堆优化的Dijkstra算法

//最小堆实现Dijkstra算法
#include<stdio.h>
#include<stdlib.h>
#define MAXVEX 100 //最大顶点数
#define MAXSIZE 20
#define OK 1
#define ERROR 0
typedef char VertexType;     //顶点
typedef int EdgeType;   //权值
#define INFINITY 65535 /*用65535来代表∞*/
#define UNVISITED -1 //标记未访问
#define VISITED 1 //标记未访问

//Dist的存储结构
typedef struct
{
int index;  //顶点的索引值
int length; //当前最短路径长度
int pre;    //路径最后经过的顶点
}Dist;

typedef struct
{
int from;   //边的始点
int to; //边的终点
    EdgeType weight;    //权重
}Edge;  //边的结构

//图的结构
typedef struct
{
int numVertex;  //顶点个数
int numEdge;    //边的个数
    VertexType vexs[MAXVEX];    /*顶点表*/
int Indegree[MAXVEX];   //顶点入度
int Mark[MAXVEX];   //标记是否被访问过
    EdgeType arc[MAXVEX][MAXVEX];   //边表
    Dist D[MAXVEX];
}Graph;

typedef int Status;
typedef Edge ElemType;  //定义为Edge类型



//最小堆的存储结构
typedef struct 
{
    ElemType heapArray[MAXSIZE];
int length;
}MinHeap;

//返回依附于顶点的第一条边
Edge FirstEdge(Graph * G,int oneVertex);

//返回与preEdge有相同顶点的下一条边
Edge NextEdge(Graph * G,Edge preEdge);

//判断是否为边
bool IsEdge(Edge oneEdge);



//初始化堆数组
Status Init_heapArray(Graph * G,MinHeap * M,int s)
{
for(Edge e=FirstEdge(G,s);IsEdge(e);e=NextEdge(G,e))
    {
        M->heapArray[M->length]=e;
        M->length++;
    }
return OK;
}



//对最小堆初始化
Status Init_MinHeap(Graph * G,MinHeap * M,int s)
{
    M->length=0;
    Init_heapArray(G,M,s);
return OK;

}



int MinHeap_Leftchild(int pos)  //返回左孩子的下标
{
return 2*pos+1;
}


int MinHeap_Rightchild(int pos) //返回右孩子的下标
{
return 2*pos+2;
}


int MinHeap_Parent(int pos) //返回双亲的下标
{
return (pos-1)/2;
}



void MinHeap_SiftDown(MinHeap * M,int left)
{
int i=left; //标识父结点
int j=MinHeap_Leftchild(i); //用于记录关键值较小的子结点
    ElemType temp=M->heapArray[i];  //保存父结点
while(j<M->length)  //过筛
    {
if((j<M->length-1)&&(M->heapArray[j].weight>M->heapArray[j+1].weight))  //若有右子结点，且小于左子结点
        {
            j++;    //j指向右子结点
        }
if(temp.weight>M->heapArray[j].weight)  //如果父结点大于子结点的值则交换位置
        {
            M->heapArray[i]=M->heapArray[j];
            i=j;
            j=MinHeap_Leftchild(j);
        }
else    //堆序性满足时则跳出
        {
break;
        }
    }
    M->heapArray[i]=temp;
}


void MinHeap_SiftUp(MinHeap * M,int position)   //从position开始向上调整
{
int temppos=position;
    ElemType temp=M->heapArray[temppos];    //记录当前元素
while((temppos>0) && (M->heapArray[MinHeap_Parent(temppos)].weight>temp.weight))    //temppos>0,结束于根结点
    {
        M->heapArray[temppos]=M->heapArray[MinHeap_Parent(temppos)];
        temppos=MinHeap_Parent(temppos);
    }
    M->heapArray[temppos]=temp;
}


void Swap(MinHeap * M,int data1,int data2)
{
    ElemType temp;
    temp=M->heapArray[data1];
    M->heapArray[data1]=M->heapArray[data2];
    M->heapArray[data2]=temp;
}


//建立最小堆
void Create_MinHeap(MinHeap * M)
{
for(int i=M->length/2-1;i>=0;i--)
    {
        MinHeap_SiftDown(M,i);
    }
}



//插入元素
Status MinHeap_Insert(MinHeap * M,ElemType NewElem)
{
if(M->length==MAXSIZE)
    {
return ERROR;
    }
    M->heapArray[M->length]=NewElem;
    MinHeap_SiftUp(M,M->length);
    M->length++;
return OK;
}



//删除最小堆的最小值
Status MinHeap_Delete(MinHeap * M,ElemType * MinElem)
{
if(M->length==0)
    {
printf("不能删除，堆已空！\n");
return ERROR;
    }
else
    {

        Swap(M,0,--M->length);
if(M->length>1)
        {
            MinHeap_SiftDown(M,0);
        }
        *MinElem=M->heapArray[M->length];
return OK;
    }
}




