Floyd算法的介绍

算法的特点

弗洛伊德算法是解决任意两点间的最短路径的一种算法，可以正确处理有向图或有向图或负权(但不可存在负权回路)的最短路径问题，同时也被用于计算有向图的传递闭包。

算法的思路

通过Floyd计算图G=(V,E)中各个顶点的最短路径时，需要引入两个矩阵，矩阵S中的元素a[i][j]表示顶点i(第i个顶点)到顶点j(第j个顶点)的距离。矩阵P中的元素b[i][j]，表示顶点i到顶点j经过了b[i][j]记录的值所表示的顶点。

假设图G中顶点个数为N，则需要对矩阵D和矩阵P进行N次更新。初始时，矩阵D中顶点a[i][j]的距离为顶点i到顶点j的权值；如果i和j不相邻，则a[i][j]=∞，矩阵P的值为顶点b[i][j]的j的值。 接下来开始，对矩阵D进行N次更新。第1次更新时，如果”a[i][j]的距离” > “a[i][0]+a[0][j]”(a[i][0]+a[0][j]表示”i与j之间经过第1个顶点的距离”)，则更新a[i][j]为”a[i][0]+a[0][j]”,更新b[i][j]=b[i][0]。 同理，第k次更新时，如果”a[i][j]的距离” > “a[i][k-1]+a[k-1][j]”，则更新a[i][j]为”a[i][k-1]+a[k-1][j]”,b[i][j]=b[i][k-1]。更新N次之后，操作完成！

补充：以下面图为例子，b[i][j]中存储的是Vi~Vj之间的中介点，b[i][j]初始值为j，比如V0~V3最短路径是V0-->V2-->V1-->v3，在计算最短路径时转换为V0-->V2的距离加上V2-->V3的最短距离，接下来类似于递归，V2-->V3的最短路径就是以V1为中介点，V2-->V1的距离加上V1-->V3的距离。因此，b[0][3]=2

实例说明

将整体分为两个步骤

1.计算metrixD矩阵(两顶点之间的最短距离)和P矩阵(两顶点的中介点)

#include #include

void Create_metrixD_P(int** metrixD, int **P ,int VerNum, intEdgNum)

{int x, y, Weight, edg_count = 0;inti, j, k;for (i = 0; i < VerNum; ++i) {for (j = 0; j < VerNum; ++j) {

metrixD[i][j]=INT_MAX;

P[i][j]=j;

}

}while (edg_count

scanf("%d%d%d", &x, &y, &Weight);

metrixD[x- 1][y - 1] =Weight;

edg_count++;

}

}//Floyd algorithm

void Floyd(int **metirxD, int **P, intVerNum) {int n, x, y, temp = 0;//The triple loop looks for shortest paths and weights

for (n = 0; n < VerNum; ++n) {for (x = 0; x < VerNum; ++x) {for (y = 0; y < VerNum; ++y) {//The distance between two vertices is compared to the distance through a vertex

temp = (metirxD[x][n] == INT_MAX || metirxD[n][y] == INT_MAX) ? INT_MAX : (metirxD[x][n] +metirxD[n][y]);if (temp

metirxD[x][y] =temp;

P[x][y]=n;

}

}

}

}

}void Show_metrixD_P(int** metrixD, int **P, intVerNum)

{intx, y;

printf("metrixD:");for (x = 0; x < VerNum; ++x) {for (y = 0; y < VerNum; ++y) {if (metrixD[x][y] ==INT_MAX) {

printf("∞");

}else{

printf("%d", metrixD[x][y]);

}

}

printf("");

}

printf("P:");for (x = 0; x < VerNum; ++x) {for (y = 0; y < VerNum; ++y) {

printf("%d", P[x][y]);

}

printf("");

}

}int main(void)

{intVerNum, EdgNum, i;int** metrixD, **P;

printf("Enter the number of vertices and edges:");

scanf("%d%d", &VerNum, &EdgNum);

metrixD= (int**)malloc(VerNum * sizeof(int));

P= (int**)malloc(VerNum * sizeof(int));for (i = 0; i < VerNum; ++i) {

metrixD[i]= (int*)malloc(VerNum * sizeof(int));

P[i]= (int*)malloc(VerNum * sizeof(int));

}

printf("Input vertices and weights:");

Create_metrixD_P(metrixD, P, VerNum, EdgNum);

Floyd(metrixD, P, VerNum);

Show_metrixD_P(metrixD, P, VerNum);for (i = 0; i < VerNum; ++i) {free(metrixD[i]);free(P[i]);

}free(metrixD);free(P);return 0;

