数据结构之最短路径（Dijkstra 和 Floyed）

最新推荐文章于 2025-06-23 21:10:47 发布

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#数据结构 #最短路径 #图

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本文通过C++代码展示了Dijkstra算法和Floyd算法的实现过程，旨在帮助读者理解这两种用于解决图中路径问题的经典算法。代码示例中包含了初始化图、执行算法并输出结果的完整流程。

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原理直接百度，这里只给出c++实现

#include <iostream>
using namespace std;
#define MAXVEX 6
#define INF 10000
void Dijkstra(int cost[][MAXVEX], int numsOfNode, int sourceNode);
void Folyed( int cost[][MAXVEX], int numsOfNode);
void main()
{
	int cost[][MAXVEX]=
	{
		{0,50,10,INF,INF,INF},
		{INF,0,15,50,10,INF},
		{20,INF,0,15,INF,INF},
		{INF,20,INF,0,35,INF},
		{INF,INF,INF,30,0,INF},
		{INF,INF,INF,3,INF,0}
	};

	//Dijkstra(cost,MAXVEX,1);
	//cout<<endl;

	/*
	//每队顶点之间的最短距离，调用MAXVEX次Dijkstra
	for (int i = 0; i < MAXVEX ; i++ )
	{
		cout<<"The source is:"<<i<<endl;
		Dijkstra(cost,MAXVEX,i);
		cout<<endl;
	}
	*/

	Folyed(cost,MAXVEX);
	cout<<endl;
}

/*
// Dijkstra是指单源最短路径，即固定一个顶点，求其分别到其他所有顶点的最短路径
void Dijkstra(int cost[][MAXVEX], int numsOfNode, int sourceNode)
{
	int dist[MAXVEX];  //dist[]中存放源结点到其他结点的最短路径
	int path[MAXVEX];	// path[]中存放最短路径的中当前结点的前一个结点，可以用来求解路径的长度
	int s[MAXVEX];		//s[i]中存放0或1，0表示未找到源结点到节点Vi的最短路径，1表示已找到
	for (int i = 0; i < numsOfNode; i++)
	{
		dist[i] = cost[sourceNode][i];
		s[i]=0;
		if ( cost[sourceNode][i] < INF )
			path[i] = sourceNode;
		else
			path[i] = -1;
	}
	s[sourceNode] = 1;		//源结点的标号放入s中
	path[sourceNode] = 0;
	for ( int i = 0; i < numsOfNode; i++ )
	{
		int mindis = INF;
		int sign = -1;
		for ( int j =0; j < numsOfNode; j++ )		//选出不在s中，且具有最小距离的顶点sign
			if( s[j] == 0 && dist[j] < mindis )
			{
				sign = j;
				mindis = dist[j];
			}
		if( sign != -1)
		{
			s[sign] = 1;	//将sign加入到s中
			for( int j = 0; j < numsOfNode; j++ )   //修改不在s中结点的距离
				if( s[j] == 0)
					if( cost[sign][j] < INF && dist[sign] + cost[sign][j] < dist[j] )
					{
						dist[j] = dist[sign] + cost[sign][j];
						path[j] = sign;
					}
		}
	}
	//cout<<endl;
	//cout<<"The Dijkstra as follow:"<<endl;
	for ( int i = 0; i<numsOfNode; i++)
	{
		if( i != sourceNode )
		{
			cout<<sourceNode<<"->"<<i<<":";
			if ( s[i] == 1)
			{
				cout<<"The length of path is: "<<dist[i];
				int pre = i;
				cout<<" The Reverse Path is: ";
				while( pre != sourceNode )
				{
					cout<<pre<<",";
					pre = path[pre];
				}
				cout<<pre<<endl;
			}
			else
				cout<<" The path is not exis!"<<endl;
		}
	}
}
*/


void Folyed( int cost[][MAXVEX], int numsOfNode)
{
	int dist[MAXVEX][MAXVEX],path[MAXVEX][MAXVEX];
	for(int i = 0; i < MAXVEX; i++)
	{
		for (int j = 0; j < MAXVEX; j++)
		{
			dist[i][j] = cost[i][j];
			path[i][j] = -1;
		}
	}
	for (int k = 0; k < MAXVEX; k++)
	{
		for(int i = 0; i < MAXVEX; i++)
		{
			for (int j = 0; j < MAXVEX; j++)
			{
				if ( dist[i][j] > dist[i][k] + dist[k][j])
				{
					dist[i][j] = dist[i][k] + dist[k][j];
					path[i][j] = k;
				}
			}
		}
	}
	cout<<endl;
	cout<< "The Folyed as follow:"<<endl;
	cout<<endl;
	for(int i = 0; i < MAXVEX; i++)
	{
		for (int j = 0; j < MAXVEX; j++)
		{
			if(i != j)
			{
				cout<<i<<"->"<<j<<",";
				if (dist[i][j] == INF)
					cout<<"The path is not exis"<<endl;
				else
				{
					cout<<" The length of path is:"<<dist[i][j]<<", ";
					cout<<"The path is: "<<i<<",";
					int pre = path[i][j];
					while( pre != -1 )
					{
						cout<<pre<<",";
						pre = path[pre][j];
					}
					cout<<j<<endl;
				}
			}
		}
		cout<<endl;
	}
}

运行结果：

Dijkstra：

Floyed：