图的搜索(遍历)

一、定义

图的搜索算法的目标是:从某个指定源点开始,遍历图中其它顶点,并作相应的处理。

API定义:

二、深度优先搜索(DFS)

基本思想:
深度优先搜索基于递归的思想:

  1. 首先以一个未被访问过的顶点作为起始顶点;
  2. 沿当前顶点的边走到一个未被访问过的顶点;
  3. 当已经没有未被访问过的顶点时,则回到上一个顶点,继续试探访问别的顶点,直到所有顶点都被访问过。

源码:

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

   /**
    * Computes the vertices in graph {@code G} that are
    * connected to the source vertex {@code s}.
    * @param G the graph
    * @param s the source vertex
    * @throws IllegalArgumentException unless {@code 0 <= s < V}
    */
   public DepthFirstSearch(Graph G, int s) {
       marked = new boolean[G.V()];
       validateVertex(s);
       dfs(G, s);
   }

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

   /**
    * Is there a path between the source vertex {@code s} and vertex {@code v}?
    * @param v the vertex
    * @return {@code true} if there is a path, {@code false} otherwise
    * @throws IllegalArgumentException unless {@code 0 <= v < V}
    */
   public boolean marked(int v) {
       validateVertex(v);
       return marked[v];
   }

   /**
    * Returns the number of vertices connected to the source vertex {@code s}.
    * @return the number of vertices connected to the source vertex {@code s}
    */
   public int count() {
       return count;
   }

   // throw an IllegalArgumentException unless {@code 0 <= v < V}
   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));
   }

   /**
    * Unit tests the {@code DepthFirstSearch} data type.
    *
    * @param args the command-line arguments
    */
   public static void main(String[] args) {
       In in = new In(args[0]);
       Graph G = new Graph(in);
       int s = Integer.parseInt(args[1]);
       DepthFirstSearch search = new DepthFirstSearch(G, s);
       for (int v = 0; v < G.V(); v++) {
           if (search.marked(v))
               StdOut.print(v + " ");
       }

       StdOut.println();
       if (search.count() != G.V()) StdOut.println("NOT connected");
       else                         StdOut.println("connected");
   }
}
2-1 DFS示意图

三、广度优先搜索(BFS)

基本思想:
广度优先搜索首先访问与源点最近的顶点,然后一层层向外扩展。
该算法用一个队列保存所有已经被标记过但其邻接表还未被检查过的顶点:

  1. 将源点加入队列,然后重复以下步骤直到队列为空;
  2. 取出队列中的下一个顶点v并标记它;
  3. 将与v相邻的所有未被标记过的顶点加入队列。

源码:

public class BreadthFirstPaths {
    private static final int INFINITY = Integer.MAX_VALUE;
    private boolean[] marked;  // marked[v] = is there an s-v path
    private int[] edgeTo;      // edgeTo[v] = previous edge on shortest s-v path
    private int[] distTo;      // distTo[v] = number of edges shortest s-v path

    /**
     * Computes the shortest path between the source vertex {@code s}
     * and every other vertex in the graph {@code G}.
     * @param G the graph
     * @param s the source vertex
     * @throws IllegalArgumentException unless {@code 0 <= s < V}
     */
    public BreadthFirstPaths(Graph G, int s) {
        marked = new boolean[G.V()];
        distTo = new int[G.V()];
        edgeTo = new int[G.V()];
        validateVertex(s);
        bfs(G, s);

        assert check(G, s);
    }

    /**
     * Computes the shortest path between any one of the source vertices in {@code sources}
     * and every other vertex in graph {@code G}.
     * @param G the graph
     * @param sources the source vertices
     * @throws IllegalArgumentException unless {@code 0 <= s < V} for each vertex
     *         {@code s} in {@code sources}
     */
    public BreadthFirstPaths(Graph G, Iterable<Integer> sources) {
        marked = new boolean[G.V()];
        distTo = new int[G.V()];
        edgeTo = new int[G.V()];
        for (int v = 0; v < G.V(); v++)
            distTo[v] = INFINITY;
        validateVertices(sources);
        bfs(G, sources);
    }

    // breadth-first search from a single source
    private void bfs(Graph G, int s) {
        Queue<Integer> q = new Queue<Integer>();
        for (int v = 0; v < G.V(); v++)
            distTo[v] = INFINITY;
        distTo[s] = 0;
        marked[s] = true;
        q.enqueue(s);

        while (!q.isEmpty()) {
            int v = q.dequeue();
            for (int w : G.adj(v)) {
                if (!marked[w]) {
                    edgeTo[w] = v;
                    distTo[w] = distTo[v] + 1;
                    marked[w] = true;
                    q.enqueue(w);
                }
            }
        }
    }

