POJ2594 Treasure Exploratio —— 最小路径覆盖 + 传递闭包

题目链接：https://vjudge.net/problem/POJ-2594

Treasure Exploration

Time Limit: 6000MS		Memory Limit: 65536K
Total Submissions: 9005		Accepted: 3680

Description

Have you ever read any book about treasure exploration? Have you ever see any film about treasure exploration? Have you ever explored treasure? If you never have such experiences, you would never know what fun treasure exploring brings to you. 
Recently, a company named EUC (Exploring the Unknown Company) plan to explore an unknown place on Mars, which is considered full of treasure. For fast development of technology and bad environment for human beings, EUC sends some robots to explore the treasure. 
To make it easy, we use a graph, which is formed by N points (these N points are numbered from 1 to N), to represent the places to be explored. And some points are connected by one-way road, which means that, through the road, a robot can only move from one end to the other end, but cannot move back. For some unknown reasons, there is no circle in this graph. The robots can be sent to any point from Earth by rockets. After landing, the robot can visit some points through the roads, and it can choose some points, which are on its roads, to explore. You should notice that the roads of two different robots may contain some same point. 
For financial reason, EUC wants to use minimal number of robots to explore all the points on Mars. 
As an ICPCer, who has excellent programming skill, can your help EUC?

Input

The input will consist of several test cases. For each test case, two integers N (1 <= N <= 500) and M (0 <= M <= 5000) are given in the first line, indicating the number of points and the number of one-way roads in the graph respectively. Each of the following M lines contains two different integers A and B, indicating there is a one-way from A to B (0 < A, B <= N). The input is terminated by a single line with two zeros.

Output

For each test of the input, print a line containing the least robots needed.

Sample Input

1 0
2 1
1 2
2 0
0 0

Sample Output

1
1
2

Source

POJ Monthly--2005.08.28,Li Haoyuan

题解：

求最小路径覆盖。但与以往不同的是：一个点可以在多条路径上，即一个点可以被走多次，那怎么办呢？

利用Flyod算法求出传递闭包：如果A可以间接走到B，那么我们就直接把AB连起来。

这样，我们就可以按照常规的方法去求最小路径覆盖了。

代码如下：

#include <iostream>
#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <string>
#include <vector>
#include <map>
#include <set>
#include <queue>
#include <sstream>
#include <algorithm>
using namespace std;
const int INF = 2e9;
const int MOD = 1e9+7;
const int MAXN = 500+10;

int n;
int M[MAXN][MAXN], link[MAXN];
bool vis[MAXN];

bool dfs(int u)
{
    for(int i = 1; i<=n; i++)
    if(M[u][i] && !vis[i])
    {
        vis[i] = true;
        if(link[i]==-1 || dfs(link[i]))
        {
            link[i] = u;
            return true;
        }
    }
    return false;
}

int hungary()
{
    int ret = 0;
    memset(link, -1, sizeof(link));
    for(int i = 1; i<=n; i++)
    {
        memset(vis, 0, sizeof(vis));
        if(dfs(i)) ret++;
    }
    return ret;
}

void Flyod()
{
    for(int k = 1; k<=n; k++)
        for(int i = 1; i<=n; i++)
            for(int j = 1; j<=n; j++)
                M[i][j] = M[i][j]|(M[i][k]&&M[k][j]);
}

int main()
{
    int m;
    while(scanf("%d%d", &n, &m) && (n||m))
    {
        memset(M, false, sizeof(M));
        for(int i = 1; i<=m; i++)
        {
            int u, v;
            scanf("%d%d", &u, &v);
            M[u][v] = true;
        }

        Flyod();    //求出传递闭包
        int cnt = hungary();
        printf("%d
", n-cnt);
    }
}