POJ 3660 Floyd +传递闭包问题
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http://poj.org/problem?id=3660
Description
N (1 ≤ N ≤ 100) cows, conveniently numbered 1..N, are participating in a programming contest. As we all know, some cows code better than others. Each cow has a certain constant skill rating that is unique among the competitors.
The contest is conducted in several head-to-head rounds, each between two cows. If cow A has a greater skill level than cow B (1 ≤ A ≤ N; 1 ≤ B ≤ N; A ≠ B), then cow A will always beat cow B.
Farmer John is trying to rank the cows by skill level. Given a list the results of M (1 ≤ M ≤ 4,500) two-cow rounds, determine the number of cows whose ranks can be precisely determined from the results. It is guaranteed that the results of the rounds will not be contradictory.
Input
* Line 1: Two space-separated integers: N and M
* Lines 2..M+1: Each line contains two space-separated integers that describe the competitors and results (the first integer, A, is the winner) of a single round of competition: A and B
Output
* Line 1: A single integer representing the number of cows whose ranks can be determined
Sample Input
5 54 34 23 21 22 5
Sample Output
2有n头牛,编号为1,2……n。其中厉害的牛要排在前面,输入A->B,则A比B厉害。求有几个牛的位置可以确定。
分析:如果有x头牛可以打败它,它可以打败y头牛,若x+y=n-1,则这头牛的位置可以唯一确定。
思路一:用floyd算发出每两个点之间的距离,最后统计时,若dis[a][b]之间无路且dis[b][a]之间无路,则该点位置不能确定。最后用点个数减去不能确定点的个数即可。
代码如下:
#include <iostream>#include <string.h>#include <stdio.h>using namespace std;const int N=110,MAX=1e9+1;int dis[N][N];int n,m;void floyd(){ for(int k=1;k<=n;++k) for(int i=1;i<=n;i++) for(int j=1;j<=n;j++) dis[i][j]=min(dis[i][j],dis[i][k]+dis[k][j]);}int main(){ while(~scanf("%d%d",&n,&m)) { for(int i=1;i<=n;i++) for(int j=1;j<=n;j++) dis[i][j]=MAX; int x,y; for(int i=1;i<=m;i++) { scanf("%d%d",&x,&y); dis[x][y]=1; } floyd(); int ans=0; for(int i=1;i<=n;i++) for(int j=1;j<=n;j++) { if(i==j) continue; if(dis[i][j]==MAX&&dis[j][i]==MAX) { ans++; break; } } printf("%d\n",n-ans); } return 0;}
方法二:
抽象为简单的floyd传递闭包算法,在加上每个顶点的出度与入度 (出度+入度=顶点数-1,则能够确定其编号)。
传递闭包的定义:G的传递闭包定义为G*=(V,E*),其中E={(i,j):图G中存在一条从i到j的路径}。
注:有关传递闭包问题可以看这里:http://www.cnblogs.com/lpshou/archive/2012/04/27/2473109.html
代码:
#include <iostream>#include <string.h>#include <stdio.h>using namespace std;const int maxnum=101;int map[maxnum][maxnum];int n,m;void floyd(){ int i,j,k; for(k=1;k<=n;k++) for(i=1;i<=n;i++) for(j=1;j<=n;j++) map[i][j]=map[i][j] || (map[i][k]&&map[k][j]);//判断i和j是否连通 /* for(int i=1;i<=n;i++) { for(int j=1;j<=m;j++) printf("%d ",map[i][j]); printf("\n"); }*/}int main(){ scanf("%d%d",&n,&m); memset(map,0,sizeof(map)); int i,j,u,v; for(i=0;i<m;i++) { scanf("%d%d",&u,&v); map[u][v]=1; } /*for(int i=1;i<=n;i++) { for(int j=1;j<=m;j++) printf("%d ",map[i][j]); printf("\n"); } printf("\n");*/ floyd(); int ans,res=0; for(i=1;i<=n;i++) { ans=0; for(j=1;j<=n;j++) { if(i==j) continue; else if(map[i][j] || map[j][i]) ans++; } if(ans==n-1)//出度+入度=n-1 res++; } printf("%d\n",res); return 0;}
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