From 668706f8ba7d1d304afe0050639ce52849ade516 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Jo=C3=A3o=20Oliveira?= Date: Tue, 20 Feb 2024 01:07:35 -0300 Subject: [PATCH 1/5] segtrees melhoradas --- .../Operation-Queue/README.md | 2 +- .../Operation-Stack/README.md | 2 +- .../Estruturas-de-Dados/Ordered-Set/README.md | 20 +-- .../Segment-Tree/Segment-Tree-2D/README.md | 9 +- .../README.md | 11 +- .../Segment-Tree-Beats-Max-Update/README.md | 9 +- .../Segment-Tree-Esparsa/README.md | 10 +- .../Segment-Tree-Esparsa/seg_tree_sparse.cpp | 83 ++++++----- .../Segment-Tree-Kadani/README.md | 9 +- .../Segment-Tree-Kadani/seg_tree_kadani.cpp | 104 ++++++++------ .../Segment-Tree/Segment-Tree-Lazy/README.md | 9 +- .../Segment-Tree-Lazy/seg_tree_lazy.cpp | 131 ++++++++++++------ .../Segment-Tree/Segment-Tree/README.md | 8 +- .../Segment-Tree/Segment-Tree/seg_tree.cpp | 93 ++++++++----- .../Sparse-Table/Sparse-Table/README.md | 2 - .../Sparse-Table/sparse_table.cpp | 2 +- Codigos/Grafos/2-SAT/README.md | 2 +- PDF/Almanaque.pdf | Bin 737952 -> 742229 bytes 18 files changed, 282 insertions(+), 224 deletions(-) diff --git a/Codigos/Estruturas-de-Dados/Operation-Queue/README.md b/Codigos/Estruturas-de-Dados/Operation-Queue/README.md index 01260651..e03f83a1 100644 --- a/Codigos/Estruturas-de-Dados/Operation-Queue/README.md +++ b/Codigos/Estruturas-de-Dados/Operation-Queue/README.md @@ -1,5 +1,5 @@ # [Operation Queue](op_queue.cpp) -Fila que armazena o resultado do operatório dos itens (ou seja, dado uma fila, responde qual é o elemento mínimo, por exemplo). É uma extensão da $std::queue$, permitindo todos os métodos já presentes nela, com a diferença de que $push$ e $pop$ agora são $add$ e $remove$, respectivamente, ambos continuam $\mathcal{O}(1)$ amortizado. A fila agora também permite a operação $get$ que retorna o resultado do operatório dos itens da fila em $\mathcal{O}(1)$ amortizado. Chamar o método $get$ em uma fila vazia é indefinido. +Fila que armazena o resultado do operatório dos itens (ou seja, dado uma fila, responde qual é o elemento mínimo, por exemplo). É uma extensão da `std::queue`, permitindo todos os métodos já presentes nela, com a diferença de que `push` e `pop` agora são `add` e `remove`, respectivamente, ambos continuam $\mathcal{O}(1)$ amortizado. A fila agora também permite a operação `get` que retorna o resultado do operatório dos itens da fila em $\mathcal{O}(1)$ amortizado. Chamar o método `get` em uma fila vazia é indefinido. **Obs**: usa a estrutura Operation Stack. diff --git a/Codigos/Estruturas-de-Dados/Operation-Stack/README.md b/Codigos/Estruturas-de-Dados/Operation-Stack/README.md index a425fb87..dff7fdde 100644 --- a/Codigos/Estruturas-de-Dados/Operation-Stack/README.md +++ b/Codigos/Estruturas-de-Dados/Operation-Stack/README.md @@ -1,3 +1,3 @@ # [Operation Stack](op_stack.cpp) -Pilha que armazena o resultado do operatório dos itens (ou seja, dado uma pilha, responde qual é o elemento mínimo, por exemplo). É uma extensão da $std::stack$, permitindo todos os métodos já presentes nela, com a diferença de que $push$ e $pop$ agora são $add$ e $remove$, respectivamente, ambos continuam $\mathcal{O}(1)$ amortizado. A pilha agora também permite a operação $get$ que retorna o resultado do operatório dos itens da pilha em $\mathcal{O}(1)$ amortizado. Chamar o método $get$ em uma pilha vazia é indefinido. +Pilha que armazena o resultado do operatório dos itens (ou seja, dado uma pilha, responde qual é o elemento mínimo, por exemplo). É uma extensão da `std::stack`, permitindo todos os métodos já presentes nela, com a diferença de que `push` e `pop` agora são `add` e `remove`, respectivamente, ambos continuam $\mathcal{O}(1)$ amortizado. A pilha agora também permite a operação `get` que retorna o resultado do operatório dos itens da pilha em $\mathcal{O}(1)$ amortizado. Chamar o método `get` em uma pilha vazia é indefinido. diff --git a/Codigos/Estruturas-de-Dados/Ordered-Set/README.md b/Codigos/Estruturas-de-Dados/Ordered-Set/README.md index b695367b..475b227b 100644 --- a/Codigos/Estruturas-de-Dados/Ordered-Set/README.md +++ b/Codigos/Estruturas-de-Dados/Ordered-Set/README.md @@ -2,24 +2,24 @@ Set com operações de busca por ordem e índice. -Pode ser usado como um $std::set$ normal, a principal diferença são duas novas operações possíveis: +Pode ser usado como um `std::set` normal, a principal diferença são duas novas operações possíveis: -- $find\_by\_order(k)$: retorna um iterador para o k-ésimo menor elemento no set (indexado em 0). -- $order\_of\_key(k)$: retorna o número de elementos menores que k. (ou seja, o índice de k no set) +- `find_by_order(k)`: retorna um iterador para o $k$-ésimo menor elemento no set (indexado em 0). +- `order_of_key(k)`: retorna o número de elementos menores que $k$. (ou seja, o índice de $k$ no set) Ambas as operações são $\mathcal{O}(log(n))$. -Também é possível criar um $ordered\_map$, funciona como um $std::map$, mas com as operações de busca por ordem e índice. $find\_by\_order(k)$ retorna um iterador para a k-ésima menor **key** no mapa (indexado em 0). $order\_of\_key(k)$ retorna o número de **keys** no mapa menores que k. (ou seja, o índice de k no map). +Também é possível criar um `ordered_map`, funciona como um `std::map`, mas com as operações de busca por ordem e índice. `find_by_order(k)` retorna um iterador para a $k$-ésima menor **key** no mapa (indexado em 0). `order_of_key(k)` retorna o número de **keys** no mapa menores que $k$. (ou seja, o índice de $k$ no map). -Para simular um $std::multiset$, há várias formas: +Para simular um `std::multiset`, há várias formas: -- Usar um $std::pair$ como elemento do set, com o primeiro elemento sendo o valor e o segundo sendo um identificador único para cada elemento. Para saber o número de elementos menores que $k$ no multiset, basta usar $order\_of\_key({k, -INF})$. +- Usar um `std::pair` como elemento do set, com o primeiro elemento