Floyd(弗洛伊德)算法(C語言)

時間 2019-11-08

標籤 floyd 弗洛伊德算法 c語言简体版

原文原文鏈接

轉載：https://blog.csdn.net/qq_35644234/article/details/60875818算法

Floyd算法的介紹

算法的特色

弗洛伊德算法是解決任意兩點間的最短路徑的一種算法，能夠正確處理有向圖或有向圖或負權（但不可存在負權迴路)的最短路徑問題，同時也被用於計算有向圖的傳遞閉包。閉包

算法的思路

經過Floyd計算圖G=(V,E)中各個頂點的最短路徑時，須要引入兩個矩陣，矩陣S中的元素a[i][j]表示頂點i(第i個頂點)到頂點j(第j個頂點)的距離。矩陣P中的元素b[i][j]，表示頂點i到頂點j通過了b[i][j]記錄的值所表示的頂點。oop

假設圖G中頂點個數爲N，則須要對矩陣D和矩陣P進行N次更新。初始時，矩陣D中頂點a[i][j]的距離爲頂點i到頂點j的權值；若是i和j不相鄰，則a[i][j]=∞，矩陣P的值爲頂點b[i][j]的j的值。接下來開始，對矩陣D進行N次更新。第1次更新時，若是」a[i][j]的距離」 > 「a[i][0]+a[0][j]」(a[i][0]+a[0][j]表示」i與j之間通過第1個頂點的距離」)，則更新a[i][j]爲」a[i][0]+a[0][j]」,更新b[i][j]=b[i][0]。同理，第k次更新時，若是」a[i][j]的距離」 > 「a[i][k-1]+a[k-1][j]」，則更新a[i][j]爲」a[i][k-1]+a[k-1][j]」,b[i][j]=b[i][k-1]。更新N次以後，操做完成！spa

補充：如下面圖爲例子，b[i][j]中存儲的是Vi~Vj之間的中介點，b[i][j]初始值爲j，好比V0~V3最短路徑是V0-->V2-->V1-->v3，在計算最短路徑時轉換爲V0-->V2的距離加上V2-->V3的最短距離，接下來相似於遞歸，V2-->V3的最短路徑就是以V1爲中介點，V2-->V1的距離加上V1-->V3的距離。所以，b[0][3]=2.net

實例說明

將總體分爲兩個步驟code

1.計算metrixD矩陣(兩頂點之間的最短距離)和P矩陣(兩頂點的中介點)orm

#include <stdio.h>
#include <stdlib.h>

void Create_metrixD_P(int** metrixD, int **P ,int VerNum, int EdgNum)
{
    int x, y, Weight, edg_count = 0;
    int i, 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 < EdgNum) {
        scanf("%d%d%d", &x, &y, &Weight);
        metrixD[x - 1][y - 1] = Weight;
        edg_count++;
    }
}

//Floyd algorithm
void Floyd(int **metirxD, int **P, int VerNum) {
    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]) {
                    //Update matrix information
                    metirxD[x][y] = temp;
                    P[x][y] = n;
                }
            }
        }
    }
}

void Show_metrixD_P(int** metrixD, int **P, int VerNum)
{
    int x, y;
    printf("metrixD:\n");
    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("\n");
    }
    printf("P:\n");
    for (x = 0; x < VerNum; ++x) {
        for (y = 0; y < VerNum; ++y) {
            printf("%d ", P[x][y]);
        }
        printf("\n");
    }
}

int main(void)
{
    int VerNum, 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.輸出頂點之間的最短距離與路徑blog

#include <stdio.h>
#include <stdlib.h>

#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
void Floyd() {
    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]) {
                    //Update matrix information
                    metirxD[x][y] = temp;
                    P[x][y] = n;
                }
            }
        }
    }
}

void Show_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("\n");
        }
    }
}

int main(void)
{
    Floyd();
    Show_Path();

    return 0;
}

完整代碼

#include <stdio.h>
#include <stdlib.h>

void Create_metrixD_P(int** metrixD, int **P ,int VerNum, int EdgNum)
{
    int x, y, Weight, edg_count = 0;
    int i, 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 < EdgNum) {
        scanf("%d%d%d", &x, &y, &Weight);
        metrixD[x - 1][y - 1] = Weight;
        edg_count++;
    }
}

//Floyd algorithm
void Floyd(int **metirxD, int **P, int VerNum) {
    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]) {
                    //Update matrix information
                    metirxD[x][y] = temp;
                    P[x][y] = n;
                }
            }
        }
    }
}

void Show_metrixD_P(int** metrixD, int **P, int VerNum)
{
    int x, y;
    printf("metrixD:\n");
    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("\n");
    }
    printf("P:\n");
    for (x = 0; x < VerNum; ++x) {
        for (y = 0; y < VerNum; ++y) {
            printf("%d ", P[x][y]);
        }
        printf("\n");
    }
}

void Show_Path(int **metirxD, int **P, int VerNum) {
    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("\n");
        }
    }
}

int main(void)
{
    int VerNum, 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;
}

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