图论——连通分量

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最新推荐文章于 2025-05-31 22:32:21 发布

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分类专栏：算法数据结构文章标签： dfs 图论连通分量 bfs

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本文深入探讨了图论中连通分量的概念，详细介绍了如何使用深度优先搜索（DFS）和广度优先搜索（BFS）算法来计算无向图的连通分量，不仅展示了计算连通分量个数的方法，还提供了获取每个连通分量中顶点的具体实现。

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文章目录

图论——连通分量

图论——连通分量

连通分量介绍

在这里插入图片描述

对于上图很显然连通分量为1，对于下图连通分量个数为2

在这里插入图片描述

DFS计算连通分量

通过上一小节dfs遍历的过程我们知道依次dfs就是一个连通分量，因为dfs只有走到无路可走才会回退，所以我们只需要记录一下dfs次数即可，有如下代码

public class UndirectedGraphDFSCC {
    private UndirectedGraph graph;
    private boolean[] visited;
    private int ccCount;//连通分量

    public UndirectedGraphDFSCC(UndirectedGraph graph){
        this.graph = graph;
        visited = new boolean[graph.vertexNum()];
        //可能有多个连通分量，所以得for
        for(int v=0;v<graph.vertexNum();v++){
            if(!visited[v]){
                dfs(v);
                ccCount++;
            }
        }
    }

    public int getCcCount(){
        return ccCount;
    }

    private void dfs(int v){
        visited[v] = true;
        for(int w:graph.adj(v)) {
            if(!visited[w]){
                dfs(w);
            }
        }
    }


    public static void main(String[] args) {
        UndirectedGraph graph = new UndirectedGraph("graph.txt");
        System.out.println(graph);
        UndirectedGraphDFSCC graphDFS = new UndirectedGraphDFSCC(graph);
        System.out.println(graphDFS.getCcCount());
    }
}

但是我们想知道的不仅仅是一个图有多少连通分量，而是每个连通分量都有哪些顶点
我们可以改变一下visited数组，改为int数组，如果数组里的值是-1，代表此下标代表的顶点未访问过，如果数组里的值是0，代表此下标代表的顶点是属于第一个连通分量的，如果数组里的值是1，代表此下标代表的顶点是属于第二个连通分量的，以此类推，有如下代码

public class UndirectedGraphCC {
    private UndirectedGraph graph;
    private int[] visited;
    private int ccCount;//连通分量

    public UndirectedGraphCC(UndirectedGraph graph){
        this.graph = graph;
        visited = new int[graph.vertexNum()];
        for(int i=0;i<visited.length;i++){
            //visited里面-1表示没有访问，0表示第一个联通分量，1表示第二个连通分量，以此类推
            visited[i] = -1;
        }
        for(int v=0;v<graph.vertexNum();v++){
            if(visited[v]==-1){
                dfs(v,ccCount);
                ccCount++;
            }
        }
    }

    private void dfs(int v,int ccId){
        visited[v] = ccId;
        for(int w:graph.adj(v)) {
            if(visited[w]==-1){
                dfs(w,ccId);
            }
        }
    }

    /**
     * 返回所有的连通分量
     * @return
     */
    public List<Integer>[] components(){
        List<Integer>[] res = new ArrayList[ccCount];
        for(int i=0;i<ccCount;i++){
            res[i] = new ArrayList<>();
        }

        for(int v=0;v<graph.vertexNum();v++){
            res[visited[v]].add(v);
        }
        return res;
    }

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

    public int getCcCount(){return ccCount;}
    @Override
    public String toString() {
        StringBuilder sb = new StringBuilder();
        for(int v:visited){
            sb.append(v+" ");
        }
        sb.append("\n连通分量:"+ccCount);
        int index = 1;
        for (List<Integer> list:components()){
            sb.append("\n第"+index+++"个连通分量的顶点:");
            for (int v:list) {
                sb.append(v+" ");
            }
        }
        return sb.toString();
    }

    public static void main(String[] args) {
        UndirectedGraph graph = new UndirectedGraph("graph.txt");
        System.out.println(graph);
        UndirectedGraphCC graphCC = new UndirectedGraphCC(graph);
        System.out.println(graphCC);
        System.out.println(graphCC.isConnected(0, 6));
        System.out.println(graphCC.isConnected(0, 5));
    }
}