sendo o valor e o segundo sendo um identificador único para cada elemento. Para saber o número de elementos menores que $k$ no multiset, basta usar `order_of_key({k, -INF})`. -- Usar um $ordered\_map$ com a key sendo o valor e o value sendo o número de ocorrências do valor no multiset. Para saber o número de elementos menores que $k$ no multiset, basta usar $order\_of\_key(k)$. +- Usar um `ordered_map` com a key sendo o valor e o value sendo o número de ocorrências do valor no multiset. Para saber o número de elementos menores que $k$ no multiset, basta usar `order_of_key(k)`. -- Criar o $ordered\_set$ substituindo o parâmetro $less$ por $less\_equal$. Isso faz com que o set aceite elementos repetidos, e $order\_of\_key(k)$ retorna o número de elementos menores ou iguais a $k$ no multiset. Porém esse método não é recomendado pois gera algumas inconsistências, como por exemplo: $upper\_bound$ funciona como $lower\_bound$ e vice-versa, $find$ sempre retorna $end()$ e $erase$ por valor não funciona, só por iterador. Dá pra usar se souber o que está fazendo. +- Criar o `ordered_set` trocando o parâmetro `less` por `less_equal`. Isso faz com que o set aceite elementos repetidos, e `order_of_key(k)` retorna o número de elementos menores ou iguais a `k` no multiset. Porém esse método não é recomendado pois gera algumas inconsistências, como por exemplo: `upper_bound` funciona como `lower_bound` e vice-versa, `find` sempre retorna `end()` e `erase` por valor não funciona, só por iterador. Dá pra usar se souber o que está fazendo. -Exemplo de uso do $ordered\_set$: +Exemplo de uso do `ordered_set`: ```cpp ordered_set X; @@ -39,7 +39,7 @@ cout << X.order_of_key(4) << endl; // 2 cout << X.order_of_key(400) << endl; // 5 ``` -Exemplo de uso do $ordered\_map$: +Exemplo de uso do `ordered_map`: ```cpp ordered_map Y; diff --git a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-2D/README.md b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-2D/README.md index 312b232a..9982458d 100644 --- a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-2D/README.md +++ b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-2D/README.md @@ -1,8 +1,3 @@ -# [Seg Tree 2D](seg_tree.cpp) +# [Segment Tree 2D](seg_tree.cpp) -Segment Tree em 2 dimensões. - -- Complexidade de tempo (Pré-processamento): $\mathcal{O}(N \cdot M)$ -- Complexidade de tempo (Consulta em intervalo): $\mathcal{O}(\log N \cdot \log M)$ -- Complexidade de tempo (Update em ponto): $\mathcal{O}(\log N \cdot \log M)$ -- Complexidade de espaço: $4 * N \cdot 4 * M = \mathcal{O}(N \cdot M)$ \ No newline at end of file +Segment Tree em 2 dimensões, suporta operações de update pontual e consulta em intervalo. A construção é $\mathcal{O}(n \cdot m)$ e as operações de consulta e update são $\mathcal{O}(log(n) \cdot log(m))$. \ No newline at end of file diff --git a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Beats-Max-And-Sum-Update/README.md b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Beats-Max-And-Sum-Update/README.md index 5f4500a1..34133c59 100644 --- a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Beats-Max-And-Sum-Update/README.md +++ b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Beats-Max-And-Sum-Update/README.md @@ -1,10 +1,3 @@ -# [Seg Tree Beats Max and Sum update](seg_tree_beats_max_and_sum_update.cpp) +# [Segment Tree Beats 2](seg_tree_beats_max_and_sum_update.cpp) -Seg Tree que suporta update de maximo, update de soma e query de soma. -Utiliza uma fila de lazy para diferenciar os updates - -- Complexidade de tempo (Pré-processamento): $\mathcal{O}(N)$ -- Complexidade de tempo (Consulta em intervalo): $\mathcal{O}(log(N))$ -- Complexidade de tempo (Update em ponto): $\mathcal{O}(log(N))$ -- Complexidade de tempo (Update em intervalo): $\mathcal{O}(log(N))$ -- Complexidade de espaço: 2 * 4 * N = $\mathcal{O}(N)$ +Segment Tree que suporta update de maximo, update de soma e query de soma. Utiliza uma fila de lazy para diferenciar os updates. A construção é $\mathcal{O}(n)$ e as operações de consulta e update são $\mathcal{O}(log(n))$. \ No newline at end of file diff --git a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Beats-Max-Update/README.md b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Beats-Max-Update/README.md index 24ac8911..5588e520 100644 --- a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Beats-Max-Update/README.md +++ b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Beats-Max-Update/README.md @@ -1,8 +1,3 @@ -# [Seg Tree Beats](seg_tree_beats.cpp) -Seg Tree que suporta update de maximo e query de soma +# [Segment Tree Beats](seg_tree_beats.cpp) -- Complexidade de tempo (Pré-processamento): $\mathcal{O}(N)$ -- Complexidade de tempo (Consulta em intervalo): $\mathcal{O}(log(N))$ -- Complexidade de tempo (Update em ponto): $\mathcal{O}(log(N))$ -- Complexidade de tempo (Update em intervalo): $\mathcal{O}(log(N))$ -- Complexidade de espaço: $2 * 4 * N = \mathcal{O}(N)$ +Segment Tree que suporta update de maximo e query de soma. A construção é $\mathcal{O}(n)$ e as operações de consulta e update são $\mathcal{O}(log(n))$. diff --git a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Esparsa/README.md b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Esparsa/README.md index c36580cd..27803acb 100644 --- a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Esparsa/README.md +++ b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Esparsa/README.md @@ -1,9 +1,3 @@ -# [Sparse Seg Tree](seg_tree_sparse.cpp) +# [Segment Tree Esparsa](seg_tree_sparse.cpp) -Seg Tree Esparsa, ou seja, uma seg tree que não guarda todos os nós, mas apenas os nós que são necessários para responder as queries, permitindo fazer queries em intervalos de tamanho arbitrário. - -Seja $LEN$ o tamanho do intervalo em que a Seg Tree foi construída: - -- Complexidade de tempo (Pré-processamento): $\mathcal{O}(1)$ -- Complexidade de tempo (Consulta em