多源最短路径Floyd算法

来源：互联网发布：加工切削参数计算软件编辑：程序博客网时间：2024/05/16 05:23

设d[i][j]为顶点 i 与顶点 j 的最短路径，设 k为i与j之间的点，那么d[i][j] = d[i][k] + d[k][j];

算法核心为：

void floyd(){    int i,j,k;    for(k = 1; k <= n; k++)    {        for(i = 1; i <= n; i++)        {            for(j = 1; j <= n; j++)            {                if(d[i][k] + d[k][j] < d[i][j])                {                    d[i][j] = d[i][k] + d[k][j];                    path[i][j] = k;                }                }                        }        }   }

path[i][j]记录的是最短路径中到j的上一个点，初始化为-1，若最后path[i][j] = -1,说明j是与源点i相邻的点

例：

#include<iostream>#include<vector>#include<windows.h>using namespace std;#define NUM 101#define INF 99999999 int d[NUM][NUM];int path[NUM][NUM];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++)            {                if(d[i][k] + d[k][j] < d[i][j])                {                    d[i][j] = d[i][k] + d[k][j];                    path[i][j] = k;                }                }                        }        }   }void findPre(int i,int j){cout<<j<<"-";    if(path[i][j] == -1)    {        cout<<i;return;    }         else     { findPre(i,path[i][j]);     }  }int main(){    int i,j;    scanf("%d%d",&n,&m);    for(i = 1; i <= n; i++)    {        for(j = 1; j <= n; j++)        {            d[i][j] = INF;            path[i][j] = -1;            }        }    for(i = 1; i <= m; i++)    {        int a,b,c;        scanf("%d%d%d",&a,&b,&c);        d[a][b] = d[b][a] = c;        }    floyd();    for(i = 1; i <= n; i++)    {        for(j = 1; j <= n; j++)        {            if(d[i][j] < INF && i == 1 && j != 1)            {                printf("%d到%d的最短路径为：%d ",i,j,d[i][j]);                printf("路径为：");                findPre(i,j);printf("\n");            }         }    }    system("pause");    return 0;        }

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