22、数据结构预算法 - 图 最短路径 弗洛伊德(Floyd)算法
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2024-03-17 10:10:28
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最短路径
弗洛伊德(Floyd)算法
弗洛伊德思想,在原有的邻接矩阵上,将任意两点之间的最短距离都给算出来
如下图:原来存的V1->V5 是V1 到V5直接相连的,现在V1 ->V0 ->V5 = 3<5,所以更新。
算法的公式
现在一下图进行分析
原来的邻接矩阵存储
需要经过3次遍历比较
经过全部遍历比较之后得到最终的结果
代码实现
#include <stdio.h>
#include "stdio.h"
#include "stdlib.h"
#include "math.h"
#include "time.h"
#define OK 1
#define ERROR 0
#define TRUE 1
#define FALSE 0
#define MAXEDGE 20
#define MAXVEX 20
#define INFINITYC 65535
typedef int Status; /* Status是函数的类型,其值是函数结果状态代码,如OK等 */
typedef struct
{
int vexs[MAXVEX];
int arc[MAXVEX][MAXVEX];
int numVertexes, numEdges;
}MGraph;
typedef int Patharc[MAXVEX][MAXVEX];
typedef int ShortPathTable[MAXVEX][MAXVEX];
/* 11.1 构成邻近矩阵 */
void CreateMGraph(MGraph *G)
{
int i, j;
/* printf("请输入边数和顶点数:"); */
G->numEdges=16;
G->numVertexes=9;
for (i = 0; i < G->numVertexes; i++)/* 初始化图 */
{
G->vexs[i]=i;
}
for (i = 0; i < G->numVertexes; i++)/* 初始化图 */
{
for ( j = 0; j < G->numVertexes; j++)
{
if (i==j)
G->arc[i][j]=0;
else
G->arc[i][j] = G->arc[j][i] = INFINITYC;
}
}
G->arc[0][1]=1;
G->arc[0][2]=5;
G->arc[1][2]=3;
G->arc[1][3]=7;
G->arc[1][4]=5;
G->arc[2][4]=1;
G->arc[2][5]=7;
G->arc[3][4]=2;
G->arc[3][6]=3;
G->arc[4][5]=3;
G->arc[4][6]=6;
G->arc[4][7]=9;
G->arc[5][7]=5;
G->arc[6][7]=2;
G->arc[6][8]=7;
G->arc[7][8]=4;
for(i = 0; i < G->numVertexes; i++)
{
for(j = i; j < G->numVertexes; j++)
{
G->arc[j][i] =G->arc[i][j];
}
}
}
/* 11. 2
Floyd算法,求网图G中各顶点v到其余顶点w的最短路径P[v][w]及带权长度D[v][w]。
Patharc 和 ShortPathTable 都是二维数组;
*/
void ShortestPath_Floyd(MGraph G, Patharc *P, ShortPathTable *D)
{
int v,w,k;
/* 1. 初始化D与P 矩阵*/
for(v=0; v<G.numVertexes; ++v)
{
for(w=0; w<G.numVertexes; ++w)
{
/* D[v][w]值即为对应点间的权值 */
(*D)[v][w]=G.arc[v][w];
/* 初始化P P[v][w] = w*/
(*P)[v][w]=w;
}
}
//2.K表示经过的中转顶点
for(k=0; k<G.numVertexes; ++k)
{
for(v=0; v<G.numVertexes; ++v)
{
for(w=0; w<G.numVertexes; ++w)
{
/*如果经过下标为k顶点路径比原两点间路径更短 */
if ((*D)[v][w]>(*D)[v][k]+(*D)[k][w])
{
/* 将当前两点间权值设为更小的一个 */
(*D)[v][w]=(*D)[v][k]+(*D)[k][w];
/* 路径设置为经过下标为k的顶点 */
(*P)[v][w]=(*P)[v][k];
}
}
}
}
}
验证
int main(void)
{
printf("Hello,最短路径弗洛伊德Floyd算法");
int v,w,k;
MGraph G;
Patharc P;
ShortPathTable D; /* 求某点到其余各点的最短路径 */
CreateMGraph(&G);
ShortestPath_Floyd(G,&P,&D);
//打印所有可能的顶点之间的最短路径以及路线值
printf("各顶点间最短路径如下:\n");
for(v=0; v<G.numVertexes; ++v)
{
