267 lines
8.8 KiB
Java
267 lines
8.8 KiB
Java
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import edu.princeton.cs.introcs.StdOut;
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/*************************************************************************
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* Compilation: javac FordFulkerson.java
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* Execution: java FordFulkerson V E
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* Dependencies: FlowNetwork.java FlowEdge.java Queue.java
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*
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* Ford-Fulkerson algorithm for computing a max flow and
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* a min cut using shortest augmenting path rule.
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*
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*********************************************************************/
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/**
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* The FordFulkerson class represents a data type for computing a
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* maximum st-flow and minimum st-cut in a flow
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* network.
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*
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* This implementation uses the Ford-Fulkerson algorithm with
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* the shortest augmenting path heuristic.
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* The constructor takes time proportional to E V ( E + V )
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* in the worst case and extra space (not including the network)
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* proportional to V , where V is the number of vertices
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* and E is the number of edges. In practice, the algorithm will
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* run much faster.
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* Afterwards, the inCut() and value() methods take
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* constant time.
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*
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* If the capacities and initial flow values are all integers, then this
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* implementation guarantees to compute an integer-valued maximum flow.
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* If the capacities and floating-point numbers, then floating-point
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* roundoff error can accumulate.
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*
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* For additional documentation, see <a href="http://algs4.cs.princeton.edu/64maxflow">Section 6.4</a>
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* Algorithms, 4th Edition by Robert Sedgewick and Kevin Wayne.
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*
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* @author Robert Sedgewick
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* @author Kevin Wayne
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*/
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public class FordFulkerson {
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private boolean[] marked; // marked[v] = true iff s->v path in residual graph
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private FlowEdge[] edgeTo; // edgeTo[v] = last edge on shortest residual s->v path
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private double value; // current value of max flow
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/**
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* Compute a maximum flow and minimum cut in the network G
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* from vertex s to vertex t .
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* @param G the flow network
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* @param s the source vertex
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* @param t the sink vertex
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* @throws IndexOutOfBoundsException unless 0 <= s < V
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* @throws IndexOutOfBoundsException unless 0 <= t < V
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* @throws IllegalArgumentException if s = t
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* @throws IllegalArgumentException if initial flow is infeasible
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*/
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public FordFulkerson(FlowNetwork G, int s, int t) {
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if (s < 0 || s >= G.V()) {
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throw new IndexOutOfBoundsException("Source s is invalid: " + s);
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}
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if (t < 0 || t >= G.V()) {
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throw new IndexOutOfBoundsException("Sink t is invalid: " + t);
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}
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if (s == t) {
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throw new IllegalArgumentException("Source equals sink");
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}
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value = excess(G, t);
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if (!isFeasible(G, s, t)) {
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throw new IllegalArgumentException("Initial flow is infeasible");
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}
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// while there exists an augmenting path, use it
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while (hasAugmentingPath(G, s, t)) {
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// compute bottleneck capacity
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double bottle = Double.POSITIVE_INFINITY;
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for (int v = t; v != s; v = edgeTo[v].other(v)) {
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bottle = Math.min(bottle, edgeTo[v].residualCapacityTo(v));
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}
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// augment flow
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for (int v = t; v != s; v = edgeTo[v].other(v)) {
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edgeTo[v].addResidualFlowTo(v, bottle);
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}
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value += bottle;
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}
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// check optimality conditions
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assert check(G, s, t);
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}
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/**
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* Returns the value of the maximum flow.
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* @return the value of the maximum flow
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*/
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public double value() {
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return value;
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}
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// is v in the s side of the min s-t cut?
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/**
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* Is vertex v on the s side of the minimum st-cut?
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* @return true if vertex v is on the s side of the micut,
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* and false if vertex v is on the t side.
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* @throws IndexOutOfBoundsException unless 0 <= v < V
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*/
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public boolean inCut(int v) {
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int V = marked.length;
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if (v < 0 || v >= V)
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throw new IndexOutOfBoundsException("vertex " + v + " is not between 0 and " + (V-1));
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return marked[v];
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}
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// is there an augmenting path?
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// if so, upon termination edgeTo[] will contain a parent-link representation of such a path
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private boolean hasAugmentingPath(FlowNetwork G, int s, int t) {
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edgeTo = new FlowEdge[G.V()];
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marked = new boolean[G.V()];
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// breadth-first search
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Queue<Integer> q = new Queue<Integer>();
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q.enqueue