照亮体育馆 Barisal Stadium

UVA10641

题目为逆时针顺序编号，这里直接将数组开两倍来处理环。（然而不知为啥开到1000也能过）

定义：

Corners[i]Corners[i]Corners[i]为体育馆点的坐标。

Lights[i]Lights[i]Lights[i]为灯的坐标及费用。

IsShineOnCur[i]IsShineOn_{Cur}[i]IsShineOnCur​[i]表示第CurCurCur盏灯是否能照到iii，用于计算范围。

RangeCurRange_{Cur}RangeCur​为第CurCurCur盏灯的照射范围，从LeftLeftLeft到RightRightRight，用于初始化dpdpdp

dp[Left][Right]dp[Left][Right]dp[Left][Right]为从第LeftLeftLeft盏灯照到第RightRightRight盏灯的最少花费。

初始化IsShineOnCur[i]\pmb{IsShineOn_{Cur}[i]}IsShineOnCur​[i]​IsShineOnCur​[i]​​IsShineOnCur​[i]：

令A、BA、BA、B两点为体育馆的两点，CCC为体育馆的一盏灯，那么CCC要想照亮边ABABAB，那么必须有0∘<∠BAC<180∘0^{\circ}< ∠BAC< 180^{\circ}0∘<∠BAC<180∘（手动转一转就知道了）



因为有：

sin(∠BAC)=AB⃗×AC⃗sin(∠BAC)=\vec{AB}×\vec{AC}sin(∠BAC)=AB×AC并且当sinθ>0sin\theta>0sinθ>0时，θ∈(0∘,180∘)\theta\in(0^{\circ},180^{\circ})θ∈(0∘,180∘)。

因此只要叉乘大于0即可。注意是逆时针的顺序。

//Cur为灯编号
void InitIsShineOn(const int& Cur) {
	memset(IsShineOn, false, sizeof(IsShineOn));
	for (int i = 1; i <= N; ++i) {
		//i以及i的下一个点位体育馆的点
		int&& Position = i % N + 1;
		//如果叉乘大于0
		IsShineOn[i] =
			(Lights[Cur].x - Corners[i].x) * (Corners[Position].y - Corners[i].y)
			>
			(Corners[Position].x - Corners[i].x) * (Lights[Cur].y - Corners[i].y);
	}
	return;
}

初始化RangeCur\pmb{Range_{Cur}}RangeCur​​RangeCur​​​RangeCur​：

扫一遍即可

void InitRange(const int& Cur) {
	if (IsShineOn[1] && IsShineOn[N]) {
		Range.Left = N;
		Range.Right = 1;
		while (IsShineOn[Range.Left]) {
			--Range.Left;
		}
		while (IsShineOn[Range.Right]) {
			++Range.Right;
		}
		++Range.Left;
		--Range.Right;
		Range.Right += N;
	}
	else {
		Range.Left = 1;
		Range.Right = N;
		while (!IsShineOn[Range.Left]) {
			++Range.Left;
		}
		while (!IsShineOn[Range.Right]) {
			--Range.Right;
		}
	}
	return;
}

dp初始化：

dp=infif(第Cur盏灯能照亮Left到Right)dp[Left][Right]=min(dp[Left][Right],Lights[i].Cost])dp=inf\\if(第Cur盏灯能照亮Left到Right)\\dp[Left][Right]=min(dp[Left][Right],Lights[i].Cost])dp=infif(第Cur盏灯能照亮Left到Right)dp[Left][Right]=min(dp[Left][Right],Lights[i].Cost])

//判断第Cur盏灯能否照亮(Left,Right)的范围
bool Judgt(const int& Left, const int& Right) {
	return
		Range.Left - N <= Left && Range.Right - N >= Right
		||
		Range.Left <= Left && Range.Right >= Right
		||
		Range.Left + N <= Left && Range.Right + N >= Right;
}
void InitDP(const int& Cur) {
	for (int Length = 0; Length < N; ++Length) {
		for (int Left = 1; Left <= 2 * N - 1; ++Left) {
			int&& Right = Left + Length;
			if (Right > 2 * N - 1) {
				break;
			}
			if (Judgt(Left, Right)) {
				dp[Left][Right] = min(dp[Left][Right], Lights[Cur].Cost);
			}
		}
	}
	return;
}

