图论11-欧拉回路与欧拉路径+Hierholzer算法实现

最新推荐文章于 2025-07-07 11:35:21 发布

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本文详细介绍了欧拉回路的概念，以及如何使用Hierholzer算法在Java中检测并生成欧拉路径。重点展示了Graph类的实现，包括添加API、删除边和深拷贝功能，以及与之相关的联通分量和欧拉回路类的示例。

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文章目录

1 欧拉回路的概念
2 欧拉回路的算法实现
3 Hierholzer算法详解
4 Hierholzer算法实现

1 欧拉回路的概念

在这里插入图片描述

2 欧拉回路的算法实现

private boolean hasEulerLoop(){

    CC cc = new CC(G);
    if(cc.count() > 1) return false;

    for(int v = 0; v < G.V(); v ++)
        if(G.degree(v) % 2 == 1) return false;
    return true;
}

3 Hierholzer算法详解

在这里插入图片描述

public ArrayList<Integer> result(){

    ArrayList<Integer> res = new ArrayList<>();
    if(!hasEulerLoop()) return res;
    
    //根据小g进行删边
    Graph g = (Graph)G.clone();

    Stack<Integer> stack = new Stack<>();
    int curv = 0; //出发点
    stack.push(curv);
    while(!stack.isEmpty()){
        if(g.degree(curv) != 0){
            stack.push(curv);
            int w = g.adj(curv).iterator().next();
            g.removeEdge(curv, w);
            curv = w;
        }
        else{
            res.add(curv);
            curv = stack.pop();
        }
    }
    return res;
}

4 Hierholzer算法实现

4.1 修改Graph，增加API

//移除边
public void removeEdge(int v, int w){
    validateVertex(v);
    validateVertex(w);
    if(adj[v].contains(w)) E --;
    adj[v].remove(w);
    adj[w].remove(v);
}

//深拷贝
@Override
public Object clone(){

    try{
        Graph cloned = (Graph) super.clone();
        cloned.adj = new TreeSet[V];
        for(int v = 0; v < V; v ++){
            cloned.adj[v] = new TreeSet<Integer>();
            for(int w: adj[v])
                cloned.adj[v].add(w);
        }
        return cloned;
    }
    catch (CloneNotSupportedException e){
        e.printStackTrace();
    }
    return null;
}

4.2 Graph.java

package Chapter08_EulerLoop_And_EulerPath;

import java.io.File;
import java.io.IOException;
import java.util.TreeSet;
import java.util.Scanner;


/// 暂时只支持无向无权图
public class Graph implements Cloneable{

    private int V;
    private int E;
    private TreeSet<Integer>[] adj;

    public Graph(String filename){

        File file = new File(filename);

        try(Scanner scanner = new Scanner(file)){

            V = scanner.nextInt();
            if(V < 0) throw new IllegalArgumentException("V must be non-negative");
            adj = new TreeSet[V];
            for(int i = 0; i < V; i ++)
                adj[i] = new TreeSet<Integer>();

            E = scanner.nextInt();
            if(E < 0) throw new IllegalArgumentException("E must be non-negative");

            for(int i = 0; i < E; i ++){
                int a = scanner.nextInt();
                validateVertex(a);
                int b = scanner.nextInt();
                validateVertex(b);

                if(a == b) throw new IllegalArgumentException("Self Loop is Detected!");
                if(adj[a].contains(b)) throw new IllegalArgumentException("Parallel Edges are Detected!");

                adj[a].add(b);
                adj[b].add(a);
            }
        }
        catch(IOException e){
            e.printStackTrace();
        }
    }

    public void validateVertex(int v){
        if(v < 0 || v >= V)
            throw new IllegalArgumentException("vertex " + v + "is invalid");
    }

    public int V(){
        return V;
    }

    public int E(){
        return E;
    }

    public boolean hasEdge(int v, int w){
        validateVertex(v);
        validateVertex(w);
        return adj[v].contains(w);
    }

    public Iterable<Integer> adj(int v){
        validateVertex(v);
        return adj[v];
    }

    public int degree(int v){
        validateVertex(v);
        return adj[v].size();
    }

    public void removeEdge(int v, int w){
        validateVertex(v);
        validateVertex(w);
        if(adj[v].contains(w)) E --;
        adj[v].remove(w);
        adj[w].remove(v);
    }