}

2.输出顶点之间的最短距离与路径

#include #include

#define VEXNUM 5

//Adjacency matrix: shows the distance between vertices

int metirxD[VEXNUM][VEXNUM] ={

INT_MAX,10, 5, INT_MAX,INT_MAX,

INT_MAX,INT_MAX,2, 1, INT_MAX,

INT_MAX,3, INT_MAX,9, 2,

INT_MAX,INT_MAX,INT_MAX,INT_MAX,4,7, INT_MAX,INT_MAX,5, INT_MAX

};//Path: passing vertex between two vertices

int P[VEXNUM][VEXNUM] ={0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4};//Floyd algorithm

voidFloyd() {int n, x, y, temp = 0;//The triple loop looks for shortest paths and weights

for (n = 0; n < VEXNUM; ++n) {for (x = 0; x < VEXNUM; ++x) {for (y = 0; y < VEXNUM; ++y) {//The distance between two vertices is compared to the distance through a vertex

temp = (metirxD[x][n] == INT_MAX || metirxD[n][y] == INT_MAX) ? INT_MAX : (metirxD[x][n] +metirxD[n][y]);if (temp

metirxD[x][y] =temp;

P[x][y]=n;

}

}

}

}

}voidShow_Path() {int x, y, temp = 0;//Output the shortest path between two vertices

for (x = 0; x < VEXNUM - 1; ++x) {for (y = x + 1; y < VEXNUM; ++y) {

printf("V%d-->V%d weight:%d path:V%d", x, y, metirxD[x][y], x);

temp=P[x][y];while (temp !=y) {

printf("-->V%d", temp);

temp=P[temp][y];

}

printf("-->V%d", y);

printf("");

}

}

}int main(void)

{

Floyd();

Show_Path();return 0;

}

完整代码

#include #include

void Create_metrixD_P(int** metrixD, int **P ,int VerNum, intEdgNum)

{int x, y, Weight, edg_count = 0;inti, j, k;for (i = 0; i < VerNum; ++i) {for (j = 0; j < VerNum; ++j) {

metrixD[i][j]=INT_MAX;

P[i][j]=j;

}

}while (edg_count

scanf("%d%d%d", &x, &y, &Weight);

metrixD[x- 1][y - 1] =Weight;

edg_count++;

}

}//Floyd algorithm

void Floyd(int **metirxD, int **P, intVerNum) {int n, x, y, temp = 0;//The triple loop looks for shortest paths and weights

for (n = 0; n < VerNum; ++n) {for (x = 0; x < VerNum; ++x) {for (y = 0; y < VerNum; ++y) {//The distance between two vertices is compared to the distance through a vertex

temp = (metirxD[x][n] == INT_MAX || metirxD[n][y] == INT_MAX) ? INT_MAX : (metirxD[x][n] +metirxD[n][y]);if (temp

metirxD[x][y] =temp;

P[x][y]=n;

}

}

}

}

}void Show_metrixD_P(int** metrixD, int **P, intVerNum)

{intx, y;

printf("metrixD:");for (x = 0; x < VerNum; ++x) {for (y = 0; y < VerNum; ++y) {if (metrixD[x][y] ==INT_MAX) {

printf("∞");

}else{

printf("%d", metrixD[x][y]);

}

}

printf("");

}

printf("P:");for (x = 0; x < VerNum; ++x) {for (y = 0; y < VerNum; ++y) {

printf("%d", P[x][y]);

}

printf("");

}

}void Show_Path(int **metirxD, int **P, intVerNum) {int x, y, temp = 0;//Output the shortest path between two vertices

for (x = 0; x < VerNum - 1; ++x) {for (y = x + 1; y < VerNum; ++y) {

printf("V%d-->V%d weight:%d path:V%d", x, y, metirxD[x][y], x);

temp=P[x][y];while (temp !=y) {

printf("-->V%d", temp);

temp=P[temp][y];

}

printf("-->V%d", y);

printf("");

}

}

}int main(void)

{intVerNum, EdgNum, i;int** metrixD, **P;

printf("Enter the number of vertices and edges:");

scanf("%d%d", &VerNum, &EdgNum);

metrixD= (int**)malloc(VerNum * sizeof(int));

P= (int**)malloc(VerNum * sizeof(int));for (i = 0; i < VerNum; ++i) {

metrixD[i]= (int*)malloc(VerNum * sizeof(int));

P[i]= (int*)malloc(VerNum * sizeof(int));

}

printf("Input vertices and weights:");

Create_metrixD_P(metrixD, P, VerNum, EdgNum);

Floyd(metrixD, P, VerNum);

Show_metrixD_P(metrixD, P, VerNum);

Show_Path(metrixD, P, VerNum);for (i = 0; i < VerNum; ++i) {free(metrixD[i]);free(P[i]);

}free(metrixD);free(P);return 0;

}