[USACO07DEC]Sightseeing Cows G

题目描述

Farmer John has decided to reward his cows for their hard work by taking them on a tour of the big city! The cows must decide how best to spend their free time.

Fortunately, they have a detailed city map showing the L (2 ≤ L ≤ 1000) major landmarks (conveniently numbered 1… L) and the P (2 ≤ P ≤ 5000) unidirectional cow paths that join them. Farmer John will drive the cows to a starting landmark of their choice, from which they will walk along the cow paths to a series of other landmarks, ending back at their starting landmark where Farmer John will pick them up and take them back to the farm. Because space in the city is at a premium, the cow paths are very narrow and so travel along each cow path is only allowed in one fixed direction.

While the cows may spend as much time as they like in the city, they do tend to get bored easily. Visiting each new landmark is fun, but walking between them takes time. The cows know the exact fun values Fi (1 ≤ Fi ≤ 1000) for each landmark i.

The cows also know about the cowpaths. Cowpath i connects landmark L1i to L2i (in the direction L1i -> L2i ) and requires time Ti (1 ≤ Ti ≤ 1000) to traverse.

In order to have the best possible day off, the cows want to maximize the average fun value per unit time of their trip. Of course, the landmarks are only fun the first time they are visited; the cows may pass through the landmark more than once, but they do not perceive its fun value again. Furthermore, Farmer John is making the cows visit at least two landmarks, so that they get some exercise during their day off.

Help the cows find the maximum fun value per unit time that they can achieve.

输入格式

• Line 1: Two space-separated integers: L and P

• Lines 2…L+1: Line i+1 contains a single one integer: Fi

• Lines L+2…L+P+1: Line L+i+1 describes cow path i with three space-separated integers: L1i , L2i , and Ti

输出格式

• Line 1: A single number given to two decimal places (do not perform explicit rounding), the maximum possible average fun per unit time, or 0 if the cows cannot plan any trip at all in accordance with the above rules.

输入输出样例

输入 #1
5 7
30
10
10
5
10
1 2 3
2 3 2
3 4 5
3 5 2
4 5 5
5 1 3
5 2 2
输出 #1
6.00

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分析
题意：一张n个点m条边的有向图，边有花费，点有价值，点可以多次经过但价值不叠加，边花费叠加
求一个环满足收益和/花费和最大
如果知道01分数规划的话，可以一眼发现就是这种题

二分答案，我们把每条边的权变为f[i]-ans*t[i]，这样一旦答案过大，会出现负环，答案过小就是正环

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程序：

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
using namespace std;

int n,m,tot=0,f[100000],head[100000];
bool vis[100000];
double w[100000],dis[100000];

struct edge
{
	int to,d,next;
}e[100000];

void add(int x,int y,int z)
{
	e[++tot].to=y;e[tot].d=z;e[tot].next=head[x];head[x]=tot;
}

int read() {
	int x=0,f=1;
	char ch=getchar();
	while(ch<'0'||ch>'9') {
		if(ch=='-')f=-1;
		ch=getchar();
	}
	while(ch>='0'&&ch<='9') {
		x=x*10+ch-'0';
		ch=getchar();
	}
	return x*f;
}

bool spfa(int x)
{
	vis[x]=true;
	for (int i=head[x];i;i=e[i].next)
	{
		int t=e[i].to;
		if (dis[t]>dis[x]+w[i])
		{
			dis[t]=dis[x]+w[i];
			if (vis[t]||spfa(t))
			{
				vis[x]=false;
				return true;
			}
		}
	}
	vis[x]=false;
	return false;
}

bool judge()
{
	for (int i=1;i<=n;i++)
		if (spfa(i)) return true;
	return false;
}

int main()
{
	n=read();m=read();
	for (int i=1;i<=n;i++)
		f[i]=read();
	for (int i=1;i<=m;i++)
	{
		int x,y,z;
		x=read();y=read();z=read();
		add(x,y,z);
	}
	double l=0,r=20000;
	while (r-l>0.0000001)
	{
		double mid=(l+r)/2;
		for (int i=1;i<=tot;i++)
			w[i]=(double)mid*e[i].d-f[e[i].to];
		if (judge()) l=mid; else r=mid;
	}
	printf("%.2lf",l);
	return 0;
}