图的环检测

一、无向图的环判断

1.1 环的定义

此处的环不包含自环和平行边。

1-1-1 自环和平行边

无向图中环的示意图如下所示：

1-1-2 无向图中环的示意图

上图中，0-6-4-5构成了环

1.2 环的判断

基本思想：
采用深度优先遍历（DFS）无向图。
如果在遍历的过程中，发现某个顶点有一条边指向已经访问过的顶点，且这个已访问过的顶点不是当前顶点的父节点（这里的父节点表示DFS遍历顺序中的父节点），则说明图包含环。

如图1-1-2中：从0开始DFS，0->6->4->5，此时顶点5的一条边指向顶点0，顶点0已经访问过，但却不是顶点5的父节点（顶点4），说明出现了环。

源码实现：

public class Cycle {
    private boolean[] marked;
    private boolean hasCycle;

    public Cycle(Graph G) {
        marked = new boolean[G.V()];
        for (int v = 0; v < G.V(); v++)
            if (!marked[v])
                dfs(G, -1, v);
    }

    /**
     * 从顶点v开始进行深度优先搜索，u表示v的前驱顶点：u->v
     */
    private void dfs(Graph G, int u, int v) {
        marked[v] = true;
        for (int w : G.adj(v)) { // w为v的后继顶点：v->w
            if (!marked[w]) { // 如果w未访问过
                dfs(G, v, w);
            } else {
                // 如果w已经访问过，但w不是当前v的前驱顶点u
                // 此时说明w有两个前驱结点
                if (w != u) {
                    hasCycle = true;
                    break;
                }
            }
        }
    }
    public boolean hasCycle() {
        return hasCycle;
    }
}

二、有向图的环路判断

2.1 环的定义

有向图中环的示意图如下所示：

2-1-1 有向图中环的示意图

1.2 环的判断

基本思想：
采用深度优先遍历（DFS）有向图。
用一个boolean数组表示顶点是否在调用栈上。如果发现某个顶点v有一条边指向已经访问过的顶点w，而w已经在深度优先调用栈上，则说明存在环。

1-2-1 有向图中环的查找

源码实现：

public class EdgeWeightedDirectedCycle {
    private boolean[] marked;             // marked[v] = has vertex v been marked?
    private DirectedEdge[] edgeTo;        // edgeTo[v] = previous edge on path to v
    private boolean[] onStack;            // onStack[v] = is vertex on the stack?
    private Stack<DirectedEdge> cycle;    // directed cycle (or null if no such cycle)

    /**
     * Determines whether the edge-weighted digraph {@code G} has a directed cycle and,
     * if so, finds such a cycle.
     * @param G the edge-weighted digraph
     */
    public EdgeWeightedDirectedCycle(EdgeWeightedDigraph G) {
        marked  = new boolean[G.V()];
        onStack = new boolean[G.V()];
        edgeTo  = new DirectedEdge[G.V()];
        for (int v = 0; v < G.V(); v++)
            if (!marked[v]) dfs(G, v);

        // check that digraph has a cycle
        assert check();
    }

    // check that algorithm computes either the topological order or finds a directed cycle
    private void dfs(EdgeWeightedDigraph G, int v) {
        onStack[v] = true;
        marked[v] = true;
        for (DirectedEdge e : G.adj(v)) {
            int w = e.to();

            // short circuit if directed cycle found
            if (cycle != null) return;

            // found new vertex, so recur
            else if (!marked[w]) {
                edgeTo[w] = e;
                dfs(G, w);
            }

            // trace back directed cycle
            else if (onStack[w]) {
                cycle = new Stack<DirectedEdge>();

                DirectedEdge f = e;
                while (f.from() != w) {
                    cycle.push(f);
                    f = edgeTo[f.from()];
                }
                cycle.push(f);

                return;
            }
        }

        onStack[v] = false;
    }

    /**
     * Does the edge-weighted digraph have a directed cycle?
     * @return {@code true} if the edge-weighted digraph has a directed cycle,
     * {@code false} otherwise
     */
    public boolean hasCycle() {
        return cycle != null;
    }

    /**
     * Returns a directed cycle if the edge-weighted digraph has a directed cycle,
     * and {@code null} otherwise.
     * @return a directed cycle (as an iterable) if the edge-weighted digraph
     *    has a directed cycle, and {@code null} otherwise
     */
    public Iterable<DirectedEdge> cycle() {
        return cycle;
    }

    // certify that digraph is either acyclic or has a directed cycle
    private boolean check() {
        // edge-weighted digraph is cyclic
        if (hasCycle()) {
            // verify cycle
            DirectedEdge first = null, last = null;
            for (DirectedEdge e : cycle()) {
                if (first == null) first = e;
                if (last != null) {
                    if (last.to() != e.from()) {
                        System.err.printf("cycle edges %s and %s not incident\n", last, e);
                        return false;
                    }
                }
                last = e;
            }
            if (last.to() != first.from()) {
                System.err.printf("cycle edges %s and %s not incident\n", last, first);
                return false;
            }
        }
        return true;
    }

    /**
     * Unit tests the {@code EdgeWeightedDirectedCycle} data type.
     *
     * @param args the command-line arguments
     */
    public static void main(String[] args) {
        // create random DAG with V vertices and E edges; then add F random edges
        int V = Integer.parseInt(args[0]);
        int E = Integer.parseInt(args[1]);
        int F = Integer.parseInt(args[2]);
        EdgeWeightedDigraph G = new EdgeWeightedDigraph(V);
        int[] vertices = new int[V];
        for (int i = 0; i < V; i++)
            vertices[i] = i;
        StdRandom.shuffle(vertices);
        for (int i = 0; i < E; i++) {
            int v, w;
            do {
                v = StdRandom.uniform(V);
                w = StdRandom.uniform(V);
            } while (v >= w);
            double weight = StdRandom.uniform();
            G.addEdge(new DirectedEdge(v, w, weight));
        }
        // add F extra edges
        for (int i = 0; i < F; i++) {
            int v = StdRandom.uniform(V);
            int w = StdRandom.uniform(V);
            double weight = StdRandom.uniform(0.0, 1.0);
            G.addEdge(new DirectedEdge(v, w, weight));
        }
        StdOut.println(G);
        // find a directed cycle
        EdgeWeightedDirectedCycle finder = new EdgeWeightedDirectedCycle(G);
        if (finder.hasCycle()) {
            StdOut.print("Cycle: ");
            for (DirectedEdge e : finder.cycle()) {
                StdOut.print(e + " ");
            }
            StdOut.println();
        }
        // or give topologial sort
        else {
            StdOut.println("No directed cycle");
        }
    }
}