b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) - left->update(v, index, insert_operation); - else - right->update(v, index, insert_operation); - } -}; + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // index, value + + Segment(int l, int r, const Tree> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) {} + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) leftIndices.insert({index, value}); + else rightIndices.insert({index, value}); + } + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left + else return right->find_k_smallest(a, b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) left->update(v, index, insert_operation); + else right->update(v, index, insert_operation); + } +}; int main() { - int n, q; cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { - indices.insert({i,0}); - } - - Segment seg(0,1e9 + 2,indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; cin >> v; - seg.update(i, v, arr[i]); - arr[i] =v; - } - - for (int i = 0; i < q; ++i) { - int t; cin >> t; - if (t == 1) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } else { - int a, b; cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; -} + int n, q; + cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } + + Segment seg(0, 1e9 + 2, indices); + vector arr(n); + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + seg.update(i, v, arr[i]); + arr[i] = v; + } + + for (int i = 0; i < q; ++i) { + int t; + cin >> t; + if (t == 1) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } else { + int a, b; + cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; +} ``` From 12f843341e6808f1a55cbba8fe81b9ce60d1a716 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Tue, 7 Jan 2025 19:42:29 +0200 Subject: [PATCH 72/79] Rename diagram.png to wavelet_tree_diagram.png --- content/6_Advanced/assets/diagram.png | Bin 2626401 -> 0 bytes .../assets/wavelet_tree_diagram.png | 1 + 2 files changed, 1 insertion(+) delete mode 100644 content/6_Advanced/assets/diagram.png create mode 100644 content/6_Advanced/assets/wavelet_tree_diagram.png diff --git a/content/6_Advanced/assets/diagram.png b/content/6_Advanced/assets/diagram.png deleted file mode 100644 index 50038d26469e5df8a947984ba6c51468fe0a68a8..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 2626401 zcmeFZbywTX_CB28?!}z~Ezsibg#se*X#HOaXzbi`}-XP;$I5@=y6cEJ*yD?D9_z=MOs=Ij?J?I3jZPGU)1~nJ2n^p 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zNOWB45GA=XcX=4btG^YLxMz_f_*ps&R5q^^)zQSt0;?bwsM6sCi{6W2c18>+v4x*_ zIjR(p{LmJ`j~%8JTV8Mn94oh1e{M_JdBa-OQ>{XPEDkwm2Wm~01z=Wm{6jYQ*J9Ca z76qMy&vlB5E&j#+JVbH3q@Ve+fFX(e0mo8D&Pq`cg$4Qk Date: Tue, 7 Jan 2025 19:48:00 +0200 Subject: [PATCH 73/79] Explained about occurrences(x).mdx --- content/6_Advanced/Wavelet.mdx | 185 +++++++++++++++++---------------- 1 file changed, 98 insertions(+), 87 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 8ed75acf5c..5c6c1d4052 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -54,6 +54,11 @@ $$ \texttt{occurrences}(l, r) = \texttt{occurrences}(r) - \texttt{occurrences}(l) \end{aligned} $$ +When occurrences(y) is the number of occurrences of x till y +aka +$$ +\text{occurrences}(y) = \sum_{i=1}^{y} \mathbb{1}(x_i = x) +$$ This reduces the problem to counting the number of occurrences in a prefix. @@ -254,99 +259,105 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // index, value - - Segment(int l, int r, const Tree> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) {} - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) leftIndices.insert({index, value}); - else rightIndices.insert({index, value}); - } - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left - else return right->find_k_smallest(a, b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) left->update(v, index, insert_operation); - else right->update(v, index, insert_operation); - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // index, value + + Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) {return;} + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) + leftIndices.insert({index, value}); + else + rightIndices.insert({index, value}); + } + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) + return left->find_k_smallest(a,b,k); // Appears in left + else + return right->find_k_smallest(a,b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) + left->update(v, index, insert_operation); + else + right->update(v, index, insert_operation); + } }; -int main() { - int n, q; - cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } - - Segment seg(0, 1e9 + 2, indices); - vector arr(n); - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - seg.update(i, v, arr[i]); - arr[i] = v; - } - - for (int i = 