最小生成树算法prim and kruskal

一.最小生成树定义：

 从不同顶点出发或搜索次序不同，可得到不同的生成树

 生成树的权：对连通网络来说，边附上权，生成树也带权，我们把生成树各边的权值总和称为生成树的权

 最小代价生成树：在一个连通网的所有生成树中， 各边的代价之和最小的那棵生成树称为该连通网的最小代价生成树（Minimum Cost Spanning Tree），简称为最小生成树(MST)。

二.最小生成树prim算法

算法思路：step1: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}(u₀属于V),TE={}开始。

              step2:在所有的u属于U，v属于V-U的边（u，v)属于E中找一条代价最小的边（u₀,v₀)并入集合TE。同时v₀并入U。

              step3:更新边（u，v)的最小值。

              step4:c重复step2 and step3直到U=V。

code:

 

  1 //MiniSpanTree_Prim.cpp

  2 //This function is to create MiniSpanTree_Prim with Prim Algorithm

  3 # include <iostream.h>

  4 # include <malloc.h>

  5 # include <conio.h>

  6 

  7 # define INFINITY 1000

  8 # define MAX_VERTEX_NUM 20

  9 # define OK 1

 10 typedef enum{DG,DN,UDG,UDN} GraphKind;

 11 typedef int EType;

 12 typedef int InfoType;

 13 typedef int VertexType;

 14 typedef int VRType;

 15 typedef int lowcost;

 16 

 17 typedef struct        //define Closedege structure

 18 {   VertexType adjvex;

 19     VRType    lowcost;

 20 }Closedge;

 21 

 22 typedef struct ArcCell    //define MGraph structure

 23 {  EType adj;

 24    InfoType *info;

 25 }ArcCell,AdjMatrix[MAX_VERTEX_NUM][MAX_VERTEX_NUM];

 26 

 27 typedef struct

 28 {  VertexType vexs[MAX_VERTEX_NUM];

 29    AdjMatrix  arcs;

 30    int vexnum,arcnum;

 31    GraphKind kind;

 32 }MGraph;

 33 

 34 int CreatUDN(MGraph &G)        //CreatUDN() sub-function

 35 {  int IncInfo,i=0,j=0,k,v1,v2,w;

 36    cout<<endl<<"Please input the number of G.vexnum (eg,G.vexnum=4) : ";

 37    cin>>G.vexnum;                  //input the number of vex

 38    cout<<"Please input the number of G.arcnum (eg,G.arcnum=4) : ";

 39    cin>>G.arcnum;        //input the number of arc

 40    for(i=0;i<G.vexnum;++i)

 41      for(j=i;j<G.vexnum;++j)

 42       {     G.arcs[i][j].adj=G.arcs[j][i].adj=INFINITY;    //initial weigh

 43      G.arcs[i][j].info=G.arcs[j][i].info=NULL;

 44       }

 45    cout<<"Please input IncInfo (0 for none)                   : ";

 46    cin>>IncInfo;        //if need information, input non-zero

 47    cout<<"Plese input arc(V1-->V2), For example: (V1=1,V2=3),(V1=2,V2=4)..."<<endl;

 48    for(k=0;k<G.arcnum;++k)    //input arc(v1,v2)

 49    {   cout<<endl<<"Please input the "<<k+1<<"th arc's v1 (0<v1<G.vexnum) : ";

 50        cin>>v1;

 51        cout<<"Please input the "<<k+1<<"th arc's v2 (0<v2<G.vexnum) : ";

 52        cin>>v2;

 53        cout<<"Please input the "<<k+1<<"th arc's weight             : ";

 54        cin>>w;

 55        i=v1;

 56        j=v2;

 57        while(i<1||i>G.vexnum||j<1||j>G.vexnum)    //if (v1,v2) illegal

 58        {   cout<<"Please input the "<<k+1<<"th arc's v1 (0<v1<G.vexnum) : ";

 59        cin>>v1;

 60        cout<<"Please input the "<<k+1<<"th arc's v2 (0<v1<G.vexnum) : ";

 61        cin>>v2;

 62        cout<<"Please input the "<<k+1<<"th arc's weight             : ";

 63        cin>>w;

 64        i=v1;

 65        j=v2;

 66        } //while end

 67        i--;

 68        j--;

 69    G.arcs[i][j].adj=G.arcs[j][i].adj=w;        //

 70    cout<<"G.arcs["<<i+1<<"]["<<j+1<<"].adj=";

 71    cout<<"G.arcs["<<j+1<<"]["<<i+1<<"].adj="<<G.arcs[j][i].adj<<endl;

 72    if(IncInfo)

 73      {   cout<<"Please input the "<<k+1<<"th arc's Info : ";

 74      cin>>*G.arcs[i][j].info;        //input information

 75      }

 76    } //for end

 77    return (OK);

 78 } //CreatUDN() end

 79 

 80 int Minimum(Closedge *closedge,int Vexnum)    //Minimum() sub-function

 81 {   int min=1,j;                        //return min (closedge[min].lowcost)

