数据结构：最小生成树（Prim算法）

算法思想：

        假设N = ( p, {e} )是连通图，TE是N上最小生成树中边的集合。算法从U = { u0 } ( u0 ∈ V )开始。重复执行下述操作：在所有u∈U，v  ∈V - U的边（u,v）∈E中找一条代价最小的边（u0, v0 ）并入集合TE， 同时v0并入U，直到U = V为止。此时TE中必有n-1条边，则T = ( V, {TE} )为N的最小生成树。

#include <iostream>
#include <string>
#include <queue>
using namespace std;

#define MAXSIZE 10  //顶点最大个数
typedef string VertexType; //顶点类型
typedef int EdgeType;  //权值类型，有向图（0，1），无向图（权值，无穷大）
#define INFINITY 0x7fffffff  
//0xffffffff赋值给有符号的整型变量，其结果为-1，因为：-2^31 + 2^0 + 2^1 + ... + 2^30 = -1
typedef struct  
{
  VertexType Vexs[MAXSIZE];  //顶点向量
  EdgeType arcs[MAXSIZE][MAXSIZE]; //邻接矩阵，可看作为边表
  int iVexNum;  //顶点个数
  int iArcNum;  //边数
}MGraph;
#define SUCCESS 1
#define UNSUCCESS 0
typedef int Status;

//创建无向图
Status CreateGraph( MGraph& MG )
{
  cout << "输入顶点个数以及边数：";
  cin >> MG.iVexNum >> MG.iArcNum;
  cout << "请输入" << MG.iVexNum << "个顶点:";
  for ( int i = 0; i < MG.iVexNum; ++i )
  {
    cin >> MG.Vexs[i];
  }

  for ( int i = 0; i < MG.iVexNum; ++i )
  {
    for ( int j = 0; j < MG.iVexNum; ++j )
    {
      MG.arcs[i][j] = INFINITY;
    }
  }

  cout << "请输入由两点构成的边及其权值:";
  for ( int i = 0; i < MG.iArcNum; ++i )
  {
    VertexType first;
    VertexType second;
    EdgeType weight;
    cin >> first >> second >> weight;
    int m = GetIndexByVertexVal( MG, first );
    int n = GetIndexByVertexVal( MG, second ); 
    if ( m == -1 || n == -1 )
      return UNSUCCESS;

    MG.arcs[m][n] = MG.arcs[n][m] = weight;
  }

  return SUCCESS;
}


//普里姆算法求解最小生成树
void MiniSpanTree_Prim( const MGraph& G )
{
  int lowcost[MAXSIZE];  //保存相关顶点间的权值
  int adjvex[MAXSIZE];//保存相关顶点下标

  lowcost[0] = 0;  //将v1加入生成树中，lowcost的值为0，说明此下标的顶点已在生成树中了
  adjvex[0] = 0;  //第一个顶点下标为0

  for ( int i = 1; i < G.iVexNum; ++i )
  {
    lowcost[i] = G.arcs[0][i];
    adjvex[i] = 0;//相关点都初始化为第一个顶点
  }

  for ( int j = 1; j < G.iVexNum; ++j )
  {
    int k = 0;
    int min = INFINITY;  //最小值
    for ( int i = 0; i < G.iVexNum; ++i )
    {
      if ( lowcost[i] != 0 && lowcost[i] < min )
      {
        min = lowcost[i];
        k = i;
      }
    }

    cout << "(" << adjvex[k] << "," << k << ")";
    lowcost[k] = 0; //将下标为k的顶点加入到最小生成树中

    //重新计算lowcost
    for ( int i = 0; i < G.iVexNum; ++i )
    {
      if ( lowcost[i] != 0 && G.arcs[k][i] < lowcost[i] )
      {
        lowcost[i] = G.arcs[k][i];
        adjvex[i] = k; //将相关顶点加入
      }
    }

  }
  
}

int main()
{
  MGraph MG;
  CreateGraph( MG );

  //普里姆算法求解最小生成树
  cout << "普里姆算法求解最小生成树:" << endl;
  MiniSpanTree_Prim( MG );
  cout << endl;

  return 0;
}