floyd算法求图中的顶点间最短路径

算法理论

创建两个数组-d数组以及p数组

d数组存放顶点间的最短距离,p数组存放被指向顶点的前驱

图示

图来自tyrantlucifer@gmail.com

我们通过将每个顶点作为中转点判断以此顶点中转后是否会让两个顶点间的路径更短

如果能,则改变d,p数组中对应的值,

举例

在加入V2作为中转点时,我们发现V1->V3本为不可达,但通过V2中转,路径比原来的短,我们则更改d数组和p数组的值

Floyd算法也被称为3for,

代码实现

接口

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

#define MAX 32767
#define num 5
typedef struct Graph
{
	int vexNum;
	int arcNum;
	char* vexs;
	int** arcs;
}Graph;
Graph* initGraph(int vexNum) ;
void createGraph(Graph* G, char* vexs, int* arcs);
void DFS(Graph* G, int* visited, int index) ;
void floyd(Graph* G);

接口实现

Graph* initGraph(int vexNum) {
	Graph* G = (Graph*)malloc(sizeof(Graph));
	G->vexs = (char*)malloc(sizeof(char) * vexNum);
	G->arcs = (int**)malloc(sizeof(int*) * vexNum);
	for (int i = 0; i < vexNum; i++) {
		G->arcs[i] = (int*)malloc(sizeof(int) * vexNum);
	}
	G->vexNum = vexNum;
	G->arcNum = 0;
	return G;
}
void createGraph(Graph* G, char* vexs, int* arcs) {
	for (int i = 0; i < G->vexNum; i++) {
		G->vexs[i] = vexs[i];
		for (int j = 0; j < G->vexNum; j++) {
			G->arcs[i][j] = *(arcs + i * G->vexNum + j);
			if (!G->arcs[i][j] && G->arcs[i][j] != MAX) {
				G->arcNum++;
			}
		}
	}
	G->arcNum /= 2;
}
void DFS(Graph* G, int* visited, int index) {
	printf("%c\t", G->vexs[index]);
	visited[index] = 1;
	for (int i = 0; i < G->vexNum; i++) {
		if (G->arcs[index][i] > 0 && G->arcs[index][i] != MAX && !visited[i]) {
			DFS(G, visited, i);
		}
	}
}
void floyd(Graph* G) {
	int d[num][num];
	int p[num][num];
	//初始化两个数组
	for (int i = 0; i < G->vexNum; i++) {
		for (int j = 0; j < G->vexNum; j++) {
			d[i][j] = G->arcs[i][j];
			if (G->arcs[i][j] > 0 && G->arcs[i][j] != MAX) {
				p[i][j] = i;
			}
			else {
				p[i][j] = -1;
			}
		}
	}
	for (int i = 0; i < G->vexNum; i++) {
		for (int j = 0; j < G->vexNum; j++) {
			printf("%d ", d[i][j]);
		}
		printf("\n");
	}

	for (int i = 0; i < G->vexNum; i++) {
		for (int j = 0; j < G->vexNum; j++) {
			printf("%d ", p[i][j]);
		}
		printf("\n");
	}
	//算法执行
	for (int i = 0; i < G->vexNum; i++) {
		for (int j = 0; j < G->vexNum; j++) {
			for (int k = 0; k < G->vexNum; k++) {
				//以i为中转点
				if (d[j][i] + d[i][k] < d[j][k]) {
					d[j][k] = d[j][i] + d[i][k];
					p[j][k] = p[i][k];//改变前驱
				}
			}
		}
	}
	for (int i = 0; i < G->vexNum; i++) {
		for (int j = 0; j < G->vexNum; j++) {
			printf("%d ", d[i][j]);
		}
		printf("\n");
	}

	for (int i = 0; i < G->vexNum; i++) {
		for (int j = 0; j < G->vexNum; j++) {
			printf("%d ", p[i][j]);
		}
		printf("\n");
	}
}

测试

void text() {
	Graph* G = initGraph(4);
	int* visited = (int*)malloc(sizeof(int) * G->vexNum);
	for (int i = 0; i < G->vexNum; i++)
		visited[i] = 0;
	int arcs[4][4] = {
			0, 1, MAX, 3,
			1, 0, 2, 2,
			MAX, 2, 0, 8,
			3, 2, 8, 0
	};
	char str[5] = "1234";
	createGraph(G, str, (int*)arcs);
	DFS(G, visited, 0);
	printf("\n");
	floyd(G);
}
int main()
{
	text();
	return 0;
}