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 graph
struct Edge
{
int src, dest, weight;
};

// a structure to represent a connected, directed and weighted graph
struct 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 edges
struct 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 solution
void 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 cycle
void 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 functions
int 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

/article/4735946.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