    // breadth-first search from multiple sources
    private void bfs(Graph G, Iterable<Integer> sources) {
        Queue<Integer> q = new Queue<Integer>();
        for (int s : sources) {
            marked[s] = true;
            distTo[s] = 0;
            q.enqueue(s);
        }
        while (!q.isEmpty()) {
            int v = q.dequeue();
            for (int w : G.adj(v)) {
                if (!marked[w]) {
                    edgeTo[w] = v;
                    distTo[w] = distTo[v] + 1;
                    marked[w] = true;
                    q.enqueue(w);
                }
            }
        }
    }

    /**
     * Is there a path between the source vertex {@code s} (or sources) and vertex {@code v}?
     * @param v the vertex
     * @return {@code true} if there is a path, and {@code false} otherwise
     * @throws IllegalArgumentException unless {@code 0 <= v < V}
     */
    public boolean hasPathTo(int v) {
        validateVertex(v);
        return marked[v];
    }

    /**
     * Returns the number of edges in a shortest path between the source vertex {@code s}
     * (or sources) and vertex {@code v}?
     * @param v the vertex
     * @return the number of edges in a shortest path
     * @throws IllegalArgumentException unless {@code 0 <= v < V}
     */
    public int distTo(int v) {
        validateVertex(v);
        return distTo[v];
    }

    /**
     * Returns a shortest path between the source vertex {@code s} (or sources)
     * and {@code v}, or {@code null} if no such path.
     * @param  v the vertex
     * @return the sequence of vertices on a shortest path, as an Iterable
     * @throws IllegalArgumentException unless {@code 0 <= v < V}
     */
    public Iterable<Integer> pathTo(int v) {
        validateVertex(v);
        if (!hasPathTo(v)) return null;
        Stack<Integer> path = new Stack<Integer>();
        int x;
        for (x = v; distTo[x] != 0; x = edgeTo[x])
            path.push(x);
        path.push(x);
        return path;
    }

    // check optimality conditions for single source
    private boolean check(Graph G, int s) {
        // check that the distance of s = 0
        if (distTo[s] != 0) {
            StdOut.println("distance of source " + s + " to itself = " + distTo[s]);
            return false;
        }
        // check that for each edge v-w dist[w] <= dist[v] + 1
        // provided v is reachable from s
        for (int v = 0; v < G.V(); v++) {
            for (int w : G.adj(v)) {
                if (hasPathTo(v) != hasPathTo(w)) {
                    StdOut.println("edge " + v + "-" + w);
                    StdOut.println("hasPathTo(" + v + ") = " + hasPathTo(v));
                    StdOut.println("hasPathTo(" + w + ") = " + hasPathTo(w));
                    return false;
                }
                if (hasPathTo(v) && (distTo[w] > distTo[v] + 1)) {
                    StdOut.println("edge " + v + "-" + w);
                    StdOut.println("distTo[" + v + "] = " + distTo[v]);
                    StdOut.println("distTo[" + w + "] = " + distTo[w]);
                    return false;
                }
            }
        }

        // check that v = edgeTo[w] satisfies distTo[w] = distTo[v] + 1
        // provided v is reachable from s
        for (int w = 0; w < G.V(); w++) {
            if (!hasPathTo(w) || w == s) continue;
            int v = edgeTo[w];
            if (distTo[w] != distTo[v] + 1) {
                StdOut.println("shortest path edge " + v + "-" + w);
                StdOut.println("distTo[" + v + "] = " + distTo[v]);
                StdOut.println("distTo[" + w + "] = " + distTo[w]);
                return false;
            }
        }
        return true;
    }

    // throw an IllegalArgumentException unless {@code 0 <= v < V}
    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));
    }

    // throw an IllegalArgumentException unless {@code 0 <= v < V}
    private void validateVertices(Iterable<Integer> vertices) {
        if (vertices == null) {
            throw new IllegalArgumentException("argument is null");
        }
        int V = marked.length;
        for (int v : vertices) {
            if (v < 0 || v >= V) {
                throw new IllegalArgumentException("vertex " + v + " is not between 0 and " + (V-1));
            }
        }
    }

    /**
     * Unit tests the {@code BreadthFirstPaths} data type.
     *
     * @param args the command-line arguments
     */
    public static void main(String[] args) {
        In in = new In(args[0]);
        Graph G = new Graph(in);
        // StdOut.println(G);

        int s = Integer.parseInt(args[1]);
        BreadthFirstPaths bfs = new BreadthFirstPaths(G, s);
        for (int v = 0; v < G.V(); v++) {
            if (bfs.hasPathTo(v)) {
                StdOut.printf("%d to %d (%d):  ", s, v, bfs.distTo(v));
                for (int x : bfs.pathTo(v)) {
                    if (x == s) StdOut.print(x);
                    else        StdOut.print("-" + x);
                }
                StdOut.println();
            }else {
                StdOut.printf("%d to %d (-):  not connected\n", s, v);
            }
        }
    }
}
3-1 BFS示意图