算法4第6章 网络流算法/FordFulkerson算法/增广路径算法讲解

 网络流算法
 问题描述:对应下图流量图的输油网络，只有一个入口和一个出口，每条管道都有相应的容量，中间每个节点的流入量和流出量要相同，怎样设计流量网络可以使出口的流量最大。
 流量图可以很自然的转换成有向图，每条边有容量限制和当前的流量，示意图如下：

 FordFulkerson算法也叫做增广路径算法可以解决最大流问题，该算法的思路是寻找一条从起点s到终点t的路径，该路径每条边的剩余流量不能为0,找出每条边剩余流量的最小值bottle，把每条边的流程增加bottle,反复此过程直到找不出这样的路径为止。这样的路径也叫做增广路径。
 增广路径中边的方向也可以与流量的方向相反，这条边中的流量不能为空，正向边是流量增加bottle,方向边是流量减少bottle,如下图的0-2-3-1-4-5路径，。
 增广路径中从起点指出的边和指向终点的边必然是正向的。

算法实现如下：，示意图如下：

采用广度优先搜索寻找增广路径，所以也叫做最短增广路径
 
public class FordFulkerson {
    private static final double FLOATING_POINT_EPSILON = 1E-11;

    private final int V;          // number of vertices
    private boolean[] marked;     // marked[v] = true iff s->v path in residual graph
    private FlowEdge[] edgeTo;    // edgeTo[v] = last edge on shortest residual s->v path
    private double value;         // current value of max flow
 
    /**
     * Compute a maximum flow and minimum cut in the network {@code G}
     * from vertex {@code s} to vertex {@code t}.
     *
     * @param  G the flow network
     * @param  s the source vertex
     * @param  t the sink vertex
     * @throws IllegalArgumentException unless {@code 0 <= s < V}
     * @throws IllegalArgumentException unless {@code 0 <= t < V}
     * @throws IllegalArgumentException if {@code s == t}
     * @throws IllegalArgumentException if initial flow is infeasible
     */
    public FordFulkerson(FlowNetwork G, int s, int t) {
        V = G.V();
        validate(s);
        validate(t);
        if (s == t)               throw new IllegalArgumentException("Source equals sink");
        if (!isFeasible(G, s, t)) throw new IllegalArgumentException("Initial flow is infeasible");

        // while there exists an augmenting path, use it
        value = excess(G, t);
        while (hasAugmentingPath(G, s, t)) {

            // compute bottleneck capacity
            double bottle = Double.POSITIVE_INFINITY;
            for (int v = t; v != s; v = edgeTo[v].other(v)) {
                bottle = Math.min(bottle, edgeTo[v].residualCapacityTo(v));
            }

            // augment flow
            for (int v = t; v != s; v = edgeTo[v].other(v)) {
                edgeTo[v].addResidualFlowTo(v, bottle);
            }

            value += bottle;
        }

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

    /**
     * Returns the value of the maximum flow.
     *
     * @return the value of the maximum flow
     */
    public double value()  {
        return value;
    }

    /**
     * Returns true if the specified vertex is on the {@code s} side of the mincut.
     *
     * @param  v vertex
     * @return {@code true} if vertex {@code v} is on the {@code s} side of the micut;
     *         {@code false} otherwise
     * @throws IllegalArgumentException unless {@code 0 <= v < V}
     */
    public boolean inCut(int v)  {
        validate(v);
        return marked[v];
    }

    // throw an IllegalArgumentException if v is outside prescibed range
    private void validate(int v)  {
        if (v < 0 || v >= V)
            throw new IllegalArgumentException("vertex " + v + " is not between 0 and " + (V-1));
    }

    // is there an augmenting path?
    // if so, upon termination edgeTo[] will contain a parent-link representation of such a path
    // this implementation finds a shortest augmenting path (fewest number of edges),
    // which performs well both in theory and in practice
    //广度优先搜索寻找增广路径
    private boolean hasAugmentingPath(FlowNetwork G, int s, int t) {
        edgeTo = new FlowEdge[G.V()];
        marked = new boolean[G.V()];

        // breadth-first search
        Queue queue = new Queue();
        queue.enqueue(s);
        marked[s] = true;
        while (!queue.isEmpty() && !marked[t]) {
            int v = queue.dequeue();

            for (FlowEdge e : G.adj(v)) {
                int w = e.other(v);

                // if residual capacity from v to w
                if (e.residualCapacityTo(w) > 0) {
                    if (!marked[w]) {
                        edgeTo[w] = e;
                        marked[w] = true;
                        queue.enqueue(w);
                    }
                }
            }
        }

        // is there an augmenting path?
        return marked[t];
    }

 