转移方程：

dp[Left][Right]=min(dp[Left][k]+dp[k+1][Right])Left≤k<Rightdp[Left][Right]=min(dp[Left][k]+dp[k+1][Right])\\Left\leq k<Rightdp[Left][Right]=min(dp[Left][k]+dp[k+1][Right])Left≤k<Right经典的区间dpdpdp

最后答案为min(dp[i][i+N−1]),(1≤1≤N)min(dp[i][i+N-1]),(1\leq 1\leq N)min(dp[i][i+N−1]),(1≤1≤N)（双倍数组处理环的答案）

AC代码：

#include<iostream>
#include<string>
#include<cstring>
#include<algorithm>
#include<vector>
#include<cmath>
#include<map>
using namespace std;
constexpr static long long inf = 0x3f3f3f3f3f3f3f3f;
struct Corner {
	double x, y;
}Corners[31];
struct Light {
	double x, y;
	long long Cost;
}Lights[1001];
struct {
	int Left, Right;
}Range;
int N, M;
bool IsShineOn[1001]{ false };
long long dp[1001][1001];
bool Input() {
	cin >> N;
	if (!N) {
		return false;
	}
	for (int i = 1; i <= N; ++i) {
		cin >> Corners[i].x >> Corners[i].y;
	}
	cin >> M;
	for (int i = 1; i <= M; ++i) {
		cin >> Lights[i].x >> Lights[i].y >> Lights[i].Cost;
	}
	return true;
}
void InitIsShineOn(const int& Cur) {
	memset(IsShineOn, false, sizeof(IsShineOn));
	for (int i = 1; i <= N; ++i) {
		int&& Position = i % N + 1;
		IsShineOn[i] =
			(Lights[Cur].x - Corners[i].x) * (Corners[Position].y - Corners[i].y)
			>
			(Corners[Position].x - Corners[i].x) * (Lights[Cur].y - Corners[i].y);
	}
	return;
}
void InitRange(const int& Cur) {
	if (IsShineOn[1] && IsShineOn[N]) {
		Range.Left = N;
		Range.Right = 1;
		while (IsShineOn[Range.Left]) {
			--Range.Left;
		}
		while (IsShineOn[Range.Right]) {
			++Range.Right;
		}
		++Range.Left;
		--Range.Right;
		Range.Right += N;
	}
	else {
		Range.Left = 1;
		Range.Right = N;
		while (!IsShineOn[Range.Left]) {
			++Range.Left;
		}
		while (!IsShineOn[Range.Right]) {
			--Range.Right;
		}
	}
	return;
}
bool Judgt(const int& Left, const int& Right) {
	return
		Range.Left - N <= Left && Range.Right - N >= Right
		||
		Range.Left <= Left && Range.Right >= Right
		||
		Range.Left + N <= Left && Range.Right + N >= Right;
}
void InitDP(const int& Cur) {
	for (int Length = 0; Length < N; ++Length) {
		for (int Left = 1; Left <= 2 * N - 1; ++Left) {
			int&& Right = Left + Length;
			if (Right > 2 * N - 1) {
				break;
			}
			if (Judgt(Left, Right)) {
				dp[Left][Right] = min(dp[Left][Right], Lights[Cur].Cost);
			}
		}
	}
	return;
}
void Compute() {
	for (int Length = 0; Length < N; ++Length) {
		for (int Left = 1; Left <= 2 * N - 1; ++Left) {
			int&& Right = Left + Length;
			if (Right > 2 * N - 1) {
				break;
			}
			for (int k = Left; k < Right; ++k) {
				if (dp[Left][k] == inf && dp[k + 1][Right] == inf) {
					continue;
				}
				dp[Left][Right] = min(dp[Left][Right], dp[Left][k] + dp[k + 1][Right]);
			}
		}
	}
}
long long DP() {
	memset(dp, 0x3f, sizeof(dp));
	for (int i = 1; i <= M; ++i) {
		InitIsShineOn(i);
		InitRange(i);
		InitDP(i);
	}
	Compute();
	long long Ans = inf;
	for (int i = 1; i <= N; ++i) {
		Ans = min(Ans, dp[i][i + N - 1]);
	}
	return Ans;
}
int main() {
	while (Input()) {
		long long&& Ans = DP();
		if (Ans == inf) {
			puts("Impossible.");
		}
		else {
			cout << Ans << endl;
		}
	}
	return 0;
}