Kruskal算法求解最小生成树的Java实现

假设一个无向图共有V（V>1）个顶点，E条边，那么它的最小生成树（如果有的话）有V个顶点，V-1条边。Krusal算法在求解最小生成树的过程中就是不断地选出一条权重最小的边，如果将这条边插入到目前的最小生成树（此时尚未最终形成）中不形成环路，则将其插入，否则再选择次小边重复以上过程，直到最后最小生成树中有V-1条边，则最小生成树求解成功，否则，原图中不包含最小生成树。

下面是《算法导论》一书中的一个Krusal算法求解最小生成树过程的一个例子。





以下是具体的Java代码实现

package Kruskal;
/**
* 边
* @author sdu20
*
*/
public class Edge {

private int v1;
private int v2;
private int weight;

/**
* 为查找最小边专门所设
* @param weight
*/
public Edge(int weight){
this.v1 = -1;
this.v2 = -1;
this.weight = weight;
}

public Edge(int v1,int v2,int weight){
this.v1 = v1;
this.v2 = v2;
this.weight = weight;
}

public int getV1(){
return v1;
}

public int getV2(){
return v2;
}

public int getWeight(){
return weight;
}

public String toString(){
String str = "[ "+v1+" , "+v2+" , "+weight+" ]";
return str;
}

public boolean equals(Edge edge){
boolean equal = this.v1==edge.getV1() && this.v2==edge.getV2() && this.weight==edge.getWeight()
|| this.v1==edge.getV2() && this.v2==edge.getV1() && this.weight==edge.getWeight();
return equal;
}
}

package Kruskal;

import java.util.*;

public class Graph {

private int vNum;
private int edgeNum;
private LinkedList<Edge>[] edgeLinks;
private LinkedList<Edge> T;	//入选的边集
private LinkedList<Integer>[] kindLists;	//用于区分是否形成环

public Graph(int vNum){
this.vNum = vNum;
this.edgeNum = 0;
edgeLinks = new LinkedList[vNum];
for(int i = 0;i<vNum;i++){
edgeLinks[i] = new LinkedList<>();
}
}

public void insertEdge(Edge edge){
int v1 = edge.getV1();
int v2 = edge.getV2();
edgeLinks[v1].add(edge);
Edge edge2 = new Edge(v2,v1,edge.getWeight());
edgeLinks[v2].add(edge2);
edgeNum++;
}

public void deleteEdge(Edge edge){
int v1 = edge.getV1();
int v2 = edge.getV2();
edgeLinks[v1].remove(edge);

for(int i = 0;i<edgeLinks[v2].size();i++){
Edge edge2 = edgeLinks[v2].get(i);
if(edge2.equals(edge)){
edgeLinks[v2].remove(edge2);
break;
}
}
edgeNum--;
}

public void bianli(){
System.out.println("共有 "+vNum+" 个顶点， "+edgeNum+" 条边。");
for(int i = 0;i<vNum;i++){
LinkedList<Edge> list = (LinkedList<Edge>) edgeLinks[i].clone();
System.out.print(i+" : [");
while(!list.isEmpty()){
Edge edge = list.pop();
System.out.print(edge.getV2()+"("+edge.getWeight()+")"+"  ");
}
System.out.println("]");
}
}

/**
* Kruskal算法实现
*/
public void Kruskal(){

T = new LinkedList<>();
kindLists = new LinkedList[vNum];
for(int i = 0;i<vNum;i++){
kindLists[i] = new LinkedList<>();
int num = i;
kindLists[i].add(num);
}

while(edgeNum>0 && T.size()!=vNum-1){
Edge edge = this.getMinEdge();
this.deleteEdge(edge);
int v1 = edge.getV1();
int v2 = edge.getV2();

int containsV1 = -1;
int containsV2 = -1;
for(int i = 0;i<vNum;i++){
LinkedList<Integer> list = (LinkedList<Integer>) kindLists[i].clone();
if(list.contains(v1)){
containsV1 = i;
}
if(list.contains(v2)){
containsV2 = i;
}
}

if(containsV1 != containsV2){
T.add(edge);
while(!kindLists[containsV2].isEmpty()){
kindLists[containsV1].add(kindLists[containsV2].pop());
}
}
}

if(T.size() == vNum-1){
System.out.println("求最小生成树成功");
int sumWeight = 0;
LinkedList<Edge> TT = (LinkedList<Edge>) T.clone();
while(!TT.isEmpty()){
Edge ee = TT.pop();
sumWeight += ee.getWeight();
System.out.println(ee.toString());
}
System.out.println("最小生成树权重之和为： "+sumWeight);
}else{
System.out.println("没有最小生成树");
}
}

public Edge getMinEdge(){

Edge minEdge = new Edge(10000);

for(int i = 0;i<vNum;i++){
LinkedList<Edge> list = (LinkedList<Edge>) edgeLinks[i].clone();
while(!list.isEmpty()){
Edge edge = list.pop();
if(minEdge.getWeight()>edge.getWeight()){
minEdge = edge;
}
}
}

if(minEdge.getWeight()==10000)
return null;
return minEdge;
}

}

package Kruskal;

public class Main {

public static void main(String[] args) {
// TODO Auto-generated method stub

bookGraph();
//randomGraph();
}

public static void bookGraph(){

Graph graph = new Graph(9);
Edge[] edges = new Edge[14];

edges[0] = new Edge(0,1,4);
edges[1] = new Edge(0,7,8);
edges[2] = new Edge(1,2,8);
edges[3] = new Edge(1,7,11);
edges[4] = new Edge(2,3,7);
edges[5] = new Edge(2,5,4);
edges[6] = new Edge(2,8,2);
edges[7] = new Edge(3,4,9);
edges[8] = new Edge(3,5,14);
edges[9] = new Edge(4,5,10);
edges[10] = new Edge(5,6,2);
edges[11] = new Edge(6,7,1);
edges[12] = new Edge(6,8,6);
edges[13] = new Edge(7,8,7);

for(int i = 0;i<14;i++){
graph.insertEdge(edges[i]);
}

graph.bianli();
graph.Kruskal();
}

/**
* 100个点，1000条边，权重为1~100的随机数
*/
public static void randomGraph(){
Graph graph = new Graph(100);

for(int i = 0;i<1000;){

int preV = (int)(Math.random()*100);
int folV = (int)(Math.random()*100);
int weight = (int)(Math.random()*100+1);
if(preV != folV){
Edge edge = new Edge(preV,folV,weight);
try{
graph.insertEdge(edge);
i++;
}catch(Exception e){
continue;
}
}
}

graph.bianli();
graph.Kruskal();
}

}

运行截图如下



该测试用例即为前面所给的例子。