    // return excess flow at vertex v
    private double excess(FlowNetwork G, int v) {
        double excess = 0.0;
        for (FlowEdge e : G.adj(v)) {
            if (v == e.from()) excess -= e.flow();
            else               excess += e.flow();
        }
        return excess;
    }

    // return excess flow at vertex v
    private boolean isFeasible(FlowNetwork G, int s, int t) {

        // check that capacity constraints are satisfied
        for (int v = 0; v < G.V(); v++) {
            for (FlowEdge e : G.adj(v)) {
                if (e.flow() < -FLOATING_POINT_EPSILON || e.flow() > e.capacity() + FLOATING_POINT_EPSILON) {
                    System.err.println("Edge does not satisfy capacity constraints: " + e);
                    return false;
                }
            }
        }

        // check that net flow into a vertex equals zero, except at source and sink
        if (Math.abs(value + excess(G, s)) > FLOATING_POINT_EPSILON) {
            System.err.println("Excess at source = " + excess(G, s));
            System.err.println("Max flow         = " + value);
            return false;
        }
        if (Math.abs(value - excess(G, t)) > FLOATING_POINT_EPSILON) {
            System.err.println("Excess at sink   = " + excess(G, t));
            System.err.println("Max flow         = " + value);
            return false;
        }
        for (int v = 0; v < G.V(); v++) {
            if (v == s || v == t) continue;
            else if (Math.abs(excess(G, v)) > FLOATING_POINT_EPSILON) {
                System.err.println("Net flow out of " + v + " doesn't equal zero");
                return false;
            }
        }
        return true;
    }

 

    // check optimality conditions
    private boolean check(FlowNetwork G, int s, int t) {

        // check that flow is feasible
        if (!isFeasible(G, s, t)) {
            System.err.println("Flow is infeasible");
            return false;
        }

        // check that s is on the source side of min cut and that t is not on source side
        if (!inCut(s)) {
            System.err.println("source " + s + " is not on source side of min cut");
            return false;
        }
        if (inCut(t)) {
            System.err.println("sink " + t + " is on source side of min cut");
            return false;
        }

        // check that value of min cut = value of max flow
        double mincutValue = 0.0;
        for (int v = 0; v < G.V(); v++) {
            for (FlowEdge e : G.adj(v)) {
                if ((v == e.from()) && inCut(e.from()) && !inCut(e.to()))
                    mincutValue += e.capacity();
            }
        }

        if (Math.abs(mincutValue - value) > FLOATING_POINT_EPSILON) {
            System.err.println("Max flow value = " + value + ", min cut value = " + mincutValue);
            return false;
        }

        return true;
    }

    /**
     * Unit tests the {@code FordFulkerson} data type.
     *
     * @param args the command-line arguments
     */
    public static void main(String[] args) {

        // create flow network with V vertices and E edges
        int V = Integer.parseInt(args[0]);
        int E = Integer.parseInt(args[1]);
        int s = 0, t = V-1;
        FlowNetwork G = new FlowNetwork(V, E);
        StdOut.println(G);

        // compute maximum flow and minimum cut
        FordFulkerson maxflow = new FordFulkerson(G, s, t);
        StdOut.println("Max flow from " + s + " to " + t);
        for (int v = 0; v < G.V(); v++) {
            for (FlowEdge e : G.adj(v)) {
                if ((v == e.from()) && e.flow() > 0)
                    StdOut.println("   " + e);
            }
        }

        // print min-cut
        StdOut.print("Min cut: ");
        for (int v = 0; v < G.V(); v++) {
            if (maxflow.inCut(v)) StdOut.print(v + " ");
        }
        StdOut.println();

        StdOut.println("Max flow value = " +  maxflow.value());
    }

}

 

public class FlowNetwork {
    private static final String NEWLINE = System.getProperty("line.separator");

    private final int V;
    private int E;
    private Bag[] adj;
    
    /**
     * Initializes an empty flow network with {@code V} vertices and 0 edges.
     * @param V the number of vertices
     * @throws IllegalArgumentException if {@code V < 0}
     */
    public FlowNetwork(int V) {
        if (V < 0) throw new IllegalArgumentException("Number of vertices in a Graph must be nonnegative");
        this.V = V;
        this.E = 0;
        adj = (Bag[]) new Bag[V];
        for (int v = 0; v < V; v++)
            adj[v] = new Bag();
    }

    /**
     * Initializes a random flow network with {@code V} vertices and E edges.
     * The capacities are integers between 0 and 99 and the flow values are zero.
     * @param V the number of vertices
     * @param E the number of edges
     * @throws IllegalArgumentException if {@code V < 0}
     * @throws IllegalArgumentException if {@code E < 0}
     */
    public FlowNetwork(int V, int E) {
        this(V);
        if (E < 0) throw new IllegalArgumentException("Number of edges must be nonnegative");
        for (int i = 0; i < E; i++) {
            int v = StdRandom.uniform(V);
            int w = StdRandom.uniform(V);
            double capacity = StdRandom.uniform(100);
            addEdge(new FlowEdge(v, w, capacity));
        }
    }

