本文详细介绍了Kruskal算法的实现过程，包括边类的定义、图类的优化，以及最小生成树的构建。通过重构代码，提高了算法效率，并展示了如何检测图中是否存在环。同时，提供了测试用例来验证算法的正确性和实用性。
这篇文章实现最小生成树的Kruskal算法
Kruskal算法：
Kruskal算法思想不同于Prim算法，Kruskal算法是一种按照连通网中边的权值的递增顺序构造最小生成树的算法。
Kruskal算法的基本步骤 :
假设G=(V,E)是一个具有n个顶点的连通网，T=(U,TE)是G的最小生成树。
令集合U的初值为U=V，即包含有G中全部顶点，集合TE的初值为TE={}。
然后，将图G中的边按权值从小到大的顺序依次选取，若选取的边使生成树T不形成回路，则把它并入TE中，保留作为T的一条边；
若选取的边使生成树T形成回路，则将其舍弃，如此进行下去，直到TE中包含有n-1条边为止，此时的T即为最小生成树。
另外对上一篇文章中的程序进行了重构优化。
展示代码如下：
点是否在边中调整到边类
package com.zas.test.tree;
/**
* 边的定义
* @author zas
*/
public class Edge {
//起点
protected Point startPoint;
//终点
protected Point endPoint;
public Edge() {
super();
}
public Edge(Point startPoint, Point endPoint) {
super();
this.startPoint = startPoint;
this.endPoint = endPoint;
}
public Point getStartPoint() {
return startPoint;
}
public void setStartPoint(Point startPoint) {
this.startPoint = startPoint;
}
public Point getEndPoint() {
return endPoint;
}
public void setEndPoint(Point endPoint) {
this.endPoint = endPoint;
}
/* (non-Javadoc)
* @see java.lang.Object#equals(java.lang.Object)
* 只有起止点相同的边才是同一条边
*/
@Override
public boolean equals(Object obj) {
if(obj instanceof Edge){
Edge outEdge = (Edge)obj;
if(outEdge.startPoint.equals(this.startPoint)){
if(outEdge.endPoint.equals(this.endPoint)){
return true;
}
}
}
return false;
}
@Override
public String toString() {
return "Edge [startPoint=" + startPoint + ", endPoint=" + endPoint
+ "]";
}
/**
* 点是否在边中
* @param p
* @param edge
* @return
*/
public boolean isPointInEdge(Point p) {
if(p.equals(this.getStartPoint()) || p.equals(this.getEndPoint())){
return true;
}
return false;
}
/**
* @param args
*/
public static void main(String[] args) {
}
}
判断图是否含有环，图顶点度的计算调整到图类
package com.zas.test.tree;
import java.util.ArrayList;
import java.util.List;
/**
* 图的定义
* @author zas
*/
public class Graph {
//图的点列表
List<Point> pointList;
//图的边列表
List<EdgeWithWeight> edgeList;
public Graph() {
super();
}
public Graph(List<Point> pointList, List<EdgeWithWeight> edgeList) {
super();
this.pointList = pointList;
this.edgeList = edgeList;
}
/**
* 添加一个点到图中
* @param p
*/
public void addPoint(Point p) {
if(pointList == null){
pointList = new ArrayList<Point>();
}
pointList.add(p);
}
/**
* 从图中删除一个点
* @param p
*/
public void removePoint(Point p){
for (Point point : pointList) {
if(p.equals(point)){
pointList.remove(point);
break;
}
}
}
/**
* 添加一条边到图中
* @param p
*/
public void addEdge(EdgeWithWeight e) {
if(edgeList == null){
edgeList = new ArrayList<EdgeWithWeight>();
}
edgeList.add(e);
}
/**
* 从图中删除一条边
* @param p
*/
public void removeEdge(EdgeWithWeight e) {
for (EdgeWithWeight edge : edgeList) {
if(e.equals(edge)){
edgeList.remove(edge);
break;
}
}
}
/**
* 点是否存在于图中
* @param p
* @return
*/
public boolean isPointInGraph(Point p) {
if(null == pointList || pointList.size() < 1){
return false;
}
for (Point point : pointList) {
if(p.equals(point)){
return true;
}
}
return false;
}
/**
* 点是否存在于图中
* @param p
* @return
*/
public boolean isEdgeInGraph(EdgeWithWeight e) {
if(null == edgeList || edgeList.size() < 1){
return false;
}
for (EdgeWithWeight edge : edgeList) {
if(e.equals(edge)){
return true;
}
}
return false;
}
public List<Point> getPointList() {
return pointList;
}
