数据结构--图--最小生成树（Prim算法）

    构造连通网的最小生成树，就是使生成树的边的权值之和最小化。常用的有Prim和Kruskal算法。先看Prim算法：假设N={V,{E}}是连通网，TE是N上最小生成树中边的集合。算法从U={u0}(uo属于V)，TE={}开始，重复执行下述操作：在所有u属于U，v属于V-U的边（u,v）属于E中找到代价最小的一条边（u0,v0）并入集合TE，同时v0并入U，直至U=V为止。此时TE中必有n-1条边，T={V,{TE}}为N的最小生成树。为实现此算法，需另设一个辅助数组closedge,以记录从U到V-U中具有最小权值的边。每次有新的顶点并入U，就要更新一次closedge。

具体代码如下：

#include <iostream>#include <queue>#include <limits.h>#include "../Header.h"using namespace std;//普里姆算法构造最小生成树const int MAX_VERTEX_NUM=20;  //最大顶点数typedef enum {DG,DN,UDG,UDN} GraphKind ;//(有向图，有向网，无向图，无向网)typedef int VRType;typedef char InfoType;typedef char VertexType;#define INFINITY INT_MAXtypedef struct ArcCell{    VRType adj;  //VRType是顶点关系类型，对于无权图，用1或者0表示顶点相邻与否，对于有权图，则为权值类型    InfoType  info;//该弧相关信息指针    ArcCell(){        adj=0;        info=0;    }}ArcCell,AdjMatrix[MAX_VERTEX_NUM][MAX_VERTEX_NUM];typedef struct MGraph{    VertexType vexs[MAX_VERTEX_NUM]; //顶点向量    AdjMatrix arcs;  //邻接矩阵    int vexnum,arcnum;  //图当前的顶点数和弧数    GraphKind kind;  //图的种类标志}MGraph;//记录从顶点集U到V-U的代价最小的边的辅助数组定义typedef struct minedge{    VertexType adjvex;    VRType lowcost;}minedge,Closedge[MAX_VERTEX_NUM];int minimum(MGraph G,Closedge closedge){    int min=1;    for(int i=1;i<G.vexnum;++i){        if(closedge[i].lowcost!=0){                min=i;                break;            }    }    for(int i=min+1;i<G.vexnum;++i){        if(closedge[i].lowcost<closedge[min].lowcost&&closedge[i].lowcost>0)            min=i;    }    return min;}int LocateVex(MGraph G,VertexType v1){    for(int i=0;i<MAX_VERTEX_NUM;++i){        if(G.vexs[i]==v1)        return i;    }    return MAX_VERTEX_NUM+1;}Status CreateUDN(MGraph &G){//采用数组（邻接矩阵）表示法，构建无向网    G.kind=UDN;  //手动赋值为无向网    int vexnumber=0,arcnumber=0;    char info;    cout<<"please input the vexnumber arcnumber and info:";    cin>>vexnumber>>arcnumber>>info;    G.vexnum=vexnumber;    G.arcnum=arcnumber;    for(int i=0;i<G.vexnum;++i){ //构造顶点向量        cout<<"please input the vertex of number "<<i<<"(type char) ";        cin>>G.vexs[i];    }    for(int i=0;i<G.vexnum;++i)  //初始化邻接矩阵        for(int j=0;j<G.vexnum;++j){            G.arcs[i][j].adj=INFINITY;            G.arcs[i][j].info=0;        }    char v1,v2;    int weight=0,i=0,j=0;    char infomation;    for(int k=0;k<G.arcnum;++k){  //初始化邻接矩阵        cout<<"please input the two vertexs of the arc and it's weight "<<k+1<<" ";        cin>>v1>>v2>>weight;        i=LocateVex(G,v1);  j=LocateVex(G,v2);        G.arcs[i][j].adj=weight;        G.arcs[j][i].adj=weight;        if(info!=48){//0的ascii码为48            cout<<"please input infomation: ";            cin>>infomation;            G.arcs[i][j].info=infomation;            G.arcs[j][i].info=infomation;        }    }    return OK;}void DisMGraph(MGraph m){    for(int i=0;i<m.vexnum;++i){        for(int j=0;j<m.vexnum;++j){            cout<<m.arcs[i][j].adj<<" ";        }        cout<<endl;    }}//普里姆算法void MiniSpanTree_Prim(MGraph G,VertexType u){    int p=LocateVex(G,u);    Closedge closedge;    for(int j=0;j<G.vexnum;++j){  //辅助数组初始化        if(j!=p)            closedge[j].adjvex=u;            closedge[j].lowcost=G.arcs[p][j].adj;    }    closedge[p].lowcost=0;    closedge[p].adjvex=u;    int k=0;    for(int i=1;i<G.vexnum;++i){        k=minimum(G,closedge);        cout<<closedge[k].adjvex<<"--"<<G.vexs[k]<<endl;        closedge[k].lowcost=0;        for(int j=0;j<G.vexnum;++j){ //更新closedge数组            if(G.arcs[k][j].adj<closedge[j].lowcost&&G.arcs[k][j].adj!=0){                closedge[j].adjvex=G.vexs[k];                closedge[j].lowcost=G.arcs[k][j].adj;            }        }    }}int main(){    MGraph m;    CreateUDN(m);    DisMGraph(m);    MiniSpanTree_Prim(m,'a');    return 0;}