    /**  
     * Initializes a flow network from an input stream.
     * The format is the number of vertices V,
     * followed by the number of edges E,
     * followed by E pairs of vertices and edge capacities,
     * with each entry separated by whitespace.
     * @param in the input stream
     * @throws IllegalArgumentException if the endpoints of any edge are not in prescribed range
     * @throws IllegalArgumentException if the number of vertices or edges is negative
     */
    public FlowNetwork(In in) {
        this(in.readInt());
        int E = in.readInt();
        if (E < 0) throw new IllegalArgumentException("number of edges must be nonnegative");
        for (int i = 0; i < E; i++) {
            int v = in.readInt();
            int w = in.readInt();
            validateVertex(v);
            validateVertex(w);
            double capacity = in.readDouble();
            addEdge(new FlowEdge(v, w, capacity));
        }
    }

    /**
     * Returns the number of vertices in the edge-weighted graph.
     * @return the number of vertices in the edge-weighted graph
     */
    public int V() {
        return V;
    }

    /**
     * Returns the number of edges in the edge-weighted graph.
     * @return the number of edges in the edge-weighted graph
     */
    public int E() {
        return E;
    }

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

    /**
     * Adds the edge {@code e} to the network.
     * @param e the edge
     * @throws IllegalArgumentException unless endpoints of edge are between
     *         {@code 0} and {@code V-1}
     */
    public void addEdge(FlowEdge e) {
        int v = e.from();
        int w = e.to();
        validateVertex(v);
        validateVertex(w);
        adj[v].add(e);
        adj[w].add(e);
        E++;
    }

    /**
     * Returns the edges incident on vertex {@code v} (includes both edges pointing to
     * and from {@code v}).
     * @param v the vertex
     * @return the edges incident on vertex {@code v} as an Iterable
     * @throws IllegalArgumentException unless {@code 0 <= v < V}
     */
    public Iterable adj(int v) {
        validateVertex(v);
        return adj[v];
    }

    // return list of all edges - excludes self loops
    public Iterable edges() {
        Bag list = new Bag();
        for (int v = 0; v < V; v++)
            for (FlowEdge e : adj(v)) {
                if (e.to() != v)
                    list.add(e);
            }
        return list;
    }

    /**
     * Returns a string representation of the flow network.
     * This method takes time proportional to E + V.
     * @return the number of vertices V, followed by the number of edges E,  
     *    followed by the V adjacency lists
     */
    public String toString() {
        StringBuilder s = new StringBuilder();
        s.append(V + " " + E + NEWLINE);
        for (int v = 0; v < V; v++) {
            s.append(v + ":  ");
            for (FlowEdge e : adj[v]) {
                if (e.to() != v) s.append(e + "  ");
            }
            s.append(NEWLINE);
        }
        return s.toString();
    }

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

}

public class FlowEdge {
    // to deal with floating-point roundoff errors
    private static final double FLOATING_POINT_EPSILON = 1E-10;

    private final int v;             // from
    private final int w;             // to
    private final double capacity;   // capacity
    private double flow;             // flow

    /**
     * Initializes an edge from vertex {@code v} to vertex {@code w} with
     * the given {@code capacity} and zero flow.
     * @param v the tail vertex
     * @param w the head vertex
     * @param capacity the capacity of the edge
     * @throws IllegalArgumentException if either {@code v} or {@code w}
     *    is a negative integer
     * @throws IllegalArgumentException if {@code capacity < 0.0}
     */
    public FlowEdge(int v, int w, double capacity) {
        if (v < 0) throw new IllegalArgumentException("vertex index must be a non-negative integer");
        if (w < 0) throw new IllegalArgumentException("vertex index must be a non-negative integer");
        if (!(capacity >= 0.0)) throw new IllegalArgumentException("Edge capacity must be non-negative");
        this.v         = v;
        this.w         = w;  
        this.capacity  = capacity;
        this.flow      = 0.0;
    }

