一.Dijkstra 算法
1.基本配置
#define OK 1
#define ERROR 0
#define TRUE 1
#define FALSE 0
#define MAXEDGE 20
#define MAXVEX 20
#define INFINITYC 65535
typedef int Status;
typedef struct
{
int vexs[MAXVEX];
int arc[MAXVEX][MAXVEX];
int numVertexes, numEdges;
}MGraph;
/*用于存储最短路径下标的数组*/
typedef int Patharc[MAXVEX];
/*用于存储到各点最短路径权值的和*/
typedef int ShortPathTable[MAXVEX];
/*10.1 创建邻近矩阵*/
void CreateMGraph(MGraph *G)
{
int i, j;
G->numEdges=16;
G->numVertexes=9;
for (i = 0; i < G->numVertexes; i++)
{
G->vexs[i]=i;
}
for (i = 0; i < G->numVertexes; i++)
{
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];
}
}
}
2.核心实现
三个辅助数组：
- 数组
final表示V0到某个顶点Vw是否已经求得了最短路径的标记，如果V0到Vw已经有结果，则final[w] = 1 - 数组
D表示V0到某个顶点Vw的路径 - 数组
p当前定点的前去定点的下标
void ShortestPath_Dijkstra(MGraph G, int v0, Patharc *P, ShortPathTable *D)
{
int v,w,k,min;
k = 0;
/*final[w] = 1 表示求得顶点V0~Vw的最短路径*/
int final[MAXVEX];
/*1.初始化数据*/
for(v=0; v<G.numVertexes; v++)
{
//全部顶点初始化为未知最短路径状态0
final[v] = 0;
//将与V0 点有连线的顶点最短路径值;
(*D)[v] = G.arc[v0][v];
//初始化路径数组p = 0;
(*P)[v] = 0;
}
//V0到V0的路径为0
(*D)[v0] = 0;
//V0到V0 是没有路径的.
final[v0] = 1;
//v0到V0是没有路径的
(*P)[v0] = -1;
//2. 开始主循环,每次求得V0到某个顶点的最短路径
for(v=1; v<G.numVertexes; v++)
{
//当前所知距离V0顶点最近的距离
min=INFINITYC;
/*3.寻找离V0最近的顶点*/
for(w=0; w<G.numVertexes; w++)
{
if(!final[w] && (*D)[w]<min)
{
k=w;
//w顶点距离V0顶点更近
min = (*D)[w];
}
}
//将目前找到最近的顶点置为1;
final[k] = 1;
/*4.把刚刚找到v0到v1最短路径的基础上,对于v1 与 其他顶点的边进行计算,得到v0与它们的当前最短距离;*/
for(w=0; w<G.numVertexes; w++)
{
//如果经过v顶点的路径比现在这条路径长度短,则更新
if(!final[w] && (min + G.arc[k][w]<(*D)[w]))
{
//找到更短路径, 则修改D[W],P[W]
//修改当前路径的长度
(*D)[w] = min + G.arc[k][w];
(*P)[w]=k;
}
}
}
}
3.调用
int main(void)
{
printf("最短路径-Dijkstra算法\n");
int i,j,v0;
MGraph G;
Patharc P;
ShortPathTable D;
v0=0;
CreateMGraph(&G);
ShortestPath_Dijkstra(G, v0, &P, &D);
printf("最短路径路线:\n");
for(i=0;i<G.numVertexes;++i)
{
printf("v%d -> v%d : ",v0,i);
j=i;
while(P[j]!=-1)
{
printf("%d ",P[j]);
j=P[j];
}
printf("\n");
}
printf("\n最短路径权值和\n");
for(i=0;i<G.numVertexes;++i)
printf("v%d -> v%d : %d \n",G.vexs[0],G.vexs[i],D[i]);
printf("\n");
return 0;
}
4.输出
最短路径-Dijkstra算法
最短路径路线:
v0 -> v0 :
v0 -> v1 : 0
v0 -> v2 : 1 0
v0 -> v3 : 4 2 1 0
v0 -> v4 : 2 1 0
v0 -> v5 : 4 2 1 0
v0 -> v6 : 3 4 2 1 0
v0 -> v7 : 6 3 4 2 1 0
v0 -> v8 : 7 6 3 4 2 1 0
最短路径权值和
v0 -> v0 : 0
v0 -> v1 : 1
v0 -> v2 : 4
v0 -> v3 : 7
v0 -> v4 : 5
v0 -> v5 : 8
v0 -> v6 : 10
v0 -> v7 : 12
v0 -> v8 : 16
二.Floyd算法
1.核心代码
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];
}
}
}
}
}
2.调用
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;
}
3.输出
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
