# 題目: UVa 11096 - Nails

# 題目說明

給N個點 (x, y平面座標)，求這些點形成的凸包(Convex Hull)周長

---

INPUT:
第一行輸入一個整數T，代表測資數
每筆測資輸入L、N，代表橡皮筋長度、點的數量
接下來有N行，每行輸入兩個變數(x, y)，為點的座標

---

OUTPUT:
輸出 MAX(凸包(Convex Hull)周長，L)

# 解題方法

將凸包算出後再兩兩計算長度，加總即為周長

# 參考程式碼 Graham's Scan

#include <iostream>
#include <vector>
#include <algorithm>
#include <math.h>
#include <iomanip>

using namespace std;

static auto fast_io = []
{
	ios::sync_with_stdio(false);
	cout.tie(nullptr);
	cin.tie(nullptr);
	return 0;
}();

struct point
{
	double x;
	double y;
	double d;
};
vector< point > V;
vector< point > ret;
int T, N;
double L;

double dist(point& a, point& b)
{
	return sqrt(pow(a.x - b.x, 2) + pow(a.y - b.y, 2));
}

double cross(point& o, point& a, point& b)
{
	return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x);
}

bool cmp(point& a, point& b)
{
	auto c = cross(V[0], a, b);
	return c == 0 ? a.d < b.d : c > 0;
}

void init()
{
	V.clear();
	ret.clear();
}

void read()
{
	double a, b;
	cin >> L >> N;
	for (int i = 0; i < N; ++i) cin >> a >> b, V.push_back({ a, b });
}

void GrahamScan()
{
	sort(V.begin(), V.end(), [](point& a, point& b)
		{ return a.y < b.y || (a.y == b.y && a.x < b.x); });
	for (int i = 1; i < N; ++i) V[i].d = dist(V[0], V[i]);
	sort(V.begin() + 1, V.end(), cmp);

	for (int i = 0; i < N; ++i)
	{
		int m = ret.size();
		while (m >= 2 && cross(ret[m - 2], ret[m - 1], V[i]) <= 0)
		{
			ret.pop_back();
			--m;
		}
		ret.emplace_back(V[i]);
	}
	ret.emplace_back(V[0]);
}

void print()
{
	double ans = 0;
	for (int i = 1; i < ret.size(); ++i) ans += dist(ret[i - 1], ret[i]);

	cout << setprecision(5) << fixed << max(ans, L) << "\n";
}

int main()
{
	cin >> T;
	while (T--)
	{
		init();
		read();
		GrahamScan();
		print();
	}
}

# 參考程式碼 Andrew's Monotone Chain

#include <iostream>
#include <vector>
#include <algorithm>
#include <math.h>
#include <iomanip>

using namespace std;

struct point
{
	double x;
	double y;
};

int T, N;
double L;
vector<point> V;
vector<point> ret;

static auto fast_io = []
{
	ios::sync_with_stdio(false);
	cout.tie(nullptr);
	cin.tie(nullptr);
	return 0;
}();

void init()
{
	V.clear();
	ret.clear();
}

void read()
{
	int a, b;
	cin >> L >> N;
	for (int i = 0; i < N; ++i) cin >> a >> b, V.push_back({ a, b });
}

double cross(point& o, point& a, point& b)
{
	return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x);
}

void Andrews_Monotone_Chain()
{
	sort(V.begin(), V.end(), [](point& a, point& b) 
		{ return a.y < b.y || (a.y == b.y && a.x < b.x); });

	for (int i = 0; i < N; ++i)
	{
		int m = ret.size();
		while (m >= 2 && cross(ret[m - 2], ret[m - 1], V[i]) <= 0)
		{
			ret.pop_back();
			--m;
		}
		ret.emplace_back(V[i]);
	}

	for (int i = N - 2, t = ret.size() + 1; i >= 0; --i)
	{
		int m = ret.size();
		while (m >= t && cross(ret[m - 2], ret[m - 1], V[i]) <= 0)
		{
			ret.pop_back();
			--m;
		}
		ret.emplace_back(V[i]);
	}
}

double dis(point& a, point& b)
{
	return sqrt(pow(a.x - b.x, 2) + pow(a.y - b.y, 2));
}

void print()
{
	double ans = 0;
	for (int i = 0; i < ret.size() - 1; ++i) ans += dis(ret[i], ret[i + 1]);

	cout << setprecision(5) << fixed << max(ans, L) << "\n";
}

int main()
{
	cin >> T;
	while (T--)
	{
		init();
		read();
		Andrews_Monotone_Chain();
		print();
	}
}

# 參考資料

https://github.com/morris821028/UVa/blob/master/volume110/11096%20-%20Nails.cpp
https://blog.csdn.net/mobius_strip/article/details/8453882