单源最短路径Dijkstra算法，多源最短路径Floyd算法

1.单源最短路径

（1）无权图的单源最短路径

 

 

 1 /*无权单源最短路径*/
 2 void UnWeighted(LGraph Graph, Vertex S)
 3 {
 4     std::queue<Vertex> Q;
 5     Vertex V;
 6     PtrToAdjVNode W;
 7     Q.push(S);
 8     while (!Q.empty())
 9     {
10         V = Q.front();
11         Q.pop();
12         for (W = Graph->G[V].FirstEdge; W; W = W->Next)
13         if (dist[W->AdjV] != -1)
14         {
15             dist[W->AdjV] += dist[V] + 1;
16             path[W->AdjV] = V;
17             Q.push(W->AdjV);
18         }
19 
20     }
21 
22 }

 

 

 

函数：返回还未被收录顶点中dist最小者

 1 Vertex FindMinDist(MGraph Graph, int dist[], int collected[])
 2 {
 3     /*返回未被收录顶点中dist最小者*/
 4     Vertex MinV, V;
 5     int MinDist = INFINITY;
 6 
 7 
 8     for (V = 0; V < Graph->Nv; ++V)
 9     {
10         if (collected[V] == false && dist[V] < MinDist)
11         {
12             MinDist = dist[V];
13             MinV = V;                //更新对应顶点
14         }
15     }
16     if (MinDist < INFINITY)            //若找到最小值
17         return MinV;
18     else
19         return -1;
20 }

（2）有权图的单源最短路径
单源最短路径Dijkstra算法

 1 /*单源最短路径Dijkstra算法*/
 2 /*dist数组存储起点到这个顶点的最小路径长度*/
 3 bool Dijkstra(MGraph Graph, int dist[], int path[], Vertex S)
 4 {
 5     int collected[MaxVertexNum];
 6     Vertex V, W;
 7 
 8     /*初始化，此处默认邻接矩阵中不存在的边用INFINITY表示*/
 9     for (V = 0; V < Graph->Nv; V++)
10     {
11         dist[V] = Graph->G[S][V];            //用S顶点对应的行向量分别初始化dist数组
12         if (dist[V] < INFINITY)                //如果(S, V)这条边存在
13             path[V] = S;                //将V顶点的父节点初始化为S
14         else
15             path[V] = -1;                //否则初始化为-1
16         collected[V] = false;            //false表示这个顶点还未被收入集合
17     }
18 
19 
20     /*现将起点S收录集合*/
21     dist[S] = 0;                //S到S的路径长度为0
22     collected[S] = true;
23 
24     while (1)
25     {
26         V = FindMinDist(Graph, dist, collected);    //V=未被收录顶点中dist最小者
27         if (V == -1)        //如果这样的V不存在
28             break;
29         collected[V] = true;        //将V收录进集合
30         for (W = 0; W < Graph->Nv; W++)        //对V的每个邻接点
31         {
32             /*如果W是V的邻接点且未被收录*/
33             if (collected[W] == false && Graph->G[V][W] < INFINITY)
34             {
35                 if (Graph->G[V][W] < 0)        //若有负边，不能正常解决，返回错误标记
36                     return false ;
37                 if (dist[W] > dist[V] + Graph->G[V][W])
38                 {
39                     dist[W] = dist[V] + Graph->G[V][W];        //更新dist[W]
40                     path[W] = V;            //更新S到W的路径
41                 }
42             }
43         }
44     }
45     return true;
46 }

 2.多源最短路径Floyd算法

 1 /*多源最短路径*/
 2 bool Floyd(MGraph Graph, WeightType D[][MaxVertexNum], Vertex path[][MaxVertexNum])
 3 {
 4     Vertex i, j, k;
 5 
 6     /*初始化*/
 7     for (i = 0; i < Graph->Nv; i++)
 8     for (j = 0; j < Graph->Nv; j++)
 9     {
10         D[i][j] = Graph->G[i][j];
11         path[i][j] = -1;
12     }
13 
14     for (k = 0; k < Graph->Nv; k++)
15     for (i = 0; i < Graph->Nv; i++)
16     for (j = 0; j < Graph->Nv; j++)
17     {
18         if (D[i][k] + D[k][j] < D[i][j])
19             D[i][j] = D[i][k] + D[k][j];
20         if (i == j && D[i][j] < 0)        //若发现负值圈，不能正常解决，返回错误标记
21             return false;
22         path[i][j] = k;
23     }
24     return true;        //算法执行完毕，返回正确标记
25 }