 82     if(closedge[min].lowcost==0)

 83       min++;                //closedge[min].lowcost!=0

 84     for(j=0;j<Vexnum;++j)

 85       if(closedge[j].lowcost<closedge[min].lowcost

 86           &&closedge[j].lowcost>0)

 87     min=j;

 88     return (min);

 89 } //Minimim() end

 90 

 91 int LocatedVex(MGraph G,VertexType u)    //LocatedVex() sub-fuction

 92 {  return (u);

 93 }

 94 

 95 void MiniSpanTree_Prim(MGraph G,VertexType u)    //MiniSpanTree_Prim() sub-function

 96 {  int k,j,i,Vexnum=G.vexnum;

 97    k=LocatedVex(G,u);

 98    Closedge closedge[MAX_VERTEX_NUM];

 99    for(j=0;j<G.vexnum;++j)    //initial closedge[]

100      if(j!=k)

101      {    closedge[j].adjvex=u;      // (u,j)

102     closedge[j].lowcost=G.arcs[k][j].adj;

103      }

104    closedge[k].lowcost=0;    //U include k

105    for(i=1;i<G.vexnum;++i)

106    {  k=Minimum(closedge,Vexnum);    //select k=min(closedge[vi].lowcost)

107       cout<<endl<<"("<<closedge[k].adjvex+1<<","<<k+1<<")";

108       cout<<"="<<G.arcs[closedge[k].adjvex][k].adj;

109       closedge[k].lowcost=0;    //U include k

110       for(j=0;j<G.vexnum;++j)   //renew closedge[k]

111     if(G.arcs[k][j].adj<closedge[j].lowcost)

112        {  closedge[j].adjvex=k;

113           closedge[j].lowcost=G.arcs[k][j].adj;

114        } //if end

115    } //for end

116 } //Minimun() end

117 

118 void main()              //main() function

119 {   MGraph G;

120     VertexType u=0;

121     cout<<endl<<endl<<"MiniSpanTree_Prim.cpp";

122     cout<<endl<<"====================="<<endl;

123     CreatUDN(G);        //call CreatUDN(G) function

124     cout<<endl<<"The MiniSpanTree_Prim is created as follow order:";

125     MiniSpanTree_Prim(G,u);    //call MiniSpanTree_Prim() function

126     cout<<endl<<endl<<"...OK!...";

127     getch();

128 } //main() end

 

三.最小生成树kruskal算法

算法思路：step1:假设联通网N=(V,{E})，则领最小生成树的初始状态为只有n个定点而无边的非联通图T=(V,{}),同中每个定点自成一个连通分量。

              step2:在E中选择代价最小的边，若改边的定点落在T中不同的连通分量上，则将此边加入到T中，否则舍弃此边而选择下一条代价最小的边。

　　　　　step3:依次类推知道T中所有定点都在同一连通分量上。

时间复杂度：O(eloge)

#include<iostream>

#include<vector>

#include<map>

using namespace std;

class edge

{

public:

    edge(char a,char b,int wight):ma(a),mb(b),mwight(wight){}

    edge(const edge &other)

    {

        ma = other.ma;

        mb = other.mb;

        mwight = other.mwight;

    }

    edge & operator=(const edge & other)

    {

        ma = other.ma;

        mb = other.mb;

        mwight = other.mwight;

        return *this;

    }

    char getma()

    {

        return ma;

    }

    char getmb()

    {

        return mb;

    }

private:

    char ma;

    char mb;

    int mwight;

};



void  kruskal(vector<edge> & edges,map<char,int> & vertexs,vector<edge> &myedge)

{

    

    vector<edge>::iterator begin = edges.begin();

    for (;begin != edges.end(); begin++)

    {

        int vera = vertexs[begin->getma()];

        int verb = vertexs[begin->getmb()];

        if ( vera != verb)

        {

            myedge.push_back(*begin);

            map<char,int>::iterator item = vertexs.begin();

            for(;item != vertexs.end();item++)

            {

                if (item->second == vera)

                {

                    item->second = verb;

                }

            }        

        }

    }

}



void main()

{

    char ch;

    int i;

    edge edges[] = {

        edge('a','c',1),

        edge('d','f',2),

        edge('b','e',3),

        edge('c','f',4),

        edge('a','d',5),

        edge('c','d',5),

        edge('c','b',5),

        edge('a','b',6),

        edge('c','e',6),

        edge('c','f',6)

    };

    map<char,int> vertex;

    vector<edge> myedges(edges,edges+sizeof(edges)/sizeof(edge)),result;

    for( ch='a', i =0;i<6;ch++,i++)

    {

        vertex.insert(std::pair<char,int>(ch,i));

    }

    kruskal(myedges,vertex,result);

    for (vector<edge>::iterator start = result.begin(); start != result.end(); start++)

    {

        cout<<start->getma()<<"--"<<start->getmb()<<" "<<endl;

    }

}