Currency Exchange(判断有无正环)

题目描述

Several currency exchange points are working in our city. Let us suppose that each point specializes in two particular currencies and performs exchange operations only with these currencies. There can be several points specializing in the same pair of currencies. Each point has its own exchange rates, exchange rate of A to B is the quantity of B you get for 1A. Also each exchange point has some commission, the sum you have to pay for your exchange operation. Commission is always collected in source currency.
For example, if you want to exchange 100 US Dollars into Russian Rubles at the exchange point, where the exchange rate is 29.75, and the commission is 0.39 you will get (100 - 0.39) * 29.75 = 2963.3975RUR.
You surely know that there are N different currencies you can deal with in our city. Let us assign unique integer number from 1 to N to each currency. Then each exchange point can be described with 6 numbers: integer A and B - numbers of currencies it exchanges, and real R AB, C AB, R BA and C BA - exchange rates and commissions when exchanging A to B and B to A respectively.
Nick has some money in currency S and wonders if he can somehow, after some exchange operations, increase his capital. Of course, he wants to have his money in currency S in the end. Help him to answer this difficult question. Nick must always have non-negative sum of money while making his operations.

Input

The first line of the input contains four numbers: N - the number of currencies, M - the number of exchange points, S - the number of currency Nick has and V - the quantity of currency units he has. The following M lines contain 6 numbers each - the description of the corresponding exchange point - in specified above order. Numbers are separated by one or more spaces. 1<=S<=N<=100, 1<=M<=100, V is real number, 0<=V<=10 3.
For each point exchange rates and commissions are real, given with at most two digits after the decimal point, 10 -2<=rate<=10 2, 0<=commission<=10 2.
Let us call some sequence of the exchange operations simple if no exchange point is used more than once in this sequence. You may assume that ratio of the numeric values of the sums at the end and at the beginning of any simple sequence of the exchange operations will be less than 10 4.

Output

If Nick can increase his wealth, output YES, in other case output NO to the output file.

Sample Input

3 2 1 20.0
1 2 1.00 1.00 1.00 1.00
2 3 1.10 1.00 1.10 1.00

Sample Output

YES

题意：

有多种汇币，汇币之间可以相互转换，公式为 a = (b - c) * r ，求出是否存在正权回路，且货币数增加。

思路：

使用ford 的判圈方法。总共需要循环所有的边n - 1次，松弛一遍，松弛完之后如果还出现满足条件的情况则一定存在正环

代码如下：

#include<iostream>
#include<vector>
#include<cstring>
#define INF 0x3f3f3f3f
#include<queue>
#include<cstdio>
#include<cmath>
using namespace std;
int a[1005], b[1005];
double c[1005], d[1005];
double dis[1005];
int n, m, s, k;double v;
bool djs()
{memset(dis, 0, sizeof dis);  //dis表示兑换成每一种类之后的钱数dis[s] = v;    //转化s种类的钱数为vfor(int i = 1; i < n; i++){bool flag = false;for(int j = 1; j <= k; j++){if(dis[b[j]] < (dis[a[j]] - d[j])*c[j]){  //先松弛一遍dis[b[j]] = (dis[a[j]] - d[j])*c[j];flag = true;}}}for(int i = 1; i <= k; i++){        if(dis[b[i]] < (dis[a[i]] - d[i])*c[i])  //松弛完一遍后，如果还满足条件，则一定存在正环return true;}return false;  //否则，return false
}
int main()
{scanf("%d %d %d %lf",&n, &m, &s, &v);k = 1;int A, B;double C, D, E, F;for(int i = 0; i < m; i++){cin >> A >> B >> C >> D >> E >> F;a[k] = A;b[k] = B;c[k] = C;d[k++] = D;a[k] = B;b[k] = A;c[k] = E;d[k++] = F;}if(djs()) cout << "YES" << endl;else cout << "NO" <<endl;return 0;
}