for(w=v+1; w<G.numVertexes; w++)
{
printf("v%d-v%d weight: %d ",v,w,D[v][w]);
//获得第一个路径顶点下标
k=P[v][w];
//打印源点
printf(" path: %d",v);
//如果路径顶点下标不是终点
while(k!=w)
{
//打印路径顶点
printf(" -> %d",k);
//获得下一个路径顶点下标
k=P[k][w];
}
//打印终点
printf(" -> %d\n",w);
}
printf("\n");
}
//打印最终变换后的最短路径D数组
printf("最短路径D数组\n");
for(v=0; v<G.numVertexes; ++v)
{
for(w=0; w<G.numVertexes; ++w)
{
printf("%d\t",D[v][w]);
}
printf("\n");
}
//打印最终变换后的最短路径P数组
printf("最短路径P数组\n");
for(v=0; v<G.numVertexes; ++v)
{
for(w=0; w<G.numVertexes; ++w)
{
printf("%d ",P[v][w]);
}
printf("\n");
}
return 0;
}
结果
Hello,最短路径弗洛伊德Floyd算法各顶点间最短路径如下:
v0-v1 weight: 1 path: 0 -> 1
v0-v2 weight: 4 path: 0 -> 1 -> 2
v0-v3 weight: 7 path: 0 -> 1 -> 2 -> 4 -> 3
v0-v4 weight: 5 path: 0 -> 1 -> 2 -> 4
v0-v5 weight: 8 path: 0 -> 1 -> 2 -> 4 -> 5
v0-v6 weight: 10 path: 0 -> 1 -> 2 -> 4 -> 3 -> 6
v0-v7 weight: 12 path: 0 -> 1 -> 2 -> 4 -> 3 -> 6 -> 7
v0-v8 weight: 16 path: 0 -> 1 -> 2 -> 4 -> 3 -> 6 -> 7 -> 8
v1-v2 weight: 3 path: 1 -> 2
v1-v3 weight: 6 path: 1 -> 2 -> 4 -> 3
v1-v4 weight: 4 path: 1 -> 2 -> 4
v1-v5 weight: 7 path: 1 -> 2 -> 4 -> 5
v1-v6 weight: 9 path: 1 -> 2 -> 4 -> 3 -> 6
v1-v7 weight: 11 path: 1 -> 2 -> 4 -> 3 -> 6 -> 7
v1-v8 weight: 15 path: 1 -> 2 -> 4 -> 3 -> 6 -> 7 -> 8
v2-v3 weight: 3 path: 2 -> 4 -> 3
v2-v4 weight: 1 path: 2 -> 4
v2-v5 weight: 4 path: 2 -> 4 -> 5
v2-v6 weight: 6 path: 2 -> 4 -> 3 -> 6
v2-v7 weight: 8 path: 2 -> 4 -> 3 -> 6 -> 7
v2-v8 weight: 12 path: 2 -> 4 -> 3 -> 6 -> 7 -> 8
v3-v4 weight: 2 path: 3 -> 4
v3-v5 weight: 5 path: 3 -> 4 -> 5
v3-v6 weight: 3 path: 3 -> 6
v3-v7 weight: 5 path: 3 -> 6 -> 7
v3-v8 weight: 9 path: 3 -> 6 -> 7 -> 8
v4-v5 weight: 3 path: 4 -> 5
v4-v6 weight: 5 path: 4 -> 3 -> 6
v4-v7 weight: 7 path: 4 -> 3 -> 6 -> 7
v4-v8 weight: 11 path: 4 -> 3 -> 6 -> 7 -> 8
v5-v6 weight: 7 path: 5 -> 7 -> 6
v5-v7 weight: 5 path: 5 -> 7
v5-v8 weight: 9 path: 5 -> 7 -> 8
v6-v7 weight: 2 path: 6 -> 7
v6-v8 weight: 6 path: 6 -> 7 -> 8
v7-v8 weight: 4 path: 7 -> 8
最短路径D数组
0 1 4 7 5 8 10 12 16
1 0 3 6 4 7 9 11 15
4 3 0 3 1 4 6 8 12
7 6 3 0 2 5 3 5 9
5 4 1 2 0 3 5 7 11
8 7 4 5 3 0 7 5 9
10 9 6 3 5 7 0 2 6
12 11 8 5 7 5 2 0 4
16 15 12 9 11 9 6 4 0
最短路径P数组
0 1 1 1 1 1 1 1 1
0 1 2 2 2 2 2 2 2
1 1 2 4 4 4 4 4 4
4 4 4 3 4 4 6 6 6
2 2 2 3 4 5 3 3 3
4 4 4 4 4 5 7 7 7
3 3 3 3 3 7 6 7 7
6 6 6 6 6 5 6 7 8
7 7 7 7 7 7 7 7 8
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