intervalo): $\mathcal{O}(log(LEN))$ -- Complexidade de tempo (Update em ponto): $\mathcal{O}(log(LEN))$ +Segment Tree Esparsa, ou seja, não armazena todos os nós da árvore, apenas os necessários, dessa forma ela suporta operações em intervalos arbitrários. A construção é $\mathcal{O}(1)$ e as operações de consulta e update são $\mathcal{O}(log(L))$, onde $L$ é o tamanho do intervalo. A implementação suporta operações de consulta em intervalo e update pontual. Está implementada para soma, mas pode ser facilmente modificada para outras operações. diff --git a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Esparsa/seg_tree_sparse.cpp b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Esparsa/seg_tree_sparse.cpp index fe35d7d6..7d97da46 100644 --- a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Esparsa/seg_tree_sparse.cpp +++ b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Esparsa/seg_tree_sparse.cpp @@ -1,44 +1,63 @@ -const int SEGMAX = 8e6 + 5; // should be Q * log(DIR-ESQ+1) -const ll ESQ = 0, DIR = 1e9 + 7; +template struct SegTree { + const ll neutral = 0; + struct node { + ll val; + int L, R; + node(ll v) : val(v), L(-1), R(-1) { } + }; + ll merge(ll a, ll b) { return a + b; } -struct seg { - ll tree[SEGMAX]; - int R[SEGMAX], L[SEGMAX], - ptr = 2; // 0 is NULL; 1 is First Root - ll op(ll a, ll b) { return (a + b) % MOD; } - int le(int i) { - if (L[i] == 0) { - L[i] = ptr++; + vector t; + + int newnode() { + t.push_back(node(neutral)); + return int(t.size() - 1); + } + + SegTree() { newnode(); } + + int le(int u) { + if (t[u].L == -1) { + t[u].L = newnode(); } - return L[i]; + return t[u].L; } - int ri(int i) { - if (R[i] == 0) { - R[i] = ptr++; + + int ri(int u) { + if (t[u].R == -1) { + t[u].R = newnode(); } - return R[i]; + return t[u].R; } - ll query(ll l, ll r, int n = 1, ll esq = ESQ, ll dir = DIR) { - if (r < esq || dir < l) { - return 0; + + ll query(int u, ll l, ll r, ll L, ll R) { + if (l > R || r < L) { + return neutral; } - if (l <= esq && dir <= r) { - return tree[n]; + if (l >= L && r <= R) { + return t[u].val; } - ll mid = (esq + dir) / 2; - return op(query(l, r, le(n), esq, mid), query(l, r, ri(n), mid + 1, dir)); + ll mid = l + (r - l) / 2; + ll ql = query(le(u), l, mid, L, R); + ll qr = query(ri(u), mid + 1, r, L, R); + return merge(ql, qr); } - void update(ll x, ll v, int n = 1, ll esq = ESQ, ll dir = DIR) { - if (esq == dir) { - tree[n] = (tree[n] + v) % MOD; + ll query(ll l, ll r) { return query(0, MINL, MAXR, l, r); } + + void update(int u, ll l, ll r, ll i, ll x) { + debug(u, l, r); + if (l == r) { + t[u].val += x; // soma + // t[u].val = x; // substitui + return; + } + ll mid = l + (r - l) / 2; + if (i <= mid) { + update(le(u), l, mid, i, x); } else { - ll mid = (esq + dir) / 2; - if (x <= mid) { - update(x, v, le(n), esq, mid); - } else { - update(x, v, ri(n), mid + 1, dir); - } - tree[n] = op(tree[le(n)], tree[ri(n)]); + update(ri(u), mid + 1, r, i, x); } + t[u].val = merge(t[le(u)].val, t[ri(u)].val); } + void update(ll i, ll x) { update(0, MINL, MAXR, i, x); } }; diff --git a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Kadani/README.md b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Kadani/README.md index eabd9e6e..a4e0f599 100644 --- a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Kadani/README.md +++ b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Kadani/README.md @@ -1,8 +1,7 @@ # [Seg Tree Kadani](seg_tree_kadani.cpp) -Implementação de uma Seg Tree que suporta update de soma e query de soma máxima em intervalo. +Implementação de uma Segment Tree que suporta update de soma em intervalo e query de soma máxima de um subarray em um intervalo. A construção é $\mathcal{O}(n)$ e as operações de consulta e update são $\mathcal{O}(log(n))$. -- Complexidade de tempo (Pré-processamento): $\mathcal{O}(N)$ -- Complexidade de tempo (Consulta em intervalo): $\mathcal{O}(log(N))$ -- Complexidade de tempo (Update em ponto): $\mathcal{O}(log(N))$ -- Complexidade de espaço: $4 * N = \mathcal{O}(N)$ +É uma Seg Tree normal, a magia está na função `merge` que é a função que combina as respostas dos filhos para computar a resposta do nodo atual. A ideia do `merge` de combinar respostas e informações já computadas dos filhos é muito útil e pode ser aplicada em muitos problemas. + +**Obs**: não considera o subarray vazio como resposta. \ No newline at end of file diff --git a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Kadani/seg_tree_kadani.cpp b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Kadani/seg_tree_kadani.cpp index 8ff3b6c3..2d261ce9 100644 --- a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Kadani/seg_tree_kadani.cpp +++ b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Kadani/seg_tree_kadani.cpp @@ -1,62 +1,76 @@ -namespace seg { - const int MAX = 1e5 + 5; +struct SegTree { struct node { - ll pref, suff, sum, best; + ll sum, pref, suf, ans; }; - node new_node(ll v) { return node{v, v, v, v}; } - const node NEUTRAL = {0, 0, 0, 0}; - node tree[4 * MAX]; - node merge(node a, node b) { - ll pref = max(a.pref, a.sum + b.pref); - ll suff = max(b.suff, b.sum + a.suff); - ll sum = a.sum + b.sum; - ll best = max(a.suff + b.pref, max(a.best, b.best)); - return node{pref, suff, sum, best}; + const node neutral = {0, 0, 0, 0}; + node merge(const node &a, const node &b) { + return {a.sum + b.sum, + max(a.pref, a.sum + b.pref), + max(b.suf, b.sum + a.suf), + max({a.ans, b.ans, a.suf + b.pref})}; } int n; - int le(int n) { return 2 * n + 1; } - int ri(int n) { return 2 * n + 2; } - void build(int n, int esq, int dir, const vector &v) { - if (esq == dir) { - tree[n] = new_node(v[esq]); + vector t; + + void build(int u, int l, int r, const vector &v) { + if (l == r) { + t[u] = {v[l], v[l], v[l], v[l]}; } else { - int mid = (esq + dir) / 2; - build(le(n), esq, mid, v); - build(ri(n), mid + 1, dir, v); - tree[n] = merge(tree[le(n)], tree[ri(n)]); + int mid = (l + r) >> 1; + build(u << 