算法储备之Bellman-Ford算法

Bellman-Ford算法也是求图的单源点最短路径问题。

不同于Dijkstra算法的是它能应用于带负权值边的图，并且可以判断图中是否有负循环。

该算法的时间复杂度为O(VE) more than Dijkstra算法。

V为顶点个数

dist[V]数组记录所有顶点到源点的距离

执行V-1次循环

  每次循环对每一条边执行如下操作

    if(dist[v]>dist[u]+weight of edge uv)

      dist[v]=dist[u]+weight;

执行第V次以上操作，若dist数组有改动则图中有负循环

#include <iostream>#include <vector>using namespace std;struct Edge {int src, dest, weight;Edge(int s, int d, int w) : src(s), dest(d), weight(w) { }};struct Graph {int V;vector<Edge> edge;public:Graph(int v) : V(v) { }void addEdge(int src, int dest, int weight);void BellmanFord(int src);};void Graph::addEdge(int src, int dest, int weight){edge.push_back(Edge(src, dest, weight));}void printSolution(vector<int> & dist){cout << "Vertex   Distance from Source" << endl;for (int i = 0;i < dist.size();i++)cout << i << "\t\t" << dist[i] << endl;}void Graph::BellmanFord(int src){vector<int> dist(V, INT_MAX);dist[src] = 0;for (int i = 1;i < V;i++){for (Edge e : edge){int u = e.src;int v = e.dest;int weight = e.weight;if (dist[u] != INT_MAX && dist[u] + weight < dist[v])dist[v] = dist[u] + weight;}}for (Edge e : edge){int u = e.src;int v = e.dest;int weight = e.weight;if (dist[u] != INT_MAX && dist[u] + weight < dist[v]){cout << "Graph contains negetive weight cycle" << endl;break;}}printSolution(dist);return;}int main(){Graph graph(5);graph.addEdge(0, 1, -1);graph.addEdge(0, 2, 4);graph.addEdge(1, 2, 3);graph.addEdge(1, 3, 2);graph.addEdge(1, 4, 2);graph.addEdge(3, 2, 5);graph.addEdge(3, 1, 1);graph.addEdge(4, 3, -3);graph.BellmanFord(3);return 0;}