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深度优先搜索

定义

深度优先搜索是一种经典的递归方法，要搜索一幅图，只需要用一个递归方法来遍历所有顶点。在访问其中一个顶点时：

  • 将它标记为已访问。
  • 递归地访问它的所有没有被标记过的邻居顶点。

接口

代码

public class DepthFirstSearch {
    private boolean[] marked;    // marked[v] = is there an s-v path?
    private int count;           // number of vertices connected to s

    public DepthFirstSearch(Graph G, int s) {
        marked = new boolean[G.V()];
        validateVertex(s);
        dfs(G, s);
    }

    private void dfs(Graph G, int v) {
        count++;
        marked[v] = true;
        for (int w : G.adj(v)) {
            if (!marked[w]) {
                dfs(G, w);
            }
        }
    }

    public boolean marked(int v) {
        validateVertex(v);
        return marked[v];
    }

    public int count() {
        return count;
    }

    private void validateVertex(int v) {
        int V = marked.length;
        if (v < 0 || v >= V)
            throw new IllegalArgumentException("vertex " + v + " is not between 0 and " + (V-1));
    }
}

轨迹图

寻找路径

定义

给定一幅图和一个起点S，找到从S到给定目的顶点V的路径。

接口

代码

public class DepthFirstPaths {
    private boolean[] marked;    // marked[v] = is there an s-v path?
    private int[] edgeTo;        // edgeTo[v] = last edge on s-v path
    private final int s;         // source vertex

    public DepthFirstPaths(Graph G, int s) {
        this.s = s;
        edgeTo = new int[G.V()];
        marked = new boolean[G.V()];
        validateVertex(s);
        dfs(G, s);
    }

    private void dfs(Graph G, int v) {
        marked[v] = true;
        for (int w : G.adj(v)) {
            if (!marked[w]) {
                edgeTo[w] = v;
                dfs(G, w);
            }
        }
    }

    public boolean hasPathTo(int v) {
        validateVertex(v);
        return marked[v];
    }

    public Iterable<Integer> pathTo(int v) {
        validateVertex(v);
        if (!hasPathTo(v)) return null;
        Stack<Integer> path = new Stack<Integer>();
        for (int x = v; x != s; x = edgeTo[x])
            path.push(x);
        path.push(s);
        return path;
    }

    private void validateVertex(int v) {
        int V = marked.length;
        if (v < 0 || v >= V)
            throw new IllegalArgumentException("vertex " + v + " is not between 0 and " + (V-1));
    }
}

轨迹图

复杂度

使用深度优先搜索得到从给定起点到任意顶点的路径所需的时间与路径的长度成正比。

连通分量

接口

代码

public class CC {
    private boolean[] marked;   // marked[v] = has vertex v been marked?
    private int[] id;           // id[v] = id of connected component containing v
    private int[] size;         // size[id] = number of vertices in given component
    private int count;          // number of connected components

    public CC(Graph G) {
        marked = new boolean[G.V()];
        id = new int[G.V()];
        size = new int[G.V()];
        for (int v = 0; v < G.V(); v++) {
            if (!marked[v]) {
                dfs(G, v);
                count++;
            }
        }
    }

    private void dfs(Graph G, int v) {
        marked[v] = true;
        id[v] = count;
        size[count]++;
        for (int w : G.adj(v)) {
            if (!marked[w]) {
                dfs(G, w);
            }
        }
    }

    public int id(int v) {
        return id[v];
    }

    public int size(int v) {
        return size[id[v]];
    }

    public int count() {
        return count;
    }

    public boolean connected(int v, int w) {
        return id(v) == id(w);
    }
}

轨迹图