算法(第4版)学习笔记:关于Dijkstra算法(求最短路径)的理解

以下的这个算法可以使用于有向图的所有边的权重都是非负值的场景下。在该算法中实现了最小优先队列PQ，按照顶点到起点的最短路径的长度从小到大添加到最短路径树中。最终遍历图中的每个节点，且每个节点都只会被放入到优先队列中一次(当节点被delMin之后,即表示已经找到了该节点的最短路径）。

算法的核心在while循环中，每执行一次while循环，就会做两件事：

(1)把队列中最小路径(distTo最小)的节点移除，这个节点指出。

(2)保证这个节点指出的所有节点都加入到队列中(要么这些节点已经在队列中，那就检测它们的权重是不是最小的，要么没在队列中，那就加入)。

因此在每次while循环执行时，队列中的其他节点的路径都比这个被删除节点来的大，而被删除节点的指出的那些节点，由于它们的路径是要在被删除节点的路径上再加一条边，且所有边的权重都是非负值，所以也不会比被删除节点小。所以当一个节点被删除后，队列中遗留下的节点的路径和后续再加入到队列中的节点的路径都不会比这个节点小。所以如果起点所能到达的节点一共有v个，当队列中进行了v次delMin操作后，即while循环执行了v次后，则最短路径就已经完成。

算法保证了当每个节点被放松(relax)时，它的最短路径就已经被求出来了。但是为什么要这么做呢？为什么要按照特定的顺序来遍历节点生成最短路径?

原因是由于在对某个节点的放松(relax)操作中，要用到它指出的边的权重和它自己的最短路径相加，与指向节点当前已知的最短路径做比较。在放松时，如果这个节点的最短路径和权重还不知道，放松就没有意义了。如果按照随机的方式遍历节点，每个节点在被放松时，它未必得到了自己的最短路径，那它在之后得到了最短路径之后，肯定还会被再被放松一次，之前的放松也就完全失去意义了

DijkstraSP.java

Below is the syntax highlighted version of DijkstraSP.java from §4.4 Shortest Paths.


/******************************************************************************
 *  Compilation:  javac DijkstraSP.java
 *  Execution:    java DijkstraSP input.txt s
 *  Dependencies: EdgeWeightedDigraph.java IndexMinPQ.java Stack.java DirectedEdge.java
 *  Data files:   https://algs4.cs.princeton.edu/44sp/tinyEWD.txt
 *                https://algs4.cs.princeton.edu/44sp/mediumEWD.txt
 *                https://algs4.cs.princeton.edu/44sp/largeEWD.txt
 *
 *  Dijkstra's algorithm. Computes the shortest path tree.
 *  Assumes all weights are nonnegative.
 *
 *  % java DijkstraSP tinyEWD.txt 0
 *  0 to 0 (0.00)  
 *  0 to 1 (1.05)  0->4  0.38   4->5  0.35   5->1  0.32   
 *  0 to 2 (0.26)  0->2  0.26   
 *  0 to 3 (0.99)  0->2  0.26   2->7  0.34   7->3  0.39   
 *  0 to 4 (0.38)  0->4  0.38   
 *  0 to 5 (0.73)  0->4  0.38   4->5  0.35   
 *  0 to 6 (1.51)  0->2  0.26   2->7  0.34   7->3  0.39   3->6  0.52   
 *  0 to 7 (0.60)  0->2  0.26   2->7  0.34   
 *
 *  % java DijkstraSP mediumEWD.txt 0
 *  0 to 0 (0.00)  
 *  0 to 1 (0.71)  0->44  0.06   44->93  0.07   ...  107->1  0.07   
 *  0 to 2 (0.65)  0->44  0.06   44->231  0.10  ...  42->2  0.11   
 *  0 to 3 (0.46)  0->97  0.08   97->248  0.09  ...  45->3  0.12   
 *  0 to 4 (0.42)  0->44  0.06   44->93  0.07   ...  77->4  0.11   
 *  ...
 *
 ******************************************************************************/


/**
 *  The {@code DijkstraSP} class represents a data type for solving the
 *  single-source shortest paths problem in edge-weighted digraphs
 *  where the edge weights are nonnegative.
 *  <p>
 *  This implementation uses <em>Dijkstra's algorithm</em> with a
 *  <em>binary heap</em>. The constructor takes
 *  &Theta;(<em>E</em> log <em>V</em>) time in the worst case,
 *  where <em>V</em> is the number of vertices and <em>E</em> is
 *  the number of edges. Each instance method takes &Theta;(1) time.
 *  It uses &Theta;(<em>V</em>) extra space (not including the
 *  edge-weighted digraph).
 *  <p>
 *  For additional documentation,    
 *  see <a href="https://algs4.cs.princeton.edu/44sp">Section 4.4</a> of    
 *  <i>Algorithms, 4th Edition</i> by Robert Sedgewick and Kevin Wayne. 
 *
 *  @author Robert Sedgewick
 *  @author Kevin Wayne
 */
public class DijkstraSP {
    private double[] distTo;          // distTo[v] = distance  of shortest s->v path
    private DirectedEdge[] edgeTo;    // edgeTo[v] = last edge on shortest s->v path
    private IndexMinPQ<Double> pq;    // priority queue of vertices