1, l, mid, v); + build(u << 1 | 1, mid + 1, r, v); + t[u] = merge(t[u << 1], t[u << 1 | 1]); } } - void build(const vector &v) { - n = v.size(); - build(0, 0, n - 1, v); + + void build(int _n) { // pra construir com tamanho, mas vazia + n = _n; + t.assign(n << 2, neutral); } - node query(int n, int esq, int dir, int l, int r) { - if (esq > r || dir < l) { - return NEUTRAL; - } - if (l <= esq && dir <= r) { - return tree[n]; + + void build(ll *bg, ll *en) { // pra construir com array estatico + n = int(en - bg); + t.assign(n << 2, neutral); + vector aux(n); + for (int i = 0; i < n; i++) { + aux[i] = bg[i]; } - int mid = (esq + dir) / 2; - return merge(query(le(n), esq, mid, l, r), query(ri(n), mid + 1, dir, l, r)); + build(1, 0, n - 1, aux); + } + + void build(const vector &v) { // pra construir com vector + n = int(v.size()); + t.assign(n << 2, neutral); + build(1, 0, n - 1, v); } - ll query(int l, int r) { return query(0, 0, n - 1, l, r).best; } - void update(int n, int esq, int dir, int x, ll v) { - if (esq > x || dir < x) { - return; + + node query(int u, int l, int r, int L, int R) { + if (l > R || r < L) { + return neutral; + } + if (l >= L && r <= R) { + return t[u]; } - if (esq == dir) { - tree[n] = new_node(v); + int mid = (l + r) >> 1; + node ql = query(u << 1, l, mid, L, R); + node qr = query(u << 1 | 1, mid + 1, r, L, R); + return merge(ql, qr); + } + ll query(int l, int r) { return query(1, 0, n - 1, l, r).ans; } + + void update(int u, int l, int r, int i, ll x) { + if (l == r) { + t[u] = {x, x, x, x}; } else { - int mid = (esq + dir) / 2; - if (x <= mid) { - update(le(n), esq, mid, x, v); + int mid = (l + r) >> 1; + if (i <= mid) { + update(u << 1, l, mid, i, x); } else { - update(ri(n), mid + 1, dir, x, v); + update(u << 1 | 1, mid + 1, r, i, x); } - tree[n] = merge(tree[le(n)], tree[ri(n)]); + t[u] = merge(t[u << 1], t[u << 1 | 1]); } } - void update(int x, ll v) { update(0, 0, n - 1, x, v); } -} + void update(int i, ll x) { update(1, 0, n - 1, i, x); } +}; \ No newline at end of file diff --git a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Lazy/README.md b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Lazy/README.md index 6e03788d..0fe80842 100644 --- a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Lazy/README.md +++ b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Lazy/README.md @@ -1,8 +1,3 @@ -# [Seg Tree Lazy](seg_tree_lazy.cpp) +# [Segment Tree Lazy](seg_tree_lazy.cpp) -Implementação padrão de Seg Tree com lazy update - -- Complexidade de tempo (Pré-processamento): $\mathcal{O}(N)$ -- Complexidade de tempo (Consulta em intervalo): $\mathcal{O}(log(N))$ -- Complexidade de tempo (Update em intervalo): $\mathcal{O}(log(N))$ -- Complexidade de espaço: $2 * 4 * N = \mathcal{O}(N)$ \ No newline at end of file +Implementação de uma Segment Tree com Lazy Propagation. Suporta operações de consulta em intervalo e update em intervalo, está implementada para responder consultas de soma e updates de soma em intervalo, ou atribuição em intervalo. A construção é $\mathcal{O}(n)$ e as operações de consulta e update são $\mathcal{O}(log(n))$. diff --git a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Lazy/seg_tree_lazy.cpp b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Lazy/seg_tree_lazy.cpp index 4ff48e19..cd86f3b6 100644 --- a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Lazy/seg_tree_lazy.cpp +++ b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree-Lazy/seg_tree_lazy.cpp @@ -1,60 +1,101 @@ -namespace seg { - const int MAX = 2e5 + 5; - const ll NEUTRAL = 0; // merge(a, neutral) = a +struct SegTree { ll merge(ll a, ll b) { return a + b; } - int sz; // size of the array - ll tree[4 * MAX], lazy[4 * MAX]; - int le(int n) { return 2 * n + 1; } - int ri(int n) { return 2 * n + 2; } - void push(int n, int esq, int dir) { - if (lazy[n] == 0) { - return; - } - tree[n] += lazy[n] * (dir - esq + 1); - if (esq != dir) { - lazy[le(n)] += lazy[n]; - lazy[ri(n)] += lazy[n]; + const ll neutral = 0; + + int n; + vector t, lazy; + vector replace; + + void push(int u, int l, int r) { + if (replace[u]) { + t[u] = lazy[u] * (r - l + 1); + if (l != r) { + lazy[u << 1] = lazy[u]; + lazy[u << 1 | 1] = lazy[u]; + replace[u << 1] = replace[u]; + replace[u << 1 | 1] = replace[u]; + } + } else if (lazy[u] != 0) { + t[u] += lazy[u] * (r - l + 1); + if (l != r) { + lazy[u << 1] += lazy[u]; + lazy[u << 1 | 1] += lazy[u]; + } } - lazy[n] = 0; + replace[u] = false; + lazy[u] = 0; } - void build(span v, int n, int esq, int dir) { - if (esq == dir) { - tree[n] = v[esq]; + + void build(int u, int l, int r, const vector &v) { + if (l == r) { + t[u] = v[l]; } else { - int mid = (esq + dir) / 2; - build(v, le(n), esq, mid); - build(v, ri(n), mid + 1, dir); - tree[n] = merge(tree[le(n)], tree[ri(n)]); + int mid = (l + r) / 2; + build(u << 1, l, mid, v); + build(u << 1 | 1, mid + 1, r, v); + t[u] = merge(t[u << 1], t[u << 1 | 1]); + } + } + + void build(int _n) { // pra construir com tamanho, mas vazia + n = _n; + t.assign(n << 2, neutral); + lazy.assign(n << 2, 0); + replace.assign(n << 2, false); + } + + void build(ll *bg, ll *en) { // pra construir com array estatico + n = int(en - bg); + t.assign(n << 2, neutral); + lazy.assign(n << 2, 0); + replace.assign(n << 2, false); + vector aux(n); + for (int i = 0; i < n; i++) { + aux[i] = bg[i]; } + build(1, 0, n - 1, aux); } - void build(span v) { - sz = v.size(); - build(v, 0, 0, sz - 1); + + void build(const vector &v) { // pra construir com vector + n = int(v.size()); + t.assign(n << 2, neutral); + lazy.assign(n << 2, 0); + replace.assign(n << 2, false); + build(1, 0, n - 1, v); } - ll query(int l, int r, int n = 0, int esq = 0, int dir = sz - 1) { - push(n, esq, dir); - if (esq > r || dir < l) { - return NEUTRAL; + + ll query(int u, int l, int r, int L, int R) { + push(u, l, r); + if (l > R || r < L) { + return neutral; } - if (l <= esq && dir <= r) { - return tree[n]; + if (l >= L && r <= R) { + return t[u]; } - int mid = (esq + dir) / 2; - return merge(query(l, r, le(n), esq, mid), query(l, r, ri(n), mid + 1, dir)); + int mid = (l + r) >> 1; + ll ql = query(u << 1, l, mid, L, R); + ll qr = query(u << 1 | 1, mid + 1, r, L, R); + return merge(ql, qr); } - void update(int l, int r, ll v, int n = 0, int esq = 0, int dir = sz - 1) { - push(n, esq, dir); - if (esq > r || dir < l) { + ll query(int l, int r) { return query(1, 0, n - 1, l, r); } + + void update(int u, int l, int r, int L, int R, ll val, bool repl) { + push(u, l, r); + if (l > R || r < L) { return; } - if (l <= esq && dir <= r) { - lazy[n] += v; - push(n, esq, dir); + if (l >= L && r <= R) { + lazy[u] = val; + replace[u] = repl; + push(u, l, r); } else { - int mid = (esq + dir) / 2; - update(l, r, v, le(n), esq, mid); - update(l, r, v, ri(n), mid + 1, dir); - tree[n] = merge(tree[le(n)], tree[ri(n)]); + int mid = (l + r) >> 1; + update(u << 1, l, mid, L, R, val, repl); + update(u << 1 | 1, mid + 1, r, L, R, val, repl); + t[u] = merge(t[u << 1], t[u << 1 | 1]); } } -} + void update(int l, int r, ll val, bool repl = false) { + update(1, 0, n - 1, l, r, val, repl); + } +}; \ No newline at end of file diff --git a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree/README.md b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree/README.md index 537f279d..25177b01 100644 --- a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree/README.md +++ b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree/README.md @@ -1,7 +1,3 @@ -# [Seg Tree](seg_tree.cpp) -Implementação padrão de Seg Tree +# [Segment Tree](seg_tree.cpp) -- Complexidade de tempo (Pré-processamento): $\mathcal{O}(N)$ -- Complexidade de tempo (Consulta em intervalo): $\mathcal{O}(log(N))$ -- Complexidade de tempo (Update em ponto): $\mathcal{O}(log(N))$ -- Complexidade de espaço: $4 * N = \mathcal{O}(N)$ +Implementação padrão de Segment Tree, suporta operações de consulta em intervalo e update pontual. Está implementada para soma, mas pode ser facilmente modificada para outras operações. A construção é $\mathcal{O}(n)$ e as operações de consulta e update são $\mathcal{O}(log(n))$. diff --git a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree/seg_tree.cpp b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree/seg_tree.cpp index 5bcd315f..355b59d1 100644 --- a/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree/seg_tree.cpp +++ b/Codigos/Estruturas-de-Dados/Segment-Tree/Segment-Tree/seg_tree.cpp @@ -1,50 +1,69 @@ -namespace seg { - const int MAX = 2e5 + 5; - int n; - ll tree[4 * MAX]; +struct SegTree { ll merge(ll a, ll b) { return a + b; } - int le(int n) { return 2 * n + 1; } - int ri(int n) { return 2 * n + 2; } - void build(int n, int esq, int dir, const vector &v) { - if (esq == dir) { - tree[n] = v[esq]; + const ll neutral = 0; + + int n; + vector t; + + void build(int u, int l, int r, const vector &v) { + if (l == r) { + t[u] = v[l]; } else { - int mid = (esq + dir) / 2; - build(le(n), esq, mid, v); - build(ri(n), mid + 1, dir, v); - tree[n] = merge(tree[le(n)], tree[ri(n)]); + int mid = (l + r) >> 1; + build(u << 1, l, mid, v); + build(u << 1 | 1, mid + 1, r, v); + t[u] = merge(t[u << 1], t[u << 1 | 1]); } } - void build(const vector &v) { - n = v.size(); - build(0, 0, n - 1, v); + + void build(int _n) { // pra construir com tamanho, mas vazia + n = _n; + t.assign(n << 2, neutral); } - ll query(int n, int esq, int dir, int l, int r) { - if (esq > r || dir < l) { - return 0; - } - if (l <= esq && dir <= r) { - return tree[n]; + + void build(ll *bg, ll *en) { // pra construir com array estatico + n = int(en - bg); + t.assign(n << 2, neutral); + vector aux(n); + for (int i = 0; i < n; i++) { + aux[i] = bg[i]; } - int mid = (esq + dir) / 2; - return merge(query(le(n), esq, mid, l, r), query(ri(n), mid + 1, dir, l, r)); + build(1, 0, n - 1, aux); } - ll query(int l, int r) { return query(0, 0, n - 1, l, r); } - void update(int n, int esq, int dir, int x, ll v) { - if (esq > x || dir < x) { - return; + + void build(const vector &v) { // pra construir com vector + n = int(v.size()); + t.assign(n << 2, neutral); + build(1, 0, n - 1, v); + } + + ll query(int u, int l, int r, int L, int R) { + if (l > R || r < L) { + return neutral; + } + if (l >= L && r <= R) { + return t[u]; } - if (esq == dir) { - tree[n] = v; + int mid = (l + r) >> 1; + ll ql = query(u << 1, l, mid, L, R); + ll qr = query(u << 1 | 1, mid + 1, r, L, R); + return merge(ql, qr); + } + ll query(int l, int r) { return query(1, 0, n - 1, l, r); } + + void update(int u, int l, int r, int i, ll x) { + if (l == r) { + t[u] += x; // soma + // t[u] = x; // substitui } else { - int mid = (esq + dir) / 2; - if (x <= mid) { - update(le(n), esq, mid, x, v); + int mid = (l + r) >> 1; + if (i <= mid) { + update(u << 1, l, mid, i, x); } else { - update(ri(n), mid + 1, dir, x, v); + update(u << 1 | 1, mid + 1, r, i, x); } - tree[n] = merge(tree[le(n)], tree[ri(n)]); + t[u] = merge(t[u << 1], t[u << 1 | 1]); } } - void update(int x, ll v) { update(0, 0, n - 1, x, v); } -} + void update(int i, ll x) { update(1, 0, n - 1, i, x); } +}; diff --git a/Codigos/Estruturas-de-Dados/Sparse-Table/Sparse-Table/README.md b/Codigos/Estruturas-de-Dados/Sparse-Table/Sparse-Table/README.md index 7716d799..3f6df59a 100644 --- a/Codigos/Estruturas-de-Dados/Sparse-Table/Sparse-Table/README.md +++ b/Codigos/Estruturas-de-Dados/Sparse-Table/Sparse-Table/README.md @@ -1,7 +1,5 @@ # [Sparse Table](sparse_table.cpp) -*Read in [English](README.en.md)* - Responde consultas de maneira eficiente em um conjunto de dados estáticos.