图算法(２):Bellman-Ford算法

　　Bellman-Ford算法是由理查德•贝尔曼（Richard Bellman） 和 莱斯特•福特 创立的，求解单源最短路径问题的一种算法。有时候这种算法也被称为 Moore-Bellman-Ford 算法，因为 Edward F. Moore 也为这个算法的发展做出了贡献。它的原理是对图进行V-1次松弛操作，得到所有可能的最短路径。其优于迪科斯彻算法的方面是边的权值可以为负数、实现简单，缺点是时间复杂度过高，高达O(VE)。

负边权操作：
　　与迪科斯彻算法不同的是，迪科斯彻算法的基本操作“拓展”是在深度上寻路，而“松弛”操作则是在广度上寻路，这就确定了贝尔曼-福特算法可以对负边进行操作而不会影响结果。
负权环判定：
　　因为负权环可以无限制的降低总花费，所以如果发现第n次操作仍可降低花销，就一定存在负权环。

Bellman-Ford 算法描述：
　　1）创建源顶点 v 到图中所有顶点的距离的集合 distSet，为图中的所有顶点指定一个距离值，初始均为 Infinite，源顶点距离为 0；
　　2）计算最短路径，执行 V - 1 次遍历；
对于图中的每条边：如果起点 u 的距离 d 加上边的权值 w 小于终点 v 的距离 d，则更新终点 v 的距离值 d；
　　3）检测图中是否有负权边形成了环，遍历图中的所有边，计算 u 至 v 的距离，如果对于 v 存在更小的距离，则说明存在环；

伪代码表示：

procedure BellmanFord(list vertices, list edges, vertex source)   // 该实现读入边和节点的列表，并向两个数组（distance和predecessor）中写入最短路径信息   // 步骤1：初始化图   for each vertex v in vertices:       if v is source then distance[v] := 0       else distance[v] := infinity       predecessor[v] := null   // 步骤2：重复对每一条边进行松弛操作   for i from 1 to size(vertices)-1:       for each edge (u, v) with weight w in edges:           if distance[u] + w < distance[v]:               distance[v] := distance[u] + w               predecessor[v] := u   // 步骤3：检查负权环   for each edge (u, v) with weight w in edges:       if distance[u] + w < distance[v]:           error "图包含了负权环"

