Prim算法与Kruskal算法实现最小生成树

（1）问题

如题，给出一个无向图，求出最小生成树
输入格式：

无
输出格式：

输出该最小生成树的边，用Vi—Vj表示

（2）解析

最小生成树定义：
一个有 n 个结点的连通图的生成树是原图的极小连通子图，且包含原图中的所有 n 个结点，并且有保持图连通的最少的边。最小生成树可以用kruskal（克鲁斯卡尔）算法或prim（普里姆）算法求出。

Prim算法

选点，每次选不在选定的点中，到已经选定的点的集合最小边所在的点，假设从A开始选

与{A,C}最近的点是F

与{A,C,F}最近的点是D

与{A,C,D,F}最近的是B

最后只剩下E，E与B最近，完成最小生成树

Kruskal算法

选择最小的边，避免环。一开始最小的边是AC
剩下的边排序，选择DF

继续排序，选择BE

对没选中的边排序，选择CF

总共要选择5条边，排序后选择BC

（3）设计

Prim算法伪代码

 Q←V[G]
 for each u∈Q
   do key[u]←∞
 key[r]←0
 p[r]←NIL
 while Q≠Æ
   do u←EXTRACT-MIN(Q)
      for each v∈Adj[u]
        do if v∈Q and w(u,v)<key[v]
            then p[v]←u
                 key[v]←w(u,v)

Kruskal算法伪代码

A = 0
for each vertex v ∈ G.V
	MAKE-SET(v)
sort the edges of G.E into nondecreasing order by weight w
for each edge(u,v) ∈ G.E, taken in nondecreasing order by weight
	if FIND-SET(u) ≠ FIND-SET(v)
		A = A ∪ {(u,v)}
		UNION(U,V)
return A

（4）分析

时间复杂度，Prim算法的时间复杂度为O (ElogV)，Kruskal算法的时间复杂度为O（ElogE）

（5）源码

#include <iostream>
#include <vector>
#include <queue>
#include <algorithm>
using namespace std;
#define INFINITE 0xFFFFFFFF   
#define VertexData unsigned int  //顶点数据
#define UINT  unsigned int
#define vexCounts 6  //顶点数量

char vextex[] = { 'A', 'B', 'C', 'D', 'E', 'F' };
struct node 
{
    VertexData data;
    unsigned int lowestcost;
}closedge[vexCounts]; //Prim算法中的辅助信息

typedef struct 
{
    VertexData u;
    VertexData v;
    unsigned int cost;  //边的代价
}Arc;  //原始图的边信息

void AdjMatrix(unsigned int adjMat[][vexCounts])  //邻接矩阵表示法
{
    for (int i = 0; i < vexCounts; i++)   //初始化邻接矩阵
        for (int j = 0; j < vexCounts; j++)
        {
            adjMat[i][j] = INFINITE;
        }
    adjMat[0][1] = 6; adjMat[0][2] = 1; adjMat[0][3] = 5;
    adjMat[1][0] = 6; adjMat[1][2] = 5; adjMat[1][4] = 3;
    adjMat[2][0] = 1; adjMat[2][1] = 5; adjMat[2][3] = 5; adjMat[2][4] = 6; adjMat[2][5] = 4;
    adjMat[3][0] = 5; adjMat[3][2] = 5; adjMat[3][5] = 2;
    adjMat[4][1] = 3; adjMat[4][2] = 6; adjMat[4][5] = 6;
    adjMat[5][2] = 4; adjMat[5][3] = 2; adjMat[5][4] = 6;
}

int Minmum(struct node * closedge)  //返回最小代价边
{
    unsigned int min = INFINITE;
    int index = -1;
    for (int i = 0; i < vexCounts;i++)
    {
        if (closedge[i].lowestcost < min && closedge[i].lowestcost !=0)
        {
            min = closedge[i].lowestcost;
            index = i;
        }
    }
    return index;
}