public void setPointList(List<Point> pointList) {
this.pointList = pointList;
}
public List<EdgeWithWeight> getEdgeList() {
return edgeList;
}
public void setEdgeList(List<EdgeWithWeight> edgeList) {
this.edgeList = edgeList;
}
@Override
public String toString() {
return "Graph [pointList=" + pointList + ", edgeList=" + edgeList + "]";
}
@Override
public Graph clone() {
Graph g = new Graph();
for (Point p : pointList) {
g.addPoint(p);
}
for (EdgeWithWeight e : edgeList) {
g.addEdge(e);
}
return g;
}
/**
* 检测是否产生回路
如果存在回路，则必存在一个子图，是一个环路。环路中所有顶点的度>=2。
n算法：
第一步：删除所有度<=1的顶点及相关的边，并将另外与这些边相关的其它顶点的度减一。
第二步：将度数变为1的顶点排入队列，并从该队列中取出一个顶点重复步骤一。
如果最后还有未删除顶点，则存在环，否则没有环。
n算法分析：
由于有m条边，n个顶点。
i)如果m>=n，则根据图论知识可直接判断存在环路。（证明：如果没有环路，则该图必然是k棵树 k>=1。根据树的性质，边的数目m = n-k。k>=1，所以：m<n）
ii)如果m<n 则按照上面的算法每删除一个度为0的顶点操作一次（最多n次），或每删除一个度为1的顶点（同时删一条边）操作一次（最多m次）。这两种操作的总数不会超过m+n。由于m<n，所以算法复杂度为O(n)。
* @param this
* @return
*/
public boolean isGraphHasCircle() {
//若图中没有顶点，或者只有一个顶点则没有回路
if(this.getPointList() == null || this.getPointList().size() < 2){
return false;
}
if(this.getEdgeList().size() > this.getPointList().size()){
return true;
}
Graph g = this.clone();
int pointsLeftCount = g.getPointList().size();
while(pointsLeftCount > 0){
//一次遍历如没有删除一个度小于2的点，则结束循环
boolean endFlag = true;
Point pointForRemove = null;
for (Point p : g.getPointList()) {
//计算顶点的度
if(g.getCountForPoint(p) <= 1){
//为了规避最后一个顶点被删除是的并发异常 采用标记删除
pointForRemove = p;
//删除之后从新遍历顶点
endFlag = false;
break;
}
}
if(endFlag){
break;
}else{
g.removePoint(pointForRemove);
List<EdgeWithWeight> edgeForRemoveList = new ArrayList<EdgeWithWeight>();
for (EdgeWithWeight e : g.getEdgeList()) {
if(e.isPointInEdge(pointForRemove)){
edgeForRemoveList.add(e);
}
}
for (EdgeWithWeight edgeWithWeight : edgeForRemoveList) {
g.removeEdge(edgeWithWeight);
}
}
pointsLeftCount = g.getPointList().size();
}
if(g.getPointList().size() > 0){
return true;
}else{
return false;
}
}
/**
* 计算顶点的度
* @param p
* @return
*/
private int getCountForPoint(Point p) {
int count = 0;
for (EdgeWithWeight e : this.getEdgeList()) {
if(e.isPointInEdge(p)){
count = count + 1;
}
}
return count;
}
/**
* @param args
*/
public static void main(String[] args) {
}
}
Prim算法调整
package com.zas.test.tree;
/**
* Prim算法实现的是找出一个有权重连通图中的最小生成树，即：具有最小权重且连接到所有结点的树。
* @author zas
*/
public class Prim {
//一个要找最小生成树的图
Graph graph;
public Prim(Graph graph) {
super();
this.graph = graph;
}
public Graph getGraph() {
return graph;
}
public void setGraph(Graph graph) {
this.graph = graph;
}
/**
* 首先以一个结点作为最小生成树的初始结点，然后以迭代的方式找出与最小生成树中各结点权重最小边，
* 并加入到最小生成树中。加入之后如果产生回路则跳过这条边，选择下一个结点。
* 当所有结点都加入到最小生成树中之后，就找出了连通图中的最小生成树了。
* @return
*/
public Graph prim() {
Graph minTree = new Graph();
for (Point p : graph.getPointList()) {
minTree.addPoint(p);
//获得该点的最小权重边
EdgeWithWeight edge = getMinWeightEdgeForPoit(p, minTree);
if(null != edge){
//添加该边到最小生成树
minTree.addEdge(edge);
//检测是否产生回路
if(minTree.isGraphHasCircle()){
minTree.removeEdge(edge);
}
}
}
return minTree;
}
/**
* 获取权重最小边
* @param p
* @param minTree
* @return
*/
private EdgeWithWeight getMinWeightEdgeForPoit(Point p, Graph minTree) {
EdgeWithWeight e = null;
for (EdgeWithWeight edge : graph.getEdgeList()) {
if(!minTree.isEdgeInGraph(edge)){