\ Realiza um pré-processamento para diminuir o tempo de cada consulta. diff --git a/Codigos/Estruturas-de-Dados/Sparse-Table/Sparse-Table/sparse_table.cpp b/Codigos/Estruturas-de-Dados/Sparse-Table/Sparse-Table/sparse_table.cpp index 8821103b..4568d1be 100644 --- a/Codigos/Estruturas-de-Dados/Sparse-Table/Sparse-Table/sparse_table.cpp +++ b/Codigos/Estruturas-de-Dados/Sparse-Table/Sparse-Table/sparse_table.cpp @@ -24,7 +24,7 @@ struct SparseTable { } return res; } - // O(1) Query for overlab friendly operations + // O(1) Query for overlap friendly operations // ex: max(), min(), gcd(), f(x, y) = x int query(int l, int r) { // if (l > r) return 2e9; diff --git a/Codigos/Grafos/2-SAT/README.md b/Codigos/Grafos/2-SAT/README.md index 39be982d..d96090cf 100644 --- a/Codigos/Grafos/2-SAT/README.md +++ b/Codigos/Grafos/2-SAT/README.md @@ -2,7 +2,7 @@ Resolve problema do 2-SAT. -- Complexidade de tempo (caso médio): $\mathcal{O}(N + M)$ +- Complexidade de tempo: $\mathcal{O}(N + M)$ N é o número de variáveis e M é o número de cláusulas. 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From: =?UTF-8?q?Jo=C3=A3o=20Oliveira?= Date: Tue, 20 Feb 2024 14:40:21 -0300 Subject: [PATCH 2/5] lichao fix --- Codigos/Estruturas-de-Dados/LiChao-Tree/lichao_tree.cpp | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/Codigos/Estruturas-de-Dados/LiChao-Tree/lichao_tree.cpp b/Codigos/Estruturas-de-Dados/LiChao-Tree/lichao_tree.cpp index fcd0ba2d..1204ceba 100644 --- a/Codigos/Estruturas-de-Dados/LiChao-Tree/lichao_tree.cpp +++ b/Codigos/Estruturas-de-Dados/LiChao-Tree/lichao_tree.cpp @@ -12,7 +12,7 @@ template struct LichaoTree { tree.push_back(Line()); L.push_back(-1); R.push_back(-1); - return int(tree.size() - 1); + return int(tree.size()) - 1; } LichaoTree() { newnode(); } @@ -62,4 +62,4 @@ template struct LichaoTree { return max(tree[n](x), query(x, ri(n), mid + 1, r)); } } -}; \ No newline at end of file +}; From d473dda0107c727e7901f9321d248043f7014077 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Jo=C3=A3o=20Oliveira?= Date: Tue, 20 Feb 2024 14:48:25 -0300 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a/Codigos/Geometria/Convex-Hull/README.md b/Codigos/Geometria/Convex-Hull/README.md new file mode 100644 index 00000000..d217d78f --- /dev/null +++ b/Codigos/Geometria/Convex-Hull/README.md @@ -0,0 +1,7 @@ +# [Convex Hull](convex_hull.cpp) + +Algoritmo Graham's Scan para encontrar o fecho convexo de um conjunto de pontos em \$\mathcal{O}(n \log n)\$. Retorna os pontos do fecho convexo em sentido horário. + +**Definição**: o fecho convexo de um conjunto de pontos é o menor polígono convexo que contém todos os pontos do conjunto. + +**Obs**: utiliza a primitiva `Ponto 2D`. diff --git a/Codigos/Geometria/Convex-Hull/convex_hull.cpp b/Codigos/Geometria/Convex-Hull/convex_hull.cpp new file mode 100644 index 00000000..9c4f03cd --- /dev/null +++ b/Codigos/Geometria/Convex-Hull/convex_hull.cpp @@ -0,0 +1,38 @@ +bool ccw(pt &p, pt &a, pt &b, bool include_collinear = 0) { + pt p1 = a - p; + pt p2 = b - p; + return include_collinear ? (p2 ^ p1) <= 0 : (p2 ^ p1) < 0; +} + +void sort_by_angle(vector& v) { // sorta o vetor por angulo em relacao ao pivo + pt p0 = *min_element(begin(v), end(v)); + sort(begin(v), end(v), [&](pt &l, pt &r) { // sorta clockwise + pt p1 = l - p0; + pt p2 = r - p0; + ll c1 = p1 ^ p2; + return c1 < 0 || ((c1 == 0) && p0.dist2(l) < p0.dist2(r)); + }); +} + +vector convex_hull(vector v, bool include_collinear = 0) { + int n = size(v); + + sort_by_angle(v); + + if (include_collinear) { + for (int i = n - 2; i >= 0; i--) { // reverte o ultimo lado do poligono + if (ccw(v[0], v[n - 1], v[i])) { + reverse(begin(v) + i + 1, end(v)); + break; + } + } + } + + vector ch{v[0], v[1]}; + for (int i = 2; i < n; i++) { + while (ch.size() > 2 && (ccw(ch.end()[-2], ch.end()[-1], v[i], !include_collinear))) ch.pop_back(); + ch.emplace_back(v[i]); + } + + return ch; +} \ No newline at end of file diff --git a/Codigos/Primitivas/Modular-Int/mint.cpp b/Codigos/Primitivas/Modular-Int/mint.cpp index 82b0ccfe..cdf77427 100644 --- a/Codigos/Primitivas/Modular-Int/mint.cpp +++ b/Codigos/Primitivas/Modular-Int/mint.cpp @@ -1,18 +1,20 @@ -template struct Mint { +template struct Mint { using m = Mint; int v; Mint() : v(0) { } - Mint(ll v) { - if (v < -mod or v >= 2 * mod) { - v %= mod; + Mint(ll val) { + if (val < -MOD or val >= 2 * MOD) { + val %= MOD; } - if (v >= mod) { - v -= mod; + val -= (MOD * (val >= MOD)); + val += (MOD * (val < 0)); + if (val >= MOD) { + val -= MOD; } - if (v < 0) { - v += mod; + if (val < 0) { + val += MOD; } - v = int(v); + v = int(val); } bool operator==(const m &o) const { return v == o.v; } bool operator!=(const m &o) const { return v != o.v; } @@ -28,22 +30,22 @@ template struct Mint { return res; } m &operator+=(const m &o) { - if ((v += o.v) >= mod) { - v -= mod; + if ((v += o.v) >= MOD) { + v -= MOD; } return *this; } m &operator-=(const m &o) { if ((v -= o.v) < 0) { - v += mod; + v += MOD; } return *this; } m &operator*=(const m &o) { - v = int(ll(v) * o.v % mod); + v = int(ll(v) * o.v % MOD); return *this; } - m &operator/=(const m &o) { return *this *= pwr(o, mod - 2); } + m &operator/=(const m &o) { return *this *= pwr(o, MOD - 2); } m &operator^=(ll e) { assert(e >= 0); return *this = pwr(*this, e); @@ -61,5 +63,5 @@ template struct Mint { } }; -const int mod = 998244353; -using mint = Mint; \ No newline at end of file +const int MOD = 998244353; // o MOD tem que ser primo +using mint = Mint; \ No newline at end of file diff --git a/Codigos/Primitivas/Ponto-2D/README.md b/Codigos/Primitivas/Ponto-2D/README.md new file mode 100644 index 00000000..6105b230 --- /dev/null +++ b/Codigos/Primitivas/Ponto-2D/README.md @@ -0,0 +1,3 @@ +# [Ponto 2D](point2d.cpp) + +Estrutura que representa um ponto no plano cartesiano em duas dimensões. Suporta operações de soma, subtração, multiplicação por escalar, produto escalar, produto vetorial e distância euclidiana. Pode ser usado também para representar um vetor. diff --git a/Codigos/Primitivas/Ponto-2D/point2d.cpp b/Codigos/Primitivas/Ponto-2D/point2d.cpp new file mode 100644 index 00000000..8152c212 --- /dev/null +++ b/Codigos/Primitivas/Ponto-2D/point2d.cpp @@ -0,0 +1,25 @@ +template struct point { + T x, y; + point(T _x = 0, T _y = 0) : x(_x), y(_y) { } + + using p = point; + + p operator*(const T o) { return p(o * x, o * y); } + p operator-(const p o) { return p(x - o.x, y - o.y); } + p operator+(const p o) { return p(x + o.x, y + o.y); } + T operator*(const p o) { return x * o.x + y * o.y; } + T operator^(const p o) { return x * o.y - y * o.x; } + bool operator<(const p o) const { return (x == o.x) ? y < o.y : x < o.x; } + bool operator==(const p o) const { return (x == o.x) and (y == o.y); } + bool operator!