0; i < q; ++i) { - int t; - cin >> t; - if (t == 1) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } else { - int a, b; - cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; +int main() { + int n, q; cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { + indices.insert({i,0}); + } + + Segment seg(0,1e9 + 2,indices); + vector arr(n); + + for (int i = 0; i < n; ++i) { + int v; cin >> v; + seg.update(i, v, arr[i]); + arr[i] =v; + } + + for (int i = 0; i < q; ++i) { + int t; cin >> t; + if (t == 1) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } else { + int a, b; cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; } + ``` From eae9a5c7dcf669e5541678a5f871db0ecd32eda6 Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Tue, 7 Jan 2025 17:49:12 +0000 Subject: [PATCH 74/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 182 ++++++++++++++++----------------- 1 file changed, 88 insertions(+), 94 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 5c6c1d4052..b5a7e2cff6 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -54,7 +54,7 @@ $$ \texttt{occurrences}(l, r) = \texttt{occurrences}(r) - \texttt{occurrences}(l) \end{aligned} $$ -When occurrences(y) is the number of occurrences of x till y +When occurrences(y) is the number of occurrences of x till y aka $$ \text{occurrences}(y) = \sum_{i=1}^{y} \mathbb{1}(x_i = x) @@ -259,105 +259,99 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // index, value - - Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) {return;} - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) - leftIndices.insert({index, value}); - else - rightIndices.insert({index, value}); - } - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) - return left->find_k_smallest(a,b,k); // Appears in left - else - return right->find_k_smallest(a,b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) - left->update(v, index, insert_operation); - else - right->update(v, index, insert_operation); - } -}; + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // index, value + + Segment(int l, int r, const Tree> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) {} + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) leftIndices.insert({index, value}); + else rightIndices.insert({index, value}); + } + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left + else return right->find_k_smallest(a, b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) left->update(v, index, insert_operation); + else right->update(v, index, insert_operation); + } +}; int main() { - int n, q; cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { - indices.insert({i,0}); - } - - Segment seg(0,1e9 + 2,indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; cin >> v; - seg.update(i, v, arr[i]); - arr[i] =v; - } - - for (int i = 0; i < q; ++i) { - int t; cin >> t; - if (t == 1) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } else { - int a, b; cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; -} + int n, q; + cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } + + Segment seg(0, 1e9 + 2, indices); + vector arr(n); + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + seg.update(i, v, arr[i]); + arr[i] = v; + } + + for (int i = 0; i < q; ++i) { + int t; + cin >> t; + if (t == 1) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } else { + int a, b; + cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; +} ``` From d640ae1dbdf49dcfbf501038bdabfd6a139e8385 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Tue, 7 Jan 2025 20:24:32 +0200 Subject: [PATCH 75/79] fixed latex.mdx --- content/6_Advanced/Wavelet.mdx | 186 +++++++++++++++++---------------- 1 file changed, 96 insertions(+), 90 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index b5a7e2cff6..248b0ea3db 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -39,7 +39,7 @@ To answer value-based queries efficiently, we'll create a segment tree where eac Let's say our array is: $[3,5,3,1,2,2,3,4,5,5]$ Each node has an array representing the indices of every number between $l$ and $r$ -![Wavelet Tree Visualization](./assets/diagram.png) +![Wavelet Tree Visualization](./assets/wavelet_tree_diagram.png) ### Solution - Range K-th Smallest Before we solve this problem, let's consider a simpler version where we are asked, given a range, to count the number of occurrences of value $x$. @@ -54,10 +54,10 @@ $$ \texttt{occurrences}(l, r) = \texttt{occurrences}(r) - \texttt{occurrences}(l) \end{aligned} $$ -When occurrences(y) is the number of occurrences of x till y +When $\texttt{occurrences}(y)$ is the number of occurences of x in the first y elements. aka $$ -\text{occurrences}(y) = \sum_{i=1}^{y} \mathbb{1}(x_i = x) +\text{occurrences} = \sum_{i=1}^{y} \mathbb{1}(x_i = x) $$ This reduces the problem to counting the number of occurrences in a prefix. @@ -259,99 +259,105 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // index, value - - Segment(int l, int r, const Tree> &indices) - : l(l), r(r), mid((r + l) / 2), indices(indices) {} - - // Sparse since values can go up to 1e9 - void update() { - if (children) { return; } - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) leftIndices.insert({index, value}); - else rightIndices.insert({index, value}); - } - - left = new Segment(l, mid, leftIndices); - right = new Segment(mid, r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left - else return right->find_k_smallest(a, b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) left->update(v, index, insert_operation); - else right->update(v, index, insert_operation); - } + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // index, value + + Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { + + } + + // Sparse since values can go up to 1e9 + void update() { + if (children) {return;} + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) + leftIndices.insert({index, value}); + else + rightIndices.insert({index, value}); + } + + left = new Segment(l,mid, leftIndices); + right = new Segment(mid,r, rightIndices); + } + } + + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) + return left->find_k_smallest(a,b,k); // Appears in left + else + return right->find_k_smallest(a,b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) + left->update(v, index, insert_operation); + else + right->update(v, index, insert_operation); + } }; -int main() { - int n, q; - cin >> n >> q; - Tree> indices; - for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } - - Segment seg(0, 1e9 + 2, indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; - cin >> v; - seg.update(i, v, arr[i]); - arr[i] = v; - } - - for (int i = 0; i < q; ++i) { - int t; - cin >> t; - if (t == 1) { - int a, b, k; - cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a, b, k) << " "; - } else { - int a, b; - cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; +int main() { + int n, q; cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { + indices.insert({i,0}); + } + + Segment seg(0,1e9 + 2,indices); + vector arr(n); + + for (int i = 0; i < n; ++i) { + int v; cin >> v; + seg.update(i, v, arr[i]); + arr[i] =v; + } + + for (int i = 0; i < q; ++i) { + int t; cin >> t; + if (t == 1) { + int a, b, k; cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a,b,k) << " "; + } else { + int a, b; cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; } + ``` From d73dacd85e30b52f483b4e756ee5e77d30c748bd Mon Sep 17 00:00:00 2001 From: "pre-commit-ci[bot]" <66853113+pre-commit-ci[bot]@users.noreply.github.com> Date: Tue, 7 Jan 2025 18:26:05 +0000 Subject: [PATCH 76/79] [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --- content/6_Advanced/Wavelet.mdx | 180 ++++++++++++++++----------------- 1 file changed, 87 insertions(+), 93 deletions(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index 248b0ea3db..eb35e11fc7 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -259,105 +259,99 @@ using namespace std; using namespace __gnu_pbds; template using Tree = - tree, rb_tree_tag, tree_order_statistics_node_update>; + tree, rb_tree_tag, tree_order_statistics_node_update>; struct Segment { - Segment *left = nullptr, *right = nullptr; - int l, r, mid; - bool children = false; - Tree> indices; // index, value - - Segment(int l, int r,const Tree> &indices) :l(l), r(r), mid((r + l) / 2), indices(indices) { - - } - - // Sparse since values can go up to 1e9 - void update() { - if (children) {return;} - children = true; - - if (r - l > 1) { - // Split the indices for left and right child - Tree> leftIndices, rightIndices; - for (auto [index, value] : indices) { - if (value < mid) - leftIndices.insert({index, value}); - else - rightIndices.insert({index, value}); - } - - left = new Segment(l,mid, leftIndices); - right = new Segment(mid,r, rightIndices); - } - } - - int find_k_smallest(int a, int b, int k) { - update(); - if (r - l <= 1) return l; - - int lb = left->indices.order_of_key({a, -1}); - int lr = left->indices.order_of_key({b, -1}); - int inLeft = lr - lb; - - if (k <= inLeft) - return left->find_k_smallest(a,b,k); // Appears in left - else - return right->find_k_smallest(a,b, k - inLeft); // Appears in right - } - - void update(int index, int v, int old_v) { - update(old_v, index, false); - update(v, index, true); - } - - void update(int v, int index, bool insert_operation) { - if (insert_operation) indices.insert({index, v}); - else indices.erase({index, v}); - - update(); - if (r - l <= 1) return; - - if (v < mid) - left->update(v, index, insert_operation); - else - right->update(v, index, insert_operation); - } -}; + Segment *left = nullptr, *right = nullptr; + int l, r, mid; + bool children = false; + Tree> indices; // index, value + + Segment(int l, int r, const Tree> &indices) + : l(l), r(r), mid((r + l) / 2), indices(indices) {} + + // Sparse since values can go up to 1e9 + void update() { + if (children) { return; } + children = true; + + if (r - l > 1) { + // Split the indices for left and right child + Tree> leftIndices, rightIndices; + for (auto [index, value] : indices) { + if (value < mid) leftIndices.insert({index, value}); + else rightIndices.insert({index, value}); + } + + left = new Segment(l, mid, leftIndices); + right = new Segment(mid, r, rightIndices); + } + } + int find_k_smallest(int a, int b, int k) { + update(); + if (r - l <= 1) return l; + + int lb = left->indices.order_of_key({a, -1}); + int lr = left->indices.order_of_key({b, -1}); + int inLeft = lr - lb; + + if (k <= inLeft) return left->find_k_smallest(a, b, k); // Appears in left + else return right->find_k_smallest(a, b, k - inLeft); // Appears in right + } + + void update(int index, int v, int old_v) { + update(old_v, index, false); + update(v, index, true); + } + + void update(int v, int index, bool insert_operation) { + if (insert_operation) indices.insert({index, v}); + else indices.erase({index, v}); + + update(); + if (r - l <= 1) return; + + if (v < mid) left->update(v, index, insert_operation); + else right->update(v, index, insert_operation); + } +}; int main() { - int n, q; cin >> n >> q; - - Tree> indices; - for (int i = 0; i < n; ++i) { - indices.insert({i,0}); - } - - Segment seg(0,1e9 + 2,indices); - vector arr(n); - - for (int i = 0; i < n; ++i) { - int v; cin >> v; - seg.update(i, v, arr[i]); - arr[i] =v; - } - - for (int i = 0; i < q; ++i) { - int t; cin >> t; - if (t == 1) { - int a, b, k; cin >> a >> b >> k; - k++; - cout << seg.find_k_smallest(a,b,k) << " "; - } else { - int a, b; cin >> a >> b; - seg.update(a, b, arr[i]); - arr[i] = b; - } - } - - return 0; -} + int n, q; + cin >> n >> q; + + Tree> indices; + for (int i = 0; i < n; ++i) { indices.insert({i, 0}); } + + Segment seg(0, 1e9 + 2, indices); + vector arr(n); + for (int i = 0; i < n; ++i) { + int v; + cin >> v; + seg.update(i, v, arr[i]); + arr[i] = v; + } + + for (int i = 0; i < q; ++i) { + int t; + cin >> t; + if (t == 1) { + int a, b, k; + cin >> a >> b >> k; + k++; + cout << seg.find_k_smallest(a, b, k) << " "; + } else { + int a, b; + cin >> a >> b; + seg.update(a, b, arr[i]); + arr[i] = b; + } + } + + return 0; +} ``` From af0d04fc61b9278aecb14f8cf715fa091429b5a7 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Tue, 7 Jan 2025 20:40:24 +0200 Subject: [PATCH 77/79] Delete content/6_Advanced/assets/wavelet_tree_diagram.png --- content/6_Advanced/assets/wavelet_tree_diagram.png | 1 - 1 file changed, 1 deletion(-) delete mode 100644 content/6_Advanced/assets/wavelet_tree_diagram.png diff --git a/content/6_Advanced/assets/wavelet_tree_diagram.png b/content/6_Advanced/assets/wavelet_tree_diagram.png deleted file mode 100644 index d3f5a12faa..0000000000 --- a/content/6_Advanced/assets/wavelet_tree_diagram.png +++ /dev/null @@ -1 +0,0 @@ - From ae1eb1513d1738f01e9c8008f458bd62f130c6d6 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Mon, 20 Jan 2025 20:17:46 +0200 Subject: [PATCH 78/79] Update content/6_Advanced/Wavelet.mdx Latex fix Co-authored-by: Justin Ji <68484800+TheGamingMousse@users.noreply.github.com> --- content/6_Advanced/Wavelet.mdx | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/6_Advanced/Wavelet.mdx b/content/6_Advanced/Wavelet.mdx index eb35e11fc7..d3e926bf49 100644 --- a/content/6_Advanced/Wavelet.mdx +++ b/content/6_Advanced/Wavelet.mdx @@ -134,7 +134,7 @@ Note that $\texttt{left\_occurrences}$ is the number of indices between $l$ and The answer then will be the value of the node we end up on (leaf). -In conclusion we maintain our ranges l and r as we recurse down to our child, and when we reach the child node we can return $r$ - $l$. +In conclusion we maintain our ranges $l$ and $r$ as we recurse down to our child, and when we reach the child node we can return $r - l$. ## Implementation **Time Complexity:** $\mathcal{O}(Q \cdot \log(M))$ From 8e3858b33bd36df0d94eea9a574f9ce94ea173d7 Mon Sep 17 00:00:00 2001 From: crafticat <68462458+crafticat@users.noreply.github.com> Date: Mon, 20 Jan 2025 20:20:48 +0200 Subject: [PATCH 79/79] added image back --- .../assets/wavelet_tree_diagram.png | Bin 0 -> 2626401 bytes 1 file changed, 0 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