最小生成树算法（下）——Kruskal(克鲁斯卡尔)算法

概要

在我的上一篇文章最小生成树算法（上）——Prim(普里姆)算法

主要讲解对于稠密图较为合适的Prim算法。那么在接下里这片文章中我主要讲解对于稀疏图较为合适的Kruskal算法。

Kruskal算法

Kruskal算法思想概述：

如果说Prim算法可以用让一颗小树慢慢长大，那么Kruskal算法也可以用一句话来总结：将森林合并成树。就是说它比Prim算法更直接的贪心，把每个顶点看成一棵树，那么恶整个图就是一个森林。要做的就是一步一步的把最小的边收录到最小生成树且与最小生成树里的边不构成回路。

Kruskal算法过程：

1）首先构造一个有所有顶点构成的并查集（利用路径压缩），并构成的边最小堆。

2）对最小堆进行删除操作，对得到的最小边的两顶点进行回路检测，若不存在回路，则把该边收录到最小生成树中。

3）当最小生成树中的边小于顶点个数-1且最小堆中还有边是一直重复操作2）。

//Kruskal算法
int Kruskal(){
int sum_weight = 0;
UF uf(this->Nv);            //构造并查集
int cnt = 0;
//构造边集合最小堆
MinHeap minheap(this->Ne);
minheap.Create(this->Edge_Set);
int edge_cnt = 0;
//最小生成树里未收录Nv-1条边且边集合里还有边则一直循环
while(edge_cnt != this->Nv-1 && !minheap.IsEmpty()){
Edge min_edge = minheap.DeleteMin();
int start= min_edge.getStartVertex();
int end = min_edge.getEndVertex();
if(uf.CheckCircle(start,end)){//不构成回路
//对应的边收录到最小生成树
this->MST.push_back(min_edge);
this->MST[edge_cnt++] = min_edge;
sum_weight += min_edge.getWeight();
}
}
if(edge_cnt < this->Nv-1){
sum_weight = -1;
}
return sum_weight;
}

例子

我们就一下图模型来检测Kruskal算法：



由于出在考研期间，想巩固各种数据结构的各种操作，故最小堆没有用stl的make_heap()或者优先队列，而是自己手动实现最小堆，故代码看起来很长，故有关最小堆或者最大堆的相关内容请移步我的另一篇博客：堆 。同样对于并查集的相关内容请移步我的另一篇博客：并查集。在这篇博客里有关上述两个数据结构我不做解释。

全部代码：

#include <iostream>
#include <cstring>
#include <vector>
using namespace std;

const int MAX = 65535;
const int Min_Data = -32768;

//并查集
class UF{
private:
int* path;      //父亲结点
int size;       //容量
public:
//构造函数
UF(int size){
this->size = size;
this->path = new int[size+1];
memset(this->path , -1 ,sizeof(path[0])*(size+1));
}

//查找操作,采用路径压缩
int Find(int x){
if(this->path[x] < 0){
return x;
}else{
//首先查找x的父节点path[x]，然后把根节点变成path[x],之后再返回根
return this->path[x] = this->Find(this->path[x]);
}
}

//并操作
void Union(int root1 ,int root2){
root1 = this->Find(root1);
root2 = this->Find(root2);
//集合里存的是集合个数的相反数,大集合并小集合
if(this->path[root1] < this->path[root2]){//集合1比集合2大
this->path[root1] += this->path[root2];
this->path[root2] = root1;
}else{//反之集合2比集合1大
this->path[root2] += this->path[root1];
this->path[root1] = root2;
}
}

//检查是否产生回路
bool CheckCircle(int root1, int root2){
root1 = this->Find(root1);
root2 = this->Find(root2);
if(root1 == root2){//如果产生回路
return false;
}else{//不产生回路
this->Union(root1,root2);
return true;
}
}
};

//边类
class Edge{
private:
int Start_Vertex;   //起始边
int End_Vertex;     //终止边
int Weight;         //边的权重
public:
//构造函数
void Set(int start_vertex,int end_vertex ,int weight){
this->Start_Vertex = start_vertex;
this->End_Vertex = end_vertex;
this->Weight = weight;
}

//打印边
void Print(){
cout<<this->Start_Vertex<<"\t"<<this->End_Vertex<<"\t"<<this->Weight<<endl;
}

int getWeight(){
return this->Weight;
}

int getStartVertex(){
return this->Start_Vertex;
}

int getEndVertex(){
return this->End_Vertex;
}
};

class MinHeap{
private:
vector<Edge> Edges;     //边集数组
int capacity;       //最大容量
int size;           //当前规模
public:
//构造函数
MinHeap(int size){
this->capacity = size*2;
this->size = 0;
this->Edges.reserve(this->capacity);
Edge edge;
edge.Set(0,0,Min_Data);
this->Edges.push_back(edge);
}