//初始化图
void InitGraph(Graph * G,int numVert,int numEd )    //传入顶点个数,边数
{
    G->numVertex=numVert;
    G->numEdge=numEd;
for(int i=0;i<numVert;i++)
    {
        G->Mark[i]=UNVISITED;
        G->Indegree[i]=0;
for(int j=0;j<numVert;j++)
        {
            G->arc[i][j]=INFINITY;
if(i==j)
            {
                G->arc[i][j]=0;
            }
        }
    }
return ;
}

//初始化Dist结构
void InitDist(Graph * G,int s)  //s是源点
{
for(int i=0;i<G->numVertex;i++)
    {
        G->D[i].index=i;
        G->D[i].length=INFINITY;
        G->D[i].pre=s;
    }
    G->D[s].length=0;
}



//判断是否为边
bool IsEdge(Edge oneEdge)
{
if(oneEdge.weight>0 && oneEdge.weight!=INFINITY && oneEdge.to>=0)
    {
return true;
    }
else
    {
return false;
    }
}




//建立有向图的邻接矩阵
void CreatGraph(Graph * G)
{
int i,j,k,w;
printf("请输入%d个顶点元素：\n",G->numVertex);
for(i=0;i<G->numVertex;i++)
    {
scanf(" %c",&G->vexs[i]);
    }
for(k=0;k<G->numEdge;k++)
    {
printf("请输入边(Vi,Vj)的下标Vi,Vj,和权重w:\n");
scanf("%d%d%d",&i,&j,&w);
        G->Indegree[j]++;
        G->arc[i][j]=w;
    }
}



//返回顶点个数
int VerticesNum(Graph * G)
{
return G->numVertex;
}


//返回依附于顶点的第一条边
Edge FirstEdge(Graph * G,int oneVertex)
{
    Edge firstEdge;
    firstEdge.from=oneVertex;
for(int i=0;i<G->numVertex;i++)
    {
if(G->arc[oneVertex][i]!=0 && G->arc[oneVertex][i]!=INFINITY)
        {
            firstEdge.to=i;
            firstEdge.weight=G->arc[oneVertex][i];
break;
        }

    }
return firstEdge;
}   


//返回oneEdge的终点
int ToVertex(Edge oneEdge)
{
return oneEdge.to;
}


//返回与preEdge有相同顶点的下一条边
Edge NextEdge(Graph * G,Edge preEdge)
{
    Edge myEdge;
    myEdge.from=preEdge.from;   //边的始点与preEdge的始点相同
if(preEdge.to<G->numVertex) //如果preEdge.to+1>=G->numVertex;将不存在下一条边
for(int i=preEdge.to+1;i<G->numVertex;i++)  //找下一个arc[oneVertex][i]
        {                                           //不为0的i
if(G->arc[preEdge.from][i]!=0 && G->arc[preEdge.from][i]!=INFINITY)
            {
                myEdge.to=i;
                myEdge.weight=G->arc[preEdge.from][i];
break;
            }
        }
return myEdge;
}


//将顶点的指向边插入
void Insert(Graph * G,MinHeap * M,int oneVertex)
{
for(Edge e=FirstEdge(G,oneVertex);IsEdge(e);e=NextEdge(G,e))
    {
        MinHeap_Insert(M,e);
    }
}

void print_Dist(Graph * G);



void Dijkstra(Graph * G,MinHeap * M,int s)
{
    InitDist(G,s);  //初始化Dist数组
    Init_MinHeap(G,M,s);//初始化最小堆 
for(int i=0;i<G->numVertex;i++)
    {
bool FOUND=false;
        Edge d;
        MinHeap_Delete(M,&d);
int v=d.from;
int nv=d.to;
        G->Mark[nv]=VISITED;    //标记该顶点的标记位置为VISITED
//加入v以后需要重新刷新D中v的邻接点的最短路径
int count=0;
for(Edge e=FirstEdge(G,v);IsEdge(e);e=NextEdge(G,e))
        {
            count++;
if(G->D[ToVertex(e)].length>(G->D[v].length+e.weight))
            {
                G->D[ToVertex(e)].length=G->D[v].length+e.weight;
                G->D[ToVertex(e)].pre=v;
            }
        }
        Insert(G,M,nv);
    }
}




//输出Dist[]数组
void print_Dist(Graph * G)
{
for(int i=0;i<G->numVertex;i++)
    {
printf("元素:%c index:%d length:%d pre:%d\n",G->vexs[i],G->D[i].index,G->D[i].length,G->D[i].pre);
    }
printf("\n");
}




int main()
{
    Graph G;
    MinHeap M;
int numVert,numEd;
printf("请输入顶点数和边数：\n");
scanf("%d%d",&numVert,&numEd);
    InitGraph(&G,numVert,numEd );
    CreatGraph(&G);
    Dijkstra(&G,&M,0);  //源点设置为0
    print_Dist(&G);
return 0;
}