最短路径Dijkstra算法

                                                                      最短路径Dijkstra算法

      

      本文取自《数据结构与算法》(C语言版)(第三版)。出版社是清华大学出版社。

     本博文作为学习资料整理。

附书的截图：

        

    最短路径的Dijkstra算法的基本思想是：设S为最短路径已确定的顶点集,V-S是最短距离尚未确定的顶点集。初始时,将源点V0加入到顶点集S中,即S={V0}。在当前顶点集V-S中选择一个最短路径最小的顶点来扩充顶点集S,以保证算法按路径长度递增的次序产生各顶点的最短路径。

      Dijkstra算法的步骤示意图例如以下：

                              
        其程序例如以下:

   #include<stdio.h>
   #include<stdlib.h>
   #include<string.h>
   #define MaxVertexNum 100
   const int INF=25500000;
   typedef struct node
   {
     int adjvex;
     int hostvex;
     struct node *nextrarc;
     int info;
   }EdgeNode;

   typedef struct vnode
   {
     char vexdate;
     int pos;
     EdgeNode *firstarc;
   }VertexNode;

   typedef VertexNode AdjList[MaxVertexNum];

   typedef struct
   {
     AdjList adjlist;
     int n,e;
   }ALGraph;

   int initGraph(ALGraph* aGraph);
   int mFind(char aChar, ALGraph* aGraph);
   int createHead(char aChar, ALGraph* aGraph);
   void addBody(char aChar, int aPos, ALGraph* aGraph, int weight);
   void showGraph(ALGraph* aGraph);
   EdgeNode* isEdge(VertexNode* start, VertexNode* end);
   void Dijkstra(ALGraph* aGraph, int v);
   void Dispath(int dist[], int path[], int s[], ALGraph* aGraph, int v);
   void privatePrintPath(ALGraph* aGraph, int path[], int curr, int v);

   int main(void)
   {
     char a;
     int isFinish=0;
     int headPos=-1;
     char a1='@', a2='@', a3='@', a4='@', a5='@', a6='@', a7='@';
     ALGraph g_graph;
     initGraph(&g_graph);

     printf("Input arcs like this '(start,end:weight)',end with $
");
     while(isFinish==0)
     {
        while(1)
        {
          a=getchar();
          if(a=='$'||a=='#')
          {
            if(a=='#')
              isFinish=1;
            break;
          }
          if(a==' '||a=='
')
            continue;
          a1=a2;
          a2=a3;
          a3=a4;
          a4=a5;
          a5=a6;
          a6=a7;
          a7=a;
          if(a1=='('&&a3==','&&a5==':'&&a7==')')
          {
            if((headPos=mFind(a2,&g_graph))==-1)
              headPos=createHead(a2,&g_graph);
            addBody(a4,headPos,&g_graph,a6-48);
          }
        }
     }
     Dijkstra(&g_graph,2);
     return 0;
   }

   EdgeNode* isEdge(VertexNode* start, VertexNode* end)
   {
     EdgeNode* temEdge=start->firstarc;
     while(temEdge!=NULL)
     {
       if(temEdge->adjvex==end->pos)
         return temEdge;
       temEdge=temEdge->nextrarc;
     }
     return NULL;
   }

   void Dijkstra(ALGraph* aGraph, int v)
   {
     int dist[MaxVertexNum], path[MaxVertexNum];
     int s[MaxVertexNum];
     int mindis,i,j,u;
     EdgeNode* temEdge;
     for(i=0; i<aGraph->n; i++)
     {
       dist[i]=INF;
       s[i]=0;
       path[i]=-1;
     }

     temEdge=aGraph->adjlist[v].firstarc;
     while(temEdge!=NULL)
     {
       dist[temEdge->adjvex]=temEdge->info;
       path[temEdge->adjvex]=v;
       temEdge=temEdge->nextrarc;
     }
     s[v]=1;path[v]=v;dist[v]=0;
     for(i=0; i<aGraph->n; i++)
     {
       mindis=INF;
       for(j=0; j<aGraph->n; j++)
       {
         if(s[j]==0&&dist[j]<mindis)
         {
           u=j;
           mindis=dist[j];
         }
       }
       s[u]=1;