BFS计算连通分量

public class UndirectedGraphCCBFS {
    private UndirectedGraph graph;
    private int[] visited;
    private int ccCount;

    public UndirectedGraphCCBFS(UndirectedGraph graph){
        this.graph = graph;
        visited = new int[graph.vertexNum()];
        for(int i=0;i<visited.length;i++){
            //visited里面-1表示没有访问，0表示第一个联通分量，1表示第二个连通分量，以此类推
            visited[i] = -1;
        }
        //多个联通分量必须for
        for(int v=0;v<graph.vertexNum();v++){
            if(visited[v]==-1){
                bfs(v,ccCount);
                ccCount++;
            }
        }
    }

    private void bfs(int v,int ccId){
        Queue<Integer> queue = new LinkedList();
        queue.offer(v);
        visited[v] = ccId;

        while(!queue.isEmpty()){
            int w = queue.poll();
            for(int u:graph.adj(w)){
                if(visited[u]==-1){
                    visited[u] = ccId;
                    queue.offer(u);
                }
            }
        }
    }

    /**
     * 返回所有的连通分量
     * @return
     */
    public List<Integer>[] components(){
        List<Integer>[] res = new ArrayList[ccCount];
        for(int i=0;i<ccCount;i++){
            res[i] = new ArrayList<>();
        }

        for(int v=0;v<graph.vertexNum();v++){
            res[visited[v]].add(v);
        }
        return res;
    }

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

    public int getCcCount() {
        return ccCount;
    }

    @Override
    public String toString() {
        StringBuilder sb = new StringBuilder();
        for(int v:visited){
            sb.append(v+" ");
        }
        sb.append("\n连通分量:"+ccCount);
        int index = 1;
        for (List<Integer> list:components()){
            sb.append("\n第"+index+++"个连通分量的顶点:");
            for (int v:list) {
                sb.append(v+" ");
            }
        }
        return sb.toString();
    }

    public static void main(String[] args) {
        UndirectedGraph graph = new UndirectedGraph("graph.txt");
        System.out.println(graph);
        UndirectedGraphCCBFS graphBFS = new UndirectedGraphCCBFS(graph);
        System.out.println(graphBFS);
    }
}

图文件graph.txt

建图类

public class UndirectedGraph {
    private int V;//顶点数
    private int E;//边数
    private TreeSet<Integer>[] adj;//邻接表，TreeSet数组存储

    public UndirectedGraph(String filename){
        File file = new File(filename);
        try(Scanner scanner = new Scanner(file)){
            V = scanner.nextInt();//顶点数
            if(V<=0) throw new RuntimeException("顶点个数必须大于0");
            adj = new TreeSet[V];
            for(int i=0;i<V;i++){
                adj[i] = new TreeSet<>();
            }
            E = scanner.nextInt();//边数
            if(E<0) throw new RuntimeException("边数不能为负数");
            for(int i=0;i<E;i++){
                int a = scanner.nextInt();
                validateVertex(a);
                int b = scanner.nextInt();
                validateVertex(b);
                //自环边检测
                if(a==b){
                    throw new RuntimeException("简单图不能包含自环边");
                }
                //平行边检测
                if(adj[a].contains(b)){
                    throw new RuntimeException("简单图不能包含平行边");
                }
                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 RuntimeException("顶点下标溢出");
        }
    }
    public int vertexNum(){
        return V;
    }
    public int edgeNum(){
        return E;
    }
    public boolean hasEdge(int v,int w){
        validateVertex(v);
        validateVertex(w);
        return adj[v].contains(w);
    }

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

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

    @Override
    public String toString() {
        StringBuilder sb = new StringBuilder();
        sb.append(String.format("V = %d,E = %d\n",V,E));
        for(int i=0;i<adj.length;i++){
            sb.append(i+":");
            for (Iterator<Integer> it = adj[i].iterator(); it.hasNext(); ) {
                sb.append(it.next()+" ");
            }
            sb.append("\n");
        }
        return sb.toString();
    }


    public static void main(String[] args) {
        UndirectedGraph graph = new UndirectedGraph("graph.txt");
        System.out.println(graph);
    }