最短路径——Bellman_Ford算法

Dijkstra算法是处理单源最短路径的有效算法，但它局限于边的权值非负的情况，若图中出现权值为负的边，Dijkstra算法就会失效，求出的最短路径就可能是错的。这时候，就需要使用其他的算法来求解最短路径，Bellman-Ford算法就是其中最常用的一个。该算法由美国数学家理查德•贝尔曼（Richard Bellman, 动态规划的提出者）和小莱斯特•福特（Lester Ford）发明。Bellman-Ford算法的流程如下：

给定图G(V, E)（其中V、E分别为图G的顶点集与边集），源点s，

• 数组Distant[i]记录从源点s到顶点i的路径长度，初始化数组Distant[n]为, Distant[s]为0；

•  

• 以下操作循环执行至多n-1次，n为顶点数：

• 对于每一条边e(u, v)，如果Distant[u] + w(u, v) < Distant[v]，则另Distant[v] = Distant[u]+w(u, v)。w(u, v)为边e(u,v)的权值；

• 若上述操作没有对Distant进行更新，说明最短路径已经查找完毕，或者部分点不可达，跳出循环。否则执行下次循环；

• 为了检测图中是否存在负环路，即权值之和小于0的环路。对于每一条边e(u, v)，如果存在Distant[u] + w(u, v) < Distant[v]的边，则图中存在负环路，即是说改图无法求出单源最短路径。否则数组Distant[n]中记录的就是源点s到各顶点的最短路径长度。

可知，Bellman-Ford算法寻找单源最短路径的时间复杂度为O(V*E).

下面是该算法的伪码

procedure BellmanFord(list vertices, list edges, vertex source)
   // This implementation takes in a graph, represented as lists of vertices
   // and edges, and modifies the vertices so that their distance and
   // predecessor attributes store the shortest paths.

   // Step 1: initialize graph
   for each vertex v in vertices:
       if v is source then v.distance := 0
       else v.distance := infinity
       v.predecessor := null

   // Step 2: relax edges repeatedly
   for i from 1 to size(vertices)-1:
       for each edge uv in edges: // uv is the edge from u to v
           u := uv.source
           v := uv.destination
           if u.distance + uv.weight < v.distance:
               v.distance := u.distance + uv.weight
               v.predecessor := u

   // Step 3: check for negative-weight cycles
   for each edge uv in edges:
       u := uv.source
       v := uv.destination
       if u.distance + uv.weight < v.distance:
           error "Graph contains a negative-weight cycle"

#include <iostream>
#include <stack>
using namespace std;

const int maxnum=15;
const int INF=999999;

struct Edge
{
	int u,v;
	int weight;
};

Edge edge[maxnum];
int dist[maxnum];
int path[maxnum];

int nodesum,edgesum,source;

void Create_Graph()
{
	cout<<"输入顶点数目和边数：";
	cin>>nodesum>>edgesum;
	
	for(int i=1;i<=nodesum;i++)
	{
		dist[i]=INF;
		path[i]=-1;
	}

	cout<<"输入要源点：";
	cin>>source;
	dist[source]=0;

	for(i=1;i<=edgesum;i++)
	{
		cout<<"输入第 "<<i<<" 条边的起点和终点还有对应的权值：";
		cin>>edge[i].u>>edge[i].v>>edge[i].weight;
		if(edge[i].u == source)   //对和源点相邻的边的权值进行初始化
		{
			dist[edge[i].v]=edge[i].weight;
			path[edge[i].v]=edge[i].u;
		}
	}

	cout<<"图构建完成"<<endl;
}

bool Bellman_Ford()
{
	for(int i=1;i<nodesum;i++)
	{
		for(int j=1;j<=edgesum;j++)
		{
			if(dist[edge[j].u] + edge[j].weight < dist[edge[j].v])
			{
				dist[edge[j].v]=dist[edge[j].u] + edge[j].weight;
				path[edge[j].v]=edge[j].u;
			}
		}
	}
	//下面是判断是否含有负权回路
	for(i=1;i<=edgesum;i++)
		if(dist[edge[i].v] > dist[edge[i].u]+edge[i].weight)
			return false;
	return true;
}

void Print_Path(int v)
{
	int w;
	cout<<"The Shortest weight from "<<source<<" to "<<v<<" is "<<dist[v]<<endl;
	cout<<"The path is: ";
	stack<int> s;
	w=path[v];
	while(w != -1)
	{
		s.push(w);
		w=path[w];
	}
	while(!s.empty())
	{
		cout<<s.top()<<" ";
		s.pop();
	}
	cout<<v<<endl;
}

int main()
{
	Create_Graph();
	if(Bellman_Ford())
	{
		for(int i=2;i<=nodesum;i++)
			Print_Path(i);
		cout<<endl;
	}
	else
		cout<<"oh!"<<endl;
	return 0;
}

下面是一个测试

该图如下所示，但是我将各个顶点的标号按照顺时针方向标号