       for(j=0; j<aGraph->n; j++)
       {
         if(s[j]==0)
         {
           EdgeNode* temEdge=isEdge(&(aGraph->adjlist[u]), &(aGraph->adjlist[j]));
           if(temEdge!=NULL)
           {
             if((dist[u]+temEdge->info)<dist[j])
             {
               dist[j]=dist[u]+temEdge->info;
               path[j]=u;
             }
           }
         }
       }
     }
     Dispath(dist,path,s,aGraph,v);
   }

   void Dispath(int dist[], int path[], int s[], ALGraph* aGraph, int v)
   {
     int i;
     for(i=0; i<aGraph->n; i++)
     {
       printf("%c->%c(distance is %d) 
%c", aGraph->adjlist[v].vexdate,
              aGraph->adjlist[i].vexdate,dist[i],aGraph->adjlist[v].vexdate);

       privatePrintPath(aGraph,path,i,v);
       printf("
");
     }
   }

   void privatePrintPath(ALGraph* aGraph, int path[], int curr, int v)
   {
     char temChar=aGraph->adjlist[curr].vexdate;
     int temPos=curr;
     if(curr!=v)
     {
       privatePrintPath(aGraph,path,path[temPos],v);
       printf(" ->%c",temChar);
     }
   }

   void showGraph(ALGraph* aGraph)
   {
     int i=0;
     for(i=0; i<aGraph->n; i++)
     {
       EdgeNode* pos;
       printf(" %c->",aGraph->adjlist[i]);
       pos=aGraph->adjlist[i].firstarc;
       while(pos!=NULL)
       {
         printf(" %d ",pos->adjvex);
         pos=pos->nextrarc;
       }
       printf("
 ");
     }
   }

   void addBody(char aChar, int aPos, ALGraph* aGraph, int weight)
   {
     int inversePos;
     EdgeNode* node=(EdgeNode*) malloc(sizeof(EdgeNode));
     node->info=weight;
     if((node->adjvex=mFind(aChar,aGraph))==-1)
       node->adjvex=createHead(aChar,aGraph);
     node->hostvex=aPos;
     node->nextrarc=NULL;
     if(aGraph->adjlist[aPos].firstarc==NULL)
       aGraph->adjlist[aPos].firstarc=node;
     else
     {
       EdgeNode* tail=aGraph->adjlist[aPos].firstarc;
       while(tail->nextrarc!=NULL)
         tail=tail->nextrarc;
       tail->nextrarc=node;
     }
     aGraph->e++;

     inversePos=node->adjvex;
     node=(EdgeNode*) malloc(sizeof(EdgeNode));
     node->info=weight;
     node->hostvex=inversePos;
     node->adjvex=aPos;
     node->nextrarc=NULL;
     if(aGraph->adjlist[inversePos].firstarc==NULL)
       aGraph->adjlist[inversePos].firstarc=node;
     else
     {
       EdgeNode* tail=aGraph->adjlist[inversePos].firstarc;
       while(tail->nextrarc!=NULL)
         tail=tail->nextrarc;
       tail->nextrarc=node;
     }
     aGraph->e++;
   }

  int createHead(char aChar, ALGraph* aGraph)
  {
    int currPos=aGraph->n;
    aGraph->adjlist[currPos].vexdate=aChar;
    aGraph->adjlist[currPos].pos=currPos;
    aGraph->n++;
    return currPos;
  }

  int mFind(char aChar, ALGraph* aGraph)
  {
    int i=0;
    for(i=0; i<aGraph->n; i++)
    {
      if(aChar==aGraph->adjlist[i].vexdate)
        return i;
    }
    return -1;
  }

  int initGraph(ALGraph* aGraph)
  {
    int i=0;
    aGraph->e=0;
    aGraph->n=0;
    for(i=0; i<MaxVertexNum; i++)
    {
      aGraph->adjlist[i].firstarc=NULL;
    }
    return 0;
  }

      在VC2010中 C++控制台程序执行的结果：例如以下图所看到的：