最短路徑(Dijkstra算法)

 

Dijkstra算法是典型最短路算法，用於計算一個節點到其他所有節點的最短路徑。主要特點是以起始點爲中心向外層層擴展，直到擴展到終點爲止。Dijkstra算法能得出最短路徑的最優解，但由於它遍歷計算的節點很多，所以效率低。

 

頭文件:Dijkstra.h

#ifndef DIJKSTRA_H
#define DIJKSTRA_H
#define MAXVEX 20
#define MAXEDGE 20
#define INFINITY 65535
typedef struct graph{
	int Vertex[MAXVEX];
	int Edge[MAXVEX][MAXVEX];
	int NumVertex,NumEdge;
}Graph;
typedef int ShortPathTable[MAXVEX]; //用於存儲最短路徑的頂點下標
typedef int ShortWeight[MAXVEX]; //用於存儲最短路徑的權值和 
void CreateGraph(Graph *G);
void ShortestPath_Dijkstra(Graph *G,int v0,ShortPathTable *p,ShortWeight *d);
#endif //DIJKSTRA_H

實現文件:Dijkstra.cpp

#include "Dijkstra.h"
#include <stdio.h>
void CreateGraph(Graph *G)
{
	G->NumVertex = 7;
	G->NumEdge = 12;
	for(int i = 0;i < G->NumVertex;++i) //初始化頂點信息
		G->Vertex[i] = i;
	for(int i = 0;i < G->NumVertex;++i) //初始化邊表的信息
		for(int j = 0;j < G->NumVertex;++j)
		{
			if(i == j)
				G->Edge[i][j] = 0;
			else
				G->Edge[i][j] = INFINITY;
		}
	G->Edge[0][1] = 2;
	G->Edge[0][3] = 1;
	G->Edge[0][2] = 4;
	
	G->Edge[1][3] = 3;
	G->Edge[1][4] = 10;

	G->Edge[2][3] = 2;
	G->Edge[2][5] = 5;

	G->Edge[3][5] = 8;
	G->Edge[3][4] = 2;
	G->Edge[3][6] = 4;

	G->Edge[4][6] = 6;

	G->Edge[5][6] = 1;

	for(int i = 0;i < G->NumVertex;++i)
		for(int j = i;j < G->NumVertex;++j)
			G->Edge[j][i] = G->Edge[i][j];
}
void ShortestPath_Dijkstra(Graph *G,int v0,ShortPathTable *p,ShortWeight *d)
{
	int final[MAXVEX]; //如果final[w] == 1,表示求得v0 到 vw的最短路徑
	int min;
	int k = 0;
	for(int i = 0;i < MAXVEX;++i)
	{
		final[i] = 0;  //初始化v0到所有頂點的最短路徑爲未知
		(*p)[i] = 0; //初始化路徑數組爲0
		(*d)[i] = G->Edge[v0][i]; //將與v0鄰接的所有權值初始化給d;
	}
	final[0] = 1; //v0 到 v0不需要求路徑
	(*d)[v0] = 0; //v0 到 v0的路徑爲0
	for(int i = 1;i < G->NumVertex;++i)
	{
		min = INFINITY;
		for(int v = 0;v < G->NumVertex;++v)
		{
			if(!final[v] && (*d)[v] < min)
			{
				min = (*d)[v];
				k = v;
			}
		}
		final[k] = 1;
		for(int j = 0;j < G->NumVertex;++j) //修正最短路徑及距離
		{
			if(!final[j] && min + G->Edge[k][j] < (*d)[j])
			{
				(*d)[j] = min + G->Edge[k][j];
				(*p)[j] = k;
			}
		}
	}
}

main.cpp

#include "Dijkstra.h"
#include <stdio.h>
int main()
{
	Graph G;
	CreateGraph(&G);
	ShortPathTable p;
	ShortWeight d;
	int v0 = 0;
	ShortestPath_Dijkstra(&G,v0,&p,&d);
	for(int i = 0;i < G.NumVertex;++i)
	{
		printf("V%d - V%d %d ",v0,G.Vertex[i],d[i]);
		printf("\n");
	}
	return 0;
}