    @Override
    public Object clone(){

        try{
            Graph cloned = (Graph) super.clone();
            cloned.adj = new TreeSet[V];
            for(int v = 0; v < V; v ++){
                cloned.adj[v] = new TreeSet<Integer>();
                for(int w: adj[v])
                    cloned.adj[v].add(w);
            }
            return cloned;
        }
        catch (CloneNotSupportedException e){
            e.printStackTrace();
        }
        return null;
    }

    @Override
    public String toString(){
        StringBuilder sb = new StringBuilder();

        sb.append(String.format("V = %d, E = %d\n", V, E));
        for(int v = 0; v < V; v ++){
            sb.append(String.format("%d : ", v));
            for(int w : adj[v])
                sb.append(String.format("%d ", w));
            sb.append('\n');
        }
        return sb.toString();
    }

    public static void main(String[] args){

        Graph g = new Graph("g.txt");
        System.out.print(g);
    }
}

4.3 联通分量类

package Chapter08_EulerLoop_And_EulerPath;

import java.util.ArrayList;

public class CC {

    private Graph G;
    private int[] visited;
    private int cccount = 0;

    public CC(Graph G){

        this.G = G;
        visited = new int[G.V()];
        for(int i = 0; i < visited.length; i ++)
            visited[i] = -1;

        for(int v = 0; v < G.V(); v ++)
            if(visited[v] == -1){
                dfs(v, cccount);
                cccount ++;
            }
    }

    private void dfs(int v, int ccid){

        visited[v] = ccid;
        for(int w: G.adj(v))
            if(visited[w] == -1)
                dfs(w, ccid);
    }

    public int count(){
        return cccount;
    }

    public boolean isConnected(int v, int w){
        G.validateVertex(v);
        G.validateVertex(w);
        return visited[v] == visited[w];
    }

    public ArrayList<Integer>[] components(){

        ArrayList<Integer>[] res = new ArrayList[cccount];
        for(int i = 0; i < cccount; i ++)
            res[i] = new ArrayList<Integer>();

        for(int v = 0; v < G.V(); v ++)
            res[visited[v]].add(v);
        return res;
    }

    public static void main(String[] args){

        Graph g = new Graph("g.txt");
        CC cc = new CC(g);
        System.out.println(cc.count());

        System.out.println(cc.isConnected(0, 6));
        System.out.println(cc.isConnected(5, 6));

        ArrayList<Integer>[] comp = cc.components();
        for(int ccid = 0; ccid < comp.length; ccid ++){
            System.out.print(ccid + " : ");
            for(int w: comp[ccid])
                System.out.print(w + " ");
            System.out.println();
        }
    }
}

4.4 欧拉回路类

package Chapter08_EulerLoop_And_EulerPath;

import java.util.ArrayList;
import java.util.Stack;

public class EulerLoop {

    private Graph G;

    public EulerLoop(Graph G){
        this.G = G;
    }

    private boolean hasEulerLoop(){

        CC cc = new CC(G);
        if(cc.count() > 1) return false;

        for(int v = 0; v < G.V(); v ++)
            if(G.degree(v) % 2 == 1) return false;
        return true;
    }

    public ArrayList<Integer> result(){

        ArrayList<Integer> res = new ArrayList<>();
        if(!hasEulerLoop()) return res;

        Graph g = (Graph)G.clone();

        Stack<Integer> stack = new Stack<>();
        int curv = 0;
        stack.push(curv);
        while(!stack.isEmpty()){
            if(g.degree(curv) != 0){
                stack.push(curv);
                int w = g.adj(curv).iterator().next();
                g.removeEdge(curv, w);
                curv = w;
            }
            else{
                res.add(curv);
                curv = stack.pop();
            }
        }
        return res;
    }

    public static void main(String[] args){

        Graph g = new Graph("g8.txt");
        EulerLoop el = new EulerLoop(g);
        System.out.println(el.result());

        Graph g2 = new Graph("g2.txt");
        EulerLoop el2 = new EulerLoop(g2);
        System.out.println(el2.result());
    }
}