    /**
     * Computes a shortest-paths tree from the source vertex {@code s} to every other
     * vertex in the edge-weighted digraph {@code G}.
     *
     * @param  G the edge-weighted digraph
     * @param  s the source vertex
     * @throws IllegalArgumentException if an edge weight is negative
     * @throws IllegalArgumentException unless {@code 0 <= s < V}
     */
    public DijkstraSP(EdgeWeightedDigraph G, int s) {
        for (DirectedEdge e : G.edges()) {
            if (e.weight() < 0)
                throw new IllegalArgumentException("edge " + e + " has negative weight");
        }

        distTo = new double[G.V()];
        edgeTo = new DirectedEdge[G.V()];

        validateVertex(s);

        for (int v = 0; v < G.V(); v++)
            distTo[v] = Double.POSITIVE_INFINITY;
        distTo[s] = 0.0;

        // relax vertices in order of distance from s
        pq = new IndexMinPQ<Double>(G.V());
        pq.insert(s, distTo[s]);
        while (!pq.isEmpty()) {
            int v = pq.delMin();
            for (DirectedEdge e : G.adj(v))
                relax(e);
        }

        // check optimality conditions
        assert check(G, s);
    }

    // relax edge e and update pq if changed
    private void relax(DirectedEdge e) {
        int v = e.from(), w = e.to();
        if (distTo[w] > distTo[v] + e.weight()) {
            distTo[w] = distTo[v] + e.weight();
            edgeTo[w] = e;
            if (pq.contains(w)) pq.decreaseKey(w, distTo[w]);
            else                pq.insert(w, distTo[w]);
        }
    }

    /**
     * Returns the length of a shortest path from the source vertex {@code s} to vertex {@code v}.
     * @param  v the destination vertex
     * @return the length of a shortest path from the source vertex {@code s} to vertex {@code v};
     *         {@code Double.POSITIVE_INFINITY} if no such path
     * @throws IllegalArgumentException unless {@code 0 <= v < V}
     */
    public double distTo(int v) {
        validateVertex(v);
        return distTo[v];
    }

    /**
     * Returns true if there is a path from the source vertex {@code s} to vertex {@code v}.
     *
     * @param  v the destination vertex
     * @return {@code true} if there is a path from the source vertex
     *         {@code s} to vertex {@code v}; {@code false} otherwise
     * @throws IllegalArgumentException unless {@code 0 <= v < V}
     */
    public boolean hasPathTo(int v) {
        validateVertex(v);
        return distTo[v] < Double.POSITIVE_INFINITY;
    }

    /**
     * Returns a shortest path from the source vertex {@code s} to vertex {@code v}.
     *
     * @param  v the destination vertex
     * @return a shortest path from the source vertex {@code s} to vertex {@code v}
     *         as an iterable of edges, and {@code null} if no such path
     * @throws IllegalArgumentException unless {@code 0 <= v < V}
     */
    public Iterable<DirectedEdge> pathTo(int v) {
        validateVertex(v);
        if (!hasPathTo(v)) return null;
        Stack<DirectedEdge> path = new Stack<DirectedEdge>();
        for (DirectedEdge e = edgeTo[v]; e != null; e = edgeTo[e.from()]) {
            path.push(e);
        }
        return path;
    }


    // check optimality conditions:
    // (i) for all edges e:            distTo[e.to()] <= distTo[e.from()] + e.weight()
    // (ii) for all edge e on the SPT: distTo[e.to()] == distTo[e.from()] + e.weight()
    private boolean check(EdgeWeightedDigraph G, int s) {

        // check that edge weights are nonnegative
        for (DirectedEdge e : G.edges()) {
            if (e.weight() < 0) {
                System.err.println("negative edge weight detected");
                return false;
            }
        }

        // check that distTo[v] and edgeTo[v] are consistent
        if (distTo[s] != 0.0 || edgeTo[s] != null) {
            System.err.println("distTo[s] and edgeTo[s] inconsistent");
            return false;
        }
        for (int v = 0; v < G.V(); v++) {
            if (v == s) continue;
            if (edgeTo[v] == null && distTo[v] != Double.POSITIVE_INFINITY) {
                System.err.println("distTo[] and edgeTo[] inconsistent");
                return false;
            }
        }

        // check that all edges e = v->w satisfy distTo[w] <= distTo[v] + e.weight()
        for (int v = 0; v < G.V(); v++) {
            for (DirectedEdge e : G.adj(v)) {
                int w = e.to();
                if (distTo[v] + e.weight() < distTo[w]) {
                    System.err.println("edge " + e + " not relaxed");
                    return false;
                }
            }
        }

        // check that all edges e = v->w on SPT satisfy distTo[w] == distTo[v] + e.weight()
        for (int w = 0; w < G.V(); w++) {
            if (edgeTo[w] == null) continue;
            DirectedEdge e = edgeTo[w];
            int v = e.from();
            if (w != e.to()) return false;
            if (distTo[v] + e.weight() != distTo[w]) {
                System.err.println("edge " + e + " on shortest path not tight");
                return false;
            }
        }
        return true;
    }

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

    /**
     * Unit tests the {@code DijkstraSP} data type.
     *
     * @param args the command-line arguments
     */
    public static void main(String[] args) {
        In in = new In(args[0]);
        EdgeWeightedDigraph G = new EdgeWeightedDigraph(in);
        int s = Integer.parseInt(args[1]);

        // compute shortest paths
        DijkstraSP sp = new DijkstraSP(G, s);


        // print shortest path
        for (int t = 0; t < G.V(); t++) {
            if (sp.hasPathTo(t)) {
                StdOut.printf("%d to %d (%.2f)  ", s, t, sp.distTo(t));
                for (DirectedEdge e : sp.pathTo(t)) {
                    StdOut.print(e + "   ");
                }
                StdOut.println();
            }
            else {
                StdOut.printf("%d to %d         no path\n", s, t);
            }
        }
    }

}