//把Edge
为根的子堆调整为最小堆
void PreDown(int n){
Edge edge = this->Edges
; //保存下标为n的边
int parent,child;
for(parent = n ; parent*2 <= this->size ; parent = child){
child = parent*2;
if(child != this->size){
int wl = this->Edges[child].getWeight();//左孩子权重
int wr = this->Edges[child+1].getWeight();//右孩子权重
if(wr < wl){//右孩子权重小于左孩子权重
child++;//选择右孩子
}
}
//如果根节点权重小于子堆，那么找到合适位置，跳出循环
if(edge.getWeight() < this->Edges[child].getWeight()){
break;
}else{
this->Edges[parent] = this->Edges[child];
}
}
this->Edges[parent] = edge;
}

//构造最小堆
void Create(vector<Edge> &set){
for(int i = 0 ; i < set.size() ; i++){
this->Edges[++this->size] = set[i];
}
//从最后一个元素的父亲开始，把自己为根的子堆调整为最小堆
for(int i = this->size/2 ; i >= 1 ; i--){
this->PreDown(i);
}
}

//删除操作
Edge DeleteMin(){
if(this->IsEmpty()){
Edge edge;
edge.Set(0,0,Min_Data);
return edge;
}
Edge min_edge = this->Edges[1];
this->Edges[1] = this->Edges[this->size];
this->size--;
this->PreDown(1);
return min_edge;
}

//判断最小堆是否为空
bool IsEmpty(){
return this->size == 0;
}

void Print(){
cout<<"最小堆的元素有："<<this->size<<endl;
for(int i = 1 ; i <= this->size ; i++){
cout<<this->Edges[i].getStartVertex()<<" "<<this->Edges[i].getEndVertex()<<" "<<this->Edges[i].getWeight()<<endl;
}
}
};

class Graph{
private:
int** G;                //邻接矩阵
int Nv;                 //顶点数
int Ne;                 //边数
vector<Edge> MST;               //最小生成树的边集合
vector<Edge> Edge_Set;          //边集合
public:
//构造函数
Graph(int nv ,int ne){
this->Nv = nv;
this->Ne = ne;
this->MST.reserve(this->Nv-1);
this->Edge_Set.reserve(ne);
this->G = new int*[nv+1];
for(int i = 0 ; i < nv+1 ; i++){
this->G[i] = new int[nv+1];
for(int j = 0 ; j < nv+1 ; j++){
this->G[i][j] = MAX;
}
}
cout<<"请输入边与边长："<<endl;
for(int i = 0 ; i < ne ; i++){
int v1,v2,weight;
cin>>v1>>v2>>weight;
this->G[v1][v2] = this->G[v2][v1]= weight;
Edge edge;
edge.Set(v1,v2,weight);
Edge_Set.push_back(edge);
}
}

//Kruskal算法
int Kruskal(){
int sum_weight = 0;
UF uf(this->Nv);            //构造并查集
int cnt = 0;
//构造边集合最小堆
MinHeap minheap(this->Ne);
minheap.Create(this->Edge_Set);
int edge_cnt = 0;
//最小生成树里未收录Nv-1条边且边集合里还有边则一直循环
while(edge_cnt != this->Nv-1 && !minheap.IsEmpty()){
Edge min_edge = minheap.DeleteMin();
int start= min_edge.getStartVertex();
int end = min_edge.getEndVertex();
if(uf.CheckCircle(start,end)){//不构成回路
//对应的边收录到最小生成树
this->MST.push_back(min_edge);
this->MST[edge_cnt++] = min_edge;
sum_weight += min_edge.getWeight();
}
}
if(edge_cnt < this->Nv-1){
sum_weight = -1;
}
return sum_weight;
}

void Print_Kruskal(){
cout<<"Kruskal算法构造的最小生成树的边集合为："<<endl;
cout<<"源点\t终点\t权重"<<endl;
for(int i = 0 ; i < this->Nv-1 ; i++){
Edge edge = this->MST[i];
edge.Print();
}
}
};

int main()
{
int nv,ne;
cout<<"请输入顶点数与边数:"<<endl;
cin>>nv>>ne;
Graph graph(nv,ne);
int min_weight = graph.Kruskal();
if(min_weight != -1){
cout<<"Kruskal算法生成的最小生成树的权重和为："<<endl;
cout<<min_weight<<endl;
graph.Print_Kruskal();
}

return 0;
} 

截图：