=(const p o) const { return (x != o.x) or (y != o.y); } + + T dist2(const p o) { + T dx = x - o.x, dy = y - o.y; + return dx * dx + dy * dy; + } + + friend ostream &operator<<(ostream &out, const p &a) { return out << "(" << a.x << "," << a.y << ")"; } + friend istream &operator>>(istream &in, p &a) { return in >> a.x >> a.y; } +}; + +using pt = point; \ No newline at end of file diff --git a/Codigos/String/Hashing/README.md b/Codigos/String/Hashing/README.md index 2da9c87f..700e9caf 100644 --- a/Codigos/String/Hashing/README.md +++ b/Codigos/String/Hashing/README.md @@ -1,9 +1,14 @@ # [Hashing](hashing.cpp) -Hashing para testar igualdade de duas strings. +Hashing polinomial para testar igualdade de strings (ou de vetores). Requer precomputar as potências de um primo, como indicado na função `precalc`. A implementação está com dois MODS e usa a primitiva `Mint`, pode-se alterar caso necessário. A construção é $\mathcal{O}(n)$ e a consulta é $\mathcal{O}(1)$. -A função **range(i, j)** retorna o hash da substring nesse range. -Pode ser necessário usar pares de hash para evitar colisões. +Exemplo de uso: -* Complexidade de tempo (Construção): $\mathcal{O}(N)$ -* Complexidade de tempo (Consulta de range): $\mathcal{O}(1)$ +```cpp +string s = "abacabab"; +Hashing a(s); +cout << (a(0, 1) == a(2, 3)) << endl; // 0 +cout << (a(0, 1) == a(4, 5)) << endl; // 1 +cout << (a(0, 2) == a(4, 6)) << endl; // 1 +cout << (a(0, 3) == a(4, 7)) << endl; // 0 +``` diff --git a/Codigos/String/Hashing/hashing.cpp b/Codigos/String/Hashing/hashing.cpp index 8163cc50..2d8728d2 100644 --- a/Codigos/String/Hashing/hashing.cpp +++ b/Codigos/String/Hashing/hashing.cpp @@ -1,26 +1,49 @@ -struct hashing { - const long long LIM = 1000006; - long long p, m; - vector pw, hsh; - hashing(long long _p, long long _m) : p(_p), m(_m) { - pw.resize(LIM); - hsh.resize(LIM); - pw[0] = 1; - for (int i = 1; i < LIM; i++) { - pw[i] = (pw[i - 1] * p) % m; - } +const int MOD1 = 998244353; +const int MOD2 = 1e9 + 7; +using mint1 = Mint; +using mint2 = Mint; + +struct Hash { + mint1 h1; + mint2 h2; + Hash() {} + Hash(mint1 _h1, mint2 _h2) : h1(_h1), h2(_h2) {} + bool operator==(Hash o) const { return h1 == o.h1 && h2 == o.h2; } + bool operator!=(Hash o) const { return h1 != o.h1 || h2 != o.h2; } + bool operator<(Hash o) const { return h1 == o.h1 ? h2 < o.h2 : h1 < o.h1; } + Hash operator+(Hash o) const { return {h1 + o.h1, h2 + o.h2}; } + Hash operator-(Hash o) const { return {h1 - o.h1, h2 - o.h2}; } + Hash operator*(Hash o) const { return {h1 * o.h1, h2 * o.h2}; } + Hash operator/(Hash o) const { return {h1 / o.h1, h2 / o.h2}; } +}; + +const int PRIME = 1001003; // qualquer primo na ordem do alfabeto + +const int MAXN = 1e6 + 5; + +Hash PR = {PRIME, PRIME}; +Hash invPR = {mint1(1) / PRIME, mint2(1) / PRIME}; + +Hash pot[MAXN], invpot[MAXN]; + +void precalc() { + pot[0] = invpot[0] = Hash(1, 1); + for (int i = 1; i < MAXN; i++) { + pot[i] = pot[i - 1] * PR; + invpot[i] = invpot[i - 1] * invPR; } - void set_string(string &s) { - hsh[0] = s[0]; - for (int i = 1; i < s.size(); i++) { - hsh[i] = (hsh[i - 1] * p + s[i]) % m; +} + +struct Hashing { + int N; + vector hsh; + Hashing () {} + Hashing(string s) : N(int(s.size())), hsh(N + 1) { + for (int i = 0; i < N; i++) { + hsh[i + 1] = hsh[i] + (pot[i + 1] * Hash(s[i], s[i])); } } - long long range(int esq, int dir) { - long long ans = hsh[dir]; - if (esq > 0) { - ans = (ans - (hsh[esq - 1] * pw[dir - esq + 1] % m) + m) % m; - } - return ans; + Hash operator()(int l, int r) const { + return (hsh[r + 1] - hsh[l]) * invpot[l]; } -}; +}; \ No newline at end of file From 2e14434ac31da2e75b7b1443ccde45761bc9b351 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Jo=C3=A3o=20Oliveira?= Date: Wed, 21 Feb 2024 04:40:44 -0300 Subject: [PATCH 5/5] hashing dinamico --- .../Fenwick-Tree/Fenwick/fenwick_tree.cpp | 4 +- Codigos/Geometria/Convex-Hull/convex_hull.cpp | 8 +-- Codigos/Primitivas/Ponto-2D/point2d.cpp | 6 ++- .../Hashing-Din\303\242mico/README.md" | 15 ++++++ .../dynamic_hashing.cpp" | 54 +++++++++++++++++++ .../Hashing-Est\303\241tico/README.md" | 0 .../Hashing-Est\303\241tico/hashing.cpp" | 13 +++-- 7 files changed, 86 insertions(+), 14 deletions(-) create mode 100644 "Codigos/String/Hashing/Hashing-Din\303\242mico/README.md" create mode 100644 "Codigos/String/Hashing/Hashing-Din\303\242mico/dynamic_hashing.cpp" rename Codigos/String/Hashing/README.md => "Codigos/String/Hashing/Hashing-Est\303\241tico/README.md" (100%) rename Codigos/String/Hashing/hashing.cpp => "Codigos/String/Hashing/Hashing-Est\303\241tico/hashing.cpp" (81%) diff --git a/Codigos/Estruturas-de-Dados/Fenwick-Tree/Fenwick/fenwick_tree.cpp b/Codigos/Estruturas-de-Dados/Fenwick-Tree/Fenwick/fenwick_tree.cpp index 9d2d872b..08b28c19 100644 --- a/Codigos/Estruturas-de-Dados/Fenwick-Tree/Fenwick/fenwick_tree.cpp +++ b/Codigos/Estruturas-de-Dados/Fenwick-Tree/Fenwick/fenwick_tree.cpp @@ -1,8 +1,8 @@ template struct FenwickTree { int n; vector bit, arr; - FenwickTree(int n = 0) : n(n), bit(n), arr(n) { } - FenwickTree(vector &v) : n(v.size()), bit(n), arr(v) { + FenwickTree(int _n = 0) : n(_n), bit(n), arr(n) { } + FenwickTree(vector &v) : n(int(v.size())), bit(n), arr(v) { for (int i = 0; i < n; i++) { bit[i] = arr[i]; } diff --git a/Codigos/Geometria/Convex-Hull/convex_hull.cpp b/Codigos/Geometria/Convex-Hull/convex_hull.cpp index 9c4f03cd..28d8912d 100644 --- a/Codigos/Geometria/Convex-Hull/convex_hull.cpp +++ b/Codigos/Geometria/Convex-Hull/convex_hull.cpp @@ -4,8 +4,8 @@ bool ccw(pt &p, pt &a, pt &b, bool include_collinear = 0) { return include_collinear ? (p2 ^ p1) <= 0 : (p2 ^ p1) < 0; } -void sort_by_angle(vector& v) { // sorta o vetor por angulo em relacao ao pivo - pt p0 = *min_element(begin(v), end(v)); +void sort_by_angle(vector &v) { // sorta o vetor por angulo em relacao ao pivo + pt p0 = *min_element(begin(v), end(v)); sort(begin(v), end(v), [&](pt &l, pt &r) { // sorta clockwise pt p1 = l - p0; pt p2 = r - p0; @@ -30,7 +30,9 @@ vector convex_hull(vector v, bool include_collinear = 0) { vector ch{v[0], v[1]}; for (int i = 2; i < n; i++) { - while (ch.size() > 2 && (ccw(ch.end()[-2], ch.end()[-1], v[i], !include_collinear))) ch.pop_back(); + while (ch.size() > 2 && + (ccw(ch.end()[-2], ch.end()[-1], v[i], !include_collinear))) + ch.pop_back(); ch.emplace_back(v[i]); } diff --git a/Codigos/Primitivas/Ponto-2D/point2d.cpp b/Codigos/Primitivas/Ponto-2D/point2d.cpp index 8152c212..c0b96454 100644 --- a/Codigos/Primitivas/Ponto-2D/point2d.cpp +++ b/Codigos/Primitivas/Ponto-2D/point2d.cpp @@ -1,4 +1,4 @@ -template struct point { +template struct point { T x, y; point(T _x = 0, T _y = 0) : x(_x), y(_y) { } @@ -18,7 +18,9 @@ template struct point { return dx * dx + dy * dy; } - friend ostream &operator<<(ostream &out, const p &a) { return out << "(" << a.x << "," << a.y << ")"; } + friend ostream &operator<<(ostream &out, const p &a) { + return out << "(" << a.x << "," << a.y << ")"; + } friend istream &operator>>(istream &in, p &a) { return in >> a.x >> a.y; } }; diff --git "a/Codigos/String/Hashing/Hashing-Din\303\242mico/README.md" "b/Codigos/String/Hashing/Hashing-Din\303\242mico/README.md" new file mode 100644 index 00000000..1de84207 --- /dev/null +++ "b/Codigos/String/Hashing/Hashing-Din\303\242mico/README.md" @@ -0,0 +1,15 @@ +# [Hashing Dinâmico](dynamic_hashing.cpp) + +Hashing polinomial para testar igualdade de strings (ou de vetores). Requer precomputar as potências de um primo, como indicado na função `precalc`. A implementação está com dois MODS e usa a primitiva `Mint`, pode-se alterar caso necessário. +Esse `Hash` suporta updates pontuais, utilizando-se de uma `Fenwick Tree` por baixo. A construção é $\mathcal{O}(n)$, consultas e updates são $\mathcal{O}(\log n)$. + +Exemplo de uso: + +```cpp +string s = "abacabab"; +DynamicHashing a(s); +cout << (a(0, 1) == a(2, 3)) << endl; // 0 +cout << (a(0, 1) == a(4, 5)) << endl; // 1 +a.update(0, 'c'); +cout << (a(0, 1) == a(4, 5)) << endl; // 0 +``` diff --git "a/Codigos/String/Hashing/Hashing-Din\303\242mico/dynamic_hashing.cpp" "b/Codigos/String/Hashing/Hashing-Din\303\242mico/dynamic_hashing.cpp" new file mode 100644 index 00000000..a5c12511 --- /dev/null +++ "b/Codigos/String/Hashing/Hashing-Din\303\242mico/dynamic_hashing.cpp" @@ -0,0 +1,54 @@ +const int MOD1 = 998244353; +const int MOD2 = 1e9 + 7; +using mint1 = Mint; +using mint2 = Mint; + +struct Hash { + mint1 h1; + mint2 h2; + Hash() { } + Hash(mint1 _h1, mint2 _h2) : h1(_h1), h2(_h2) { } + bool operator==(Hash o) const { return h1 == o.h1 && h2 == o.h2; } + bool operator!=(Hash o) const { return h1 != o.h1 || h2 != o.h2; } + bool operator<(Hash o) const { return h1 == o.h1 ? h2 < o.h2 : h1 < o.h1; } + Hash operator+(Hash o) const { return {h1 + o.h1, h2 + o.h2}; } + Hash operator-(Hash o) const { return {h1 - o.h1, h2 - o.h2}; } + Hash operator*(Hash o) const { return {h1 * o.h1, h2 * o.h2}; } + Hash operator/(Hash o) const { return {h1 / o.h1, h2 / o.h2}; } +}; + +const int PRIME = 1001003; // qualquer primo na ordem do alfabeto + +const int MAXN = 1e6 + 5; + +Hash PR = {PRIME, PRIME}; +Hash invPR = {mint1(1) / PRIME, mint2(1) / PRIME}; + +Hash pot[MAXN], invpot[MAXN]; + +void precalc() { + pot[0] = invpot[0] = Hash(1, 1); + for (int i = 1; i < MAXN; i++) { + pot[i] = pot[i - 1] * PR; + invpot[i] = invpot[i - 1] * invPR; + } +} + +struct DynamicHashing { + int N; + FenwickTree hsh; + DynamicHashing() { } + DynamicHashing(string s) : N(int(s.size())) { + vector v(N); + for (int i = 0; i < N; i++) { + int c = int(s[i] - 'a'); + v[i] = pot[i + 1] * Hash(c, c); + } + hsh = FenwickTree(v); + } + Hash operator()(int l, int r) { return hsh.query(l, r) * invpot[l]; } + void update(int i, char ch) { + int c = int(ch - 'a'); + hsh.updateSet(i, pot[i + 1] * Hash(c, c)); + } +}; diff --git a/Codigos/String/Hashing/README.md "b/Codigos/String/Hashing/Hashing-Est\303\241tico/README.md" similarity index 100% rename from Codigos/String/Hashing/README.md rename to "Codigos/String/Hashing/Hashing-Est\303\241tico/README.md" diff --git a/Codigos/String/Hashing/hashing.cpp "b/Codigos/String/Hashing/Hashing-Est\303\241tico/hashing.cpp" similarity index 81% rename from Codigos/String/Hashing/hashing.cpp rename to "Codigos/String/Hashing/Hashing-Est\303\241tico/hashing.cpp" index 2d8728d2..19d557c6 100644 --- a/Codigos/String/Hashing/hashing.cpp +++ "b/Codigos/String/Hashing/Hashing-Est\303\241tico/hashing.cpp" @@ -6,8 +6,8 @@ using mint2 = Mint; struct Hash { mint1 h1; mint2 h2; - Hash() {} - Hash(mint1 _h1, mint2 _h2) : h1(_h1), h2(_h2) {} + Hash() { } + Hash(mint1 _h1, mint2 _h2) : h1(_h1), h2(_h2) { } bool operator==(Hash o) const { return h1 == o.h1 && h2 == o.h2; } bool operator!=(Hash o) const { return h1 != o.h1 || h2 != o.h2; } bool operator<(Hash o) const { return h1 == o.h1 ? h2 < o.h2 : h1 < o.h1; } @@ -37,13 +37,12 @@ void precalc() { struct Hashing { int N; vector hsh; - Hashing () {} + Hashing() { } Hashing(string s) : N(int(s.size())), hsh(N + 1) { for (int i = 0; i < N; i++) { - hsh[i + 1] = hsh[i] + (pot[i + 1] * Hash(s[i], s[i])); + int c = int(s[i] - 'a'); + hsh[i + 1] = hsh[i] + (pot[i + 1] * Hash(c, c)); } } - Hash operator()(int l, int r) const { - return (hsh[r + 1] - hsh[l]) * invpot[l]; - } + Hash operator()(int l, int r) const { return (hsh[r + 1] - hsh[l]) * invpot[l]; } }; \ No newline at end of file