// A C / C++ program for Bellman-Ford's single source shortest path algorithm.#include <stdio.h>#include <stdlib.h>#include <string.h>#include <limits.h>// a structure to represent a weighted edge in graphstruct Edge{    int src, dest, weight;};// a structure to represent a connected, directed and weighted graphstruct Graph{    // V-> Number of vertices, E-> Number of edges    int V, E;    // graph is represented as an array of edges.    struct Edge* edge;};// Creates a graph with V vertices and E edgesstruct Graph* createGraph(int V, int E){    struct Graph* graph = (struct Graph*) malloc( sizeof(struct Graph) );    graph->V = V;    graph->E = E;    graph->edge = (struct Edge*) malloc( graph->E * sizeof( struct Edge ) );    return graph;}// A utility function used to print the solutionvoid printArr(int dist[], int n){    printf("Vertex   Distance from Source\n");    for (int i = 0; i < n; ++i)        printf("%d \t\t %d\n", i, dist[i]);}// The main function that finds shortest distances from src to all other// vertices using Bellman-Ford algorithm.  The function also detects negative// weight cyclevoid BellmanFord(struct Graph* graph, int src){    int V = graph->V;    int E = graph->E;    int dist[V];    // Step 1: Initialize distances from src to all other vertices as INFINITE    for (int i = 0; i < V; i++)        dist[i]   = INT_MAX;    dist[src] = 0;    // Step 2: Relax all edges |V| - 1 times. A simple shortest path from src    // to any other vertex can have at-most |V| - 1 edges    for (int i = 1; i <= V-1; i++)    {        for (int j = 0; j < E; j++)        {            int u = graph->edge[j].src;            int v = graph->edge[j].dest;            int weight = graph->edge[j].weight;            if (dist[u] != INT_MAX && dist[u] + weight < dist[v])                dist[v] = dist[u] + weight;        }    }    // Step 3: check for negative-weight cycles.  The above step guarantees    // shortest distances if graph doesn't contain negative weight cycle.    // If we get a shorter path, then there is a cycle.    for (int i = 0; i < E; i++)    {        int u = graph->edge[i].src;        int v = graph->edge[i].dest;        int weight = graph->edge[i].weight;        if (dist[u] != INT_MAX && dist[u] + weight < dist[v])            printf("Graph contains negative weight cycle");    }    printArr(dist, V);    return;}// Driver program to test above functionsint main(){    /* Let us create the graph given in above example */    int V = 5;  // Number of vertices in graph    int E = 8;  // Number of edges in graph    struct Graph* graph = createGraph(V, E);    // add edge 0-1 (or A-B in above figure)    graph->edge[0].src = 0;    graph->edge[0].dest = 1;    graph->edge[0].weight = -1;    // add edge 0-2 (or A-C in above figure)    graph->edge[1].src = 0;    graph->edge[1].dest = 2;    graph->edge[1].weight = 4;    // add edge 1-2 (or B-C in above figure)    graph->edge[2].src = 1;    graph->edge[2].dest = 2;    graph->edge[2].weight = 3;    // add edge 1-3 (or B-D in above figure)    graph->edge[3].src = 1;    graph->edge[3].dest = 3;    graph->edge[3].weight = 2;    // add edge 1-4 (or A-E in above figure)    graph->edge[4].src = 1;    graph->edge[4].dest = 4;    graph->edge[4].weight = 2;    // add edge 3-2 (or D-C in above figure)    graph->edge[5].src = 3;    graph->edge[5].dest = 2;    graph->edge[5].weight = 5;    // add edge 3-1 (or D-B in above figure)    graph->edge[6].src = 3;    graph->edge[6].dest = 1;    graph->edge[6].weight = 1;    // add edge 4-3 (or E-D in above figure)    graph->edge[7].src = 4;    graph->edge[7].dest = 3;    graph->edge[7].weight = -3;    BellmanFord(graph, 0);    return 0;}

优化：
1）循环的提前跳出：
　　在实际操作中，贝尔曼-福特算法经常会在未达到V-1次前就出解，V-1其实是最大值。于是可以在循环中设置判定，在某次循环不再进行松弛时，直接退出循环，进行负权环判定。

2）队列优化：
　　求单源最短路的SPFA算法的全称是：Shortest Path Faster Algorithm。 SPFA算法是西南交通大学段凡丁于1994年发表的。松弛操作必定只会发生在最短路径前导节点松弛成功过的节点上，用一个队列记录松弛过的节点，可以避免了冗余计算。复杂度可以降低到O(kE)，k是个比较小的系数(并且在绝大多数的图中，k<=2，然而在一些精心构造的图中可能会上升到很高)

Begin  initialize-single-source(G,s);  initialize-queue(Q);  enqueue(Q,s);  while not empty(Q) do     begin      u:=dequeue(Q);      for each v∈adj[u] do         begin          tmp:=d[v];          relax(u,v);          if (tmp<>d[v]) and (not v in Q) then            enqueue(Q,v);        end;    end;End;

参考：
https://zh.wikipedia.org/w/index.php?title=%E8%B4%9D%E5%B0%94%E6%9B%BC-%E7%A6%8F%E7%89%B9%E7%AE%97%E6%B3%95&redirect=no
http://www.cnblogs.com/hxsyl/p/3248391.html
http://www.geeksforgeeks.org/dynamic-programming-set-23-bellman-ford-algorithm/
http://www.nocow.cn/index.php/%E6%9C%80%E7%9F%AD%E8%B7%AF%E5%BE%84