From 88d5c82a29140aa45172a0b246f575771c571fcc Mon Sep 17 00:00:00 2001
From: Anthony <thonywong@rocketmail.com>
Date: Tue, 1 Nov 2016 06:31:45 +0700
Subject: [PATCH] Add materials for week 12

---
 README.md                |  1 +
 week12/ConvexPolygon.cpp | 27 ++++++++++++++++++++
 week12/GrahamScan.cpp    | 54 ++++++++++++++++++++++++++++++++++++++++
 week12/MonotoneChain.cpp | 37 +++++++++++++++++++++++++++
 4 files changed, 119 insertions(+)
 create mode 100644 week12/ConvexPolygon.cpp
 create mode 100644 week12/GrahamScan.cpp
 create mode 100644 week12/MonotoneChain.cpp

diff --git a/README.md b/README.md
index 466ecee..86baba4 100644
--- a/README.md
+++ b/README.md
@@ -15,6 +15,7 @@ Materials:
 * Week09: Advanced Dynamic Programming
 * Week10: String Algorithms
 * Week11: SQRT Decomposition and Segment Tree with Lazy Propagation
+* Week12: Geometry: Convex Hull
 
 Team:
 
diff --git a/week12/ConvexPolygon.cpp b/week12/ConvexPolygon.cpp
new file mode 100644
index 0000000..9508ba1
--- /dev/null
+++ b/week12/ConvexPolygon.cpp
@@ -0,0 +1,27 @@
+struct Point {
+	int x, y;
+	Point() { }
+	Point(int x, int y): x(x), y(y) { }
+};
+
+long long cross(Point o, Point a, Point b) {
+	return (long long) (a.x - o.x) * (b.y - o.y) - (long long) (b.x - o.x) * (a.y - o.y);
+}
+
+double area(vector<Point> v) {
+	double ans = 0.0;
+	for (int i = 0; i < (int) v.size(); ++i) {
+		int nxt = (i + 1) % v.size();
+		ans += cross(Point(0, 0), v[i], v[nxt]);
+	}
+	return fabs(ans) * 0.5;
+}
+
+double perimeter(vector<Point> v) {
+	double ans = 0.0;
+	for (int i = 0; i < (int) v.size(); ++i) {
+		int nxt = (i + 1) % v.size();
+		ans += hypot(v[i].x - v[nxt].x, v[i].y - v[nxt].y);
+	}
+	return ans;
+}
diff --git a/week12/GrahamScan.cpp b/week12/GrahamScan.cpp
new file mode 100644
index 0000000..465d67a
--- /dev/null
+++ b/week12/GrahamScan.cpp
@@ -0,0 +1,54 @@
+// Implementation of Graham Scan algorithm
+
+struct Point {
+	int x, y;
+	Point() { }
+	Point(int x, int y): x(x), y(y) { }
+	bool operator < (const Point& other) const {
+		// priority: lowest point
+		// tie: rightmost point
+		return (y == other.y) ? (x > other.x) : (y < other.y);
+	}
+};
+
+Point o;
+
+long long cross(Point o, Point a, Point b) {
+	return (long long) (a.x - o.x) * (b.y - o.y) - (long long) (a.y - o.y) * (b.x - o.x);
+}
+
+bool cmp(Point a, Point b) {
+	long long c = cross(o, a, b);
+	if (c == 0LL)
+		return hypot(a.x - o.x, a.y - o.y) < hypot(b.x - o.x, b.y - o.y);
+	return c > 0;
+	// Angle comparison
+	// return (atan2(a.y - o.y, a.x - o.x) - atan2(b.y - o.y, b.x - o.x)) < 0;
+}
+
+vector<Point> convexHull(vector<Point> v) {
+	vector<Point> ret;
+	int st = 0;
+	// find pivot
+	for (int i = 1; i < (int) v.size(); ++i) {
+		if (v[i] < v[st]) {
+			st = i;
+		}
+	}
+	swap(v[st], v[0]);
+	o = v[0];
+	// sort the remaining points
+	sort(v.begin() + 1, v.end(), cmp);
+	ret.push_back(v.back());
+	ret.push_back(o);
+	int k = 2;
+	for (int i = 1; i < (int) v.size(); ++i) {
+		while (cross(ret[k - 2], ret[k - 1], v[i]) < 0) {
+			ret.pop_back();
+			--k;
+		}
+		ret.push_back(v[i]);
+		++k;
+	}
+	return ret;
+}
diff --git a/week12/MonotoneChain.cpp b/week12/MonotoneChain.cpp
new file mode 100644
index 0000000..dad16c0
--- /dev/null
+++ b/week12/MonotoneChain.cpp
@@ -0,0 +1,37 @@
+// Implementation of Andrew's Monotone Chain algorithm
+
+struct Point {
+	int x, y;
+	Point() { }
+	Point(int x, int y): x(x), y(y) { }
+	bool operator < (const Point& other) const {
+		return x < other.x;
+	}
+};
+
+long long cross(Point o, Point a, Point b) {
+	return (long long) (a.x - o.x) * (b.y - o.y) - (long long) (a.y - o.y) * (b.x - o.x);
+}
+
+vector<Point> convexHull(vector<Point> v) {
+	vector<Point> ret;
+	sort(v.begin(), v.end());
+	int k = 0;
+	for (int i = 0; i < n; ++i) {
+		while (k > 1 && cross(ret[k - 2], ret[k - 1], v[i]) > 0) {
+			--k;
+			ret.pop_back();
+		}
+		ret.push_back(v[i]);
+		++k;
+	}
+	for (int i = n - 2, t = k; i >= 0; --i) {
+		while (k > t && cross(ret[k - 2], ret[k - 1], v[i]) > 0) {
+			--k;
+			ret.pop_back();
+		}
+		ret.push_back(v[i]);
+		++k;
+	}
+	return ret;
+}