if(edge.isPointInEdge(p)){
if(e == null){
e = edge;
}else{
if(e.compareTo(edge) == -1){
e = edge;
}
}
}
}
}
return e;
}
/**
* @param args
*/
public static void main(String[] args) {
//构建一个图
Graph graph = new Graph();
Point a = new Point("A");
Point b = new Point("B");
Point c = new Point("C");
Point d = new Point("D");
Point e = new Point("E");
Point f = new Point("F");
graph.addPoint(a);
graph.addPoint(b);
graph.addPoint(c);
graph.addPoint(d);
graph.addPoint(e);
graph.addPoint(f);
//所有边权重相同
graph.addEdge(new EdgeWithWeight(a, b, new Weight()));
graph.addEdge(new EdgeWithWeight(a, c, new Weight()));
graph.addEdge(new EdgeWithWeight(a, d, new Weight()));
graph.addEdge(new EdgeWithWeight(b, c, new Weight()));
graph.addEdge(new EdgeWithWeight(b, e, new Weight()));
graph.addEdge(new EdgeWithWeight(c, d, new Weight()));
graph.addEdge(new EdgeWithWeight(c, e, new Weight()));
graph.addEdge(new EdgeWithWeight(c, f, new Weight()));
graph.addEdge(new EdgeWithWeight(d, f, new Weight()));
graph.addEdge(new EdgeWithWeight(e, f, new Weight()));
Prim prim = new Prim(graph);
Graph minTree = prim.prim();
System.out.println(minTree);
Graph graphWithWeight = new Graph();
graphWithWeight.addPoint(a);
graphWithWeight.addPoint(b);
graphWithWeight.addPoint(c);
graphWithWeight.addPoint(d);
graphWithWeight.addPoint(e);
graphWithWeight.addPoint(f);
graphWithWeight.addEdge(new EdgeWithWeight(a, b, new Weight(6)));
graphWithWeight.addEdge(new EdgeWithWeight(a, c, new Weight(1)));
graphWithWeight.addEdge(new EdgeWithWeight(a, d, new Weight(5)));
graphWithWeight.addEdge(new EdgeWithWeight(b, c, new Weight(5)));
graphWithWeight.addEdge(new EdgeWithWeight(b, e, new Weight(3)));
graphWithWeight.addEdge(new EdgeWithWeight(c, d, new Weight(7)));
graphWithWeight.addEdge(new EdgeWithWeight(c, e, new Weight(5)));
graphWithWeight.addEdge(new EdgeWithWeight(c, f, new Weight(4)));
graphWithWeight.addEdge(new EdgeWithWeight(d, f, new Weight(2)));
graphWithWeight.addEdge(new EdgeWithWeight(e, f, new Weight(6)));
Prim primForWeigtTree = new Prim(graphWithWeight);
Graph minTreeForWeightTree = primForWeigtTree.prim();
System.out.println(minTreeForWeightTree);
}
}
Kruskal算法实现
package com.zas.test.tree;
import java.util.ArrayList;
import java.util.List;
/**
* Kruskal算法思想不同于Prim算法，
* Kruskal算法是一种按照连通网中边的权值的递增顺序构造最小生成树的算法。
* @author zas
*/
public class Kruskal {
//一个要找最小生成树的图
Graph graph;
public Kruskal() {
super();
}
public Kruskal(Graph graph) {
super();
this.graph = graph;
}
public Graph getGraph() {
return graph;
}
public void setGraph(Graph graph) {
this.graph = graph;
}
/**
* Kruskal算法的基本步骤 :
* 假设G=(V,E)是一个具有n个顶点的连通网，T=(U,TE)是G的最小生成树。
* 令集合U的初值为U=V，即包含有G中全部顶点，集合TE的初值为TE={}。
* 然后，将图G中的边按权值从小到大的顺序依次选取，若选取的边使生成树T不形成回路，则把它并入TE中，保留作为T的一条边；
* 若选取的边使生成树T形成回路，则将其舍弃，如此进行下去，直到TE中包含有n-1条边为止，此时的T即为最小生成树。
* @return
*/
public Graph kruskal() {
Graph minTree = new Graph();
//将所有顶点加入最小生成树
for (Point p : this.getGraph().getPointList()) {
minTree.addPoint(p);
}
//对原图的边按权值从小到大排序
List<EdgeWithWeight> edgeList = sortByEdgeAsc(this.graph.getEdgeList());
//加入 n - 1条最小权值边
for (int i = 0; i < edgeList.size(); i++) {
EdgeWithWeight edge = edgeList.get(i);
minTree.addEdge(edge);