void MiniSpanTree_Prim(unsigned int adjMat[][vexCounts], VertexData s)
{
    for (int i = 0; i < vexCounts;i++)
    {
        closedge[i].lowestcost = INFINITE;
    }      
    closedge[s].data = s;      //从顶点s开始
    closedge[s].lowestcost = 0;
    for (int i = 0; i < vexCounts;i++)  //初始化辅助数组
    {
        if (i != s)
        {
            closedge[i].data = s;
            closedge[i].lowestcost = adjMat[s][i];
        }
    }
    for (int e = 1; e <= vexCounts -1; e++)  //n-1条边时退出
    {
        int k = Minmum(closedge);  //选择最小代价边
        cout << vextex[closedge[k].data] << "--" << vextex[k] << endl;//加入到最小生成树
        closedge[k].lowestcost = 0; //代价置为0
        for (int i = 0; i < vexCounts;i++)  //更新v中顶点最小代价边信息
        {
            if ( adjMat[k][i] < closedge[i].lowestcost)
            {
                closedge[i].data = k;
                closedge[i].lowestcost = adjMat[k][i];
            }
        }
    }
}

void ReadArc(unsigned int  adjMat[][vexCounts],vector<Arc> &vertexArc) //保存图的边代价信息
{
    Arc * temp = NULL;
    for (unsigned int i = 0; i < vexCounts;i++)
    {
        for (unsigned int j = 0; j < i; j++)
        {
            if (adjMat[i][j]!=INFINITE)
            {
                temp = new Arc;
                temp->u = i;
                temp->v = j;
                temp->cost = adjMat[i][j];
                vertexArc.push_back(*temp);
            }
        }
    }
}

bool compare(Arc  A, Arc  B)
{
    return A.cost < B.cost ? true : false;
}

bool FindTree(VertexData u, VertexData v,vector<vector<VertexData> > &Tree)
{
    unsigned int index_u = INFINITE;
    unsigned int index_v = INFINITE;
    for (unsigned int i = 0; i < Tree.size();i++)  //检查u,v分别属于哪颗树
    {
        if (find(Tree[i].begin(), Tree[i].end(), u) != Tree[i].end())
            index_u = i;
        if (find(Tree[i].begin(), Tree[i].end(), v) != Tree[i].end())
            index_v = i;
    }
 
    if (index_u != index_v)   //u,v不在一颗树上，合并两颗树
    {
        for (unsigned int i = 0; i < Tree[index_v].size();i++)
        {
            Tree[index_u].push_back(Tree[index_v][i]);
        }
        Tree[index_v].clear();
        return true;
    }
    return false;
}

void MiniSpanTree_Kruskal(unsigned int adjMat[][vexCounts])
{
    vector<Arc> vertexArc;
    ReadArc(adjMat, vertexArc);//读取边信息
    sort(vertexArc.begin(), vertexArc.end(), compare);//边按从小到大排序
    vector<vector<VertexData> > Tree(vexCounts); //6棵独立树
    for (unsigned int i = 0; i < vexCounts; i++)
    {
        Tree[i].push_back(i);  //初始化6棵独立树的信息
    }
    for (unsigned int i = 0; i < vertexArc.size(); i++)//依次从小到大取最小代价边
    {
        VertexData u = vertexArc[i].u;  
        VertexData v = vertexArc[i].v;
        if (FindTree(u, v, Tree))//检查此边的两个顶点是否在一颗树内
        {
            cout << vextex[u] << "---" << vextex[v] << endl;//把此边加入到最小生成树中
        }   
    }
}
 
int main()
{
    unsigned int  adjMat[vexCounts][vexCounts] = { 0 };
    AdjMatrix(adjMat);   //邻接矩阵
    cout << "Prim :" << endl;
    MiniSpanTree_Prim(adjMat,0); //Prim算法，从顶点0开始.
    cout << "-------------" << endl << "Kruskal:" << endl;
    MiniSpanTree_Kruskal(adjMat);//Kruskal算法
    return 0;
}