圖算法(２):Bellman-Ford算法

　　Bellman-Ford算法是由理查德•貝爾曼（Richard Bellman） 和 萊斯特•福特 創立的，求解單源最短路徑問題的一種算法。有時候這種算法也被稱爲 Moore-Bellman-Ford 算法，因爲 Edward F. Moore 也爲這個算法的發展做出了貢獻。它的原理是對圖進行V-1次鬆弛操作，得到所有可能的最短路徑。其優於迪科斯徹算法的方面是邊的權值可以爲負數、實現簡單，缺點是時間複雜度過高，高達O(VE)。

負邊權操作：
　　與迪科斯徹算法不同的是，迪科斯徹算法的基本操作“拓展”是在深度上尋路，而“鬆弛”操作則是在廣度上尋路，這就確定了貝爾曼-福特算法可以對負邊進行操作而不會影響結果。
負權環判定：
　　因爲負權環可以無限制的降低總花費，所以如果發現第n次操作仍可降低花銷，就一定存在負權環。

Bellman-Ford 算法描述：
　　1）創建源頂點 v 到圖中所有頂點的距離的集合 distSet，爲圖中的所有頂點指定一個距離值，初始均爲 Infinite，源頂點距離爲 0；
　　2）計算最短路徑，執行 V - 1 次遍歷；
對於圖中的每條邊：如果起點 u 的距離 d 加上邊的權值 w 小於終點 v 的距離 d，則更新終點 v 的距離值 d；
　　3）檢測圖中是否有負權邊形成了環，遍歷圖中的所有邊，計算 u 至 v 的距離，如果對於 v 存在更小的距離，則說明存在環；

僞代碼表示：

procedure BellmanFord(list vertices, list edges, vertex source)
   // 該實現讀入邊和節點的列表，並向兩個數組（distance和predecessor）中寫入最短路徑信息

   // 步驟1：初始化圖
   for each vertex v in vertices:
       if v is source then distance[v] := 0
       else distance[v] := infinity
       predecessor[v] := null

   // 步驟2：重複對每一條邊進行鬆弛操作
   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

   // 步驟3：檢查負權環
   for each edge (u, v) with weight w in edges:
       if distance[u] + w < distance[v]:
           error "圖包含了負權環"

// 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;
}

優化：
1）循環的提前跳出：
　　在實際操作中，貝爾曼-福特算法經常會在未達到V-1次前就出解，V-1其實是最大值。於是可以在循環中設置判定，在某次循環不再進行鬆弛時，直接退出循環，進行負權環判定。

2）隊列優化：
　　求單源最短路的SPFA算法的全稱是：Shortest Path Faster Algorithm。 SPFA算法是西南交通大學段凡丁於1994年發表的。鬆弛操作必定只會發生在最短路徑前導節點鬆弛成功過的節點上，用一個隊列記錄鬆弛過的節點，可以避免了冗餘計算。複雜度可以降低到O(kE)，k是個比較小的係數(並且在絕大多數的圖中，k<=2，然而在一些精心構造的圖中可能會上升到很高)

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;

參考：
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