//是否含有环
if(minTree.isGraphHasCircle()){
minTree.removeEdge(edge);
}
//结束条件
if(minTree.getEdgeList().size() == minTree.getPointList().size() - 1){
break;
}
}
return minTree;
}
/**
* 对边按权值从小到大排序
* @param edgeList
* @return
*/
private List<EdgeWithWeight> sortByEdgeAsc(List<EdgeWithWeight> graphEdgeList) {
//克隆一下，防止影响到原图
List<EdgeWithWeight> edgeList = new ArrayList<EdgeWithWeight>();
for (EdgeWithWeight edgeWithWeight : graphEdgeList) {
edgeList.add(edgeWithWeight);
}
//暂时采用选择排序，如果数据规模大、有性能要求可改进
int selectedIndex = 0;
EdgeWithWeight edgeForSwap = null;
for (int i = 0; i < edgeList.size(); i++) {
selectedIndex = i;
for (int j = i + 1; j < edgeList.size(); j++) {
if(edgeList.get(j).compareTo(edgeList.get(selectedIndex)) == 1){
selectedIndex = j;
}
}
if(selectedIndex != i){
//交换
edgeForSwap = edgeList.get(selectedIndex);
edgeList.set(selectedIndex, edgeList.get(i));
edgeList.set(i, edgeForSwap);
}
}
return edgeList;
}
/**
* @param args
*/
public static void main(String[] args) {
//构建一个图
Graph graph = new Graph();
Point a = new Point("A");
Point b = new Point("B");
Point c = new Point("C");
Point d = new Point("D");
Point e = new Point("E");
Point f = new Point("F");
graph.addPoint(a);
graph.addPoint(b);
graph.addPoint(c);
graph.addPoint(d);
graph.addPoint(e);
graph.addPoint(f);
// 所有边权重相同
graph.addEdge(new EdgeWithWeight(a, b, new Weight()));
graph.addEdge(new EdgeWithWeight(a, c, new Weight()));
graph.addEdge(new EdgeWithWeight(a, d, new Weight()));
graph.addEdge(new EdgeWithWeight(b, c, new Weight()));
graph.addEdge(new EdgeWithWeight(b, e, new Weight()));
graph.addEdge(new EdgeWithWeight(c, d, new Weight()));
graph.addEdge(new EdgeWithWeight(c, e, new Weight()));
graph.addEdge(new EdgeWithWeight(c, f, new Weight()));
graph.addEdge(new EdgeWithWeight(d, f, new Weight()));
graph.addEdge(new EdgeWithWeight(e, f, new Weight()));
Kruskal kruskal = new Kruskal(graph);
Graph minTree = kruskal.kruskal();
System.out.println(minTree);
Graph graphWithWeight = new Graph();
graphWithWeight.addPoint(a);
graphWithWeight.addPoint(b);
graphWithWeight.addPoint(c);
graphWithWeight.addPoint(d);
graphWithWeight.addPoint(e);
graphWithWeight.addPoint(f);
graphWithWeight.addEdge(new EdgeWithWeight(a, b, new Weight(6)));
graphWithWeight.addEdge(new EdgeWithWeight(a, c, new Weight(1)));
graphWithWeight.addEdge(new EdgeWithWeight(a, d, new Weight(5)));
graphWithWeight.addEdge(new EdgeWithWeight(b, c, new Weight(5)));
graphWithWeight.addEdge(new EdgeWithWeight(b, e, new Weight(3)));
graphWithWeight.addEdge(new EdgeWithWeight(c, d, new Weight(7)));
graphWithWeight.addEdge(new EdgeWithWeight(c, e, new Weight(5)));
graphWithWeight.addEdge(new EdgeWithWeight(c, f, new Weight(4)));
graphWithWeight.addEdge(new EdgeWithWeight(d, f, new Weight(2)));
graphWithWeight.addEdge(new EdgeWithWeight(e, f, new Weight(6)));
Kruskal kruskalForWeigtTree = new Kruskal(graphWithWeight);
Graph minTreeForWeightTree = kruskalForWeigtTree.kruskal();
System.out.println(minTreeForWeightTree);
}
}
测试用例所用的图
总结
Prim与Kruskal算法的关键都在于查找图是否含有环。
Prim算法从点入手，再加入与其有关的不产生环路的最小权重边。
Kruskal算法则先把所有点加入图，再加入n-1条不产生环路的最小权重边。
用面向对象的思维实现两个算法之后，感觉记忆更深刻了。
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