    /**
     * Initializes an edge from vertex {@code v} to vertex {@code w} with
     * the given {@code capacity} and {@code flow}.
     * @param v the tail vertex
     * @param w the head vertex
     * @param capacity the capacity of the edge
     * @param flow the flow on the edge
     * @throws IllegalArgumentException if either {@code v} or {@code w}
     *    is a negative integer
     * @throws IllegalArgumentException if {@code capacity} is negative
     * @throws IllegalArgumentException unless {@code flow} is between
     *    {@code 0.0} and {@code capacity}.
     */
    public FlowEdge(int v, int w, double capacity, double flow) {
        if (v < 0) throw new IllegalArgumentException("vertex index must be a non-negative integer");
        if (w < 0) throw new IllegalArgumentException("vertex index must be a non-negative integer");
        if (!(capacity >= 0.0))  throw new IllegalArgumentException("edge capacity must be non-negative");
        if (!(flow <= capacity)) throw new IllegalArgumentException("flow exceeds capacity");
        if (!(flow >= 0.0))      throw new IllegalArgumentException("flow must be non-negative");
        this.v         = v;
        this.w         = w;  
        this.capacity  = capacity;
        this.flow      = flow;
    }

    /**
     * Initializes a flow edge from another flow edge.
     * @param e the edge to copy
     */
    public FlowEdge(FlowEdge e) {
        this.v         = e.v;
        this.w         = e.w;
        this.capacity  = e.capacity;
        this.flow      = e.flow;
    }

    /**
     * Returns the tail vertex of the edge.
     * @return the tail vertex of the edge
     */
    public int from() {
        return v;
    }  

    /**
     * Returns the head vertex of the edge.
     * @return the head vertex of the edge
     */
    public int to() {
        return w;
    }  

    /**
     * Returns the capacity of the edge.
     * @return the capacity of the edge
     */
    public double capacity() {
        return capacity;
    }

    /**
     * Returns the flow on the edge.
     * @return the flow on the edge
     */
    public double flow() {
        return flow;
    }

    /**
     * Returns the endpoint of the edge that is different from the given vertex
     * (unless the edge represents a self-loop in which case it returns the same vertex).
     * @param vertex one endpoint of the edge
     * @return the endpoint of the edge that is different from the given vertex
     *   (unless the edge represents a self-loop in which case it returns the same vertex)
     * @throws IllegalArgumentException if {@code vertex} is not one of the endpoints
     *   of the edge
     */
    public int other(int vertex) {
        if      (vertex == v) return w;
        else if (vertex == w) return v;
        else throw new IllegalArgumentException("invalid endpoint");
    }

    /**
     * Returns the residual capacity of the edge in the direction
     *  to the given {@code vertex}.
     * @param vertex one endpoint of the edge
     * @return the residual capacity of the edge in the direction to the given vertex
     *   If {@code vertex} is the tail vertex, the residual capacity equals
     *   {@code capacity() - flow()}; if {@code vertex} is the head vertex, the
     *   residual capacity equals {@code flow()}.
     * @throws IllegalArgumentException if {@code vertex} is not one of the endpoints of the edge
     */
    public double residualCapacityTo(int vertex) {
        if      (vertex == v) return flow;              // backward edge
        else if (vertex == w) return capacity - flow;   // forward edge
        else throw new IllegalArgumentException("invalid endpoint");
    }

    /**
     * Increases the flow on the edge in the direction to the given vertex.
     *   If {@code vertex} is the tail vertex, this increases the flow on the edge by {@code delta};
     *   if {@code vertex} is the head vertex, this decreases the flow on the edge by {@code delta}.
     * @param vertex one endpoint of the edge
     * @param delta amount by which to increase flow
     * @throws IllegalArgumentException if {@code vertex} is not one of the endpoints
     *   of the edge
     * @throws IllegalArgumentException if {@code delta} makes the flow on
     *   on the edge either negative or larger than its capacity
     * @throws IllegalArgumentException if {@code delta} is {@code NaN}
     */
    public void addResidualFlowTo(int vertex, double delta) {
        if (!(delta >= 0.0)) throw new IllegalArgumentException("Delta must be nonnegative");

        if      (vertex == v) flow -= delta;           // backward edge
        else if (vertex == w) flow += delta;           // forward edge
        else throw new IllegalArgumentException("invalid endpoint");

        // round flow to 0 or capacity if within floating-point precision
        if (Math.abs(flow) <= FLOATING_POINT_EPSILON)
            flow = 0;
        if (Math.abs(flow - capacity) <= FLOATING_POINT_EPSILON)
            flow = capacity;

        if (!(flow >= 0.0))      throw new IllegalArgumentException("Flow is negative");
        if (!(flow <= capacity)) throw new IllegalArgumentException("Flow exceeds capacity");
    }

    /**
     * Returns a string representation of the edge.
     * @return a string representation of the edge
     */
    public String toString() {
        return v + "->" + w + " " + flow + "/" + capacity;
    }

   /**
     * Unit tests the {@code FlowEdge} data type.
     *
     * @param args the command-line arguments
     */
    public static void main(String[] args) {
        FlowEdge e = new FlowEdge(12, 23, 4.56);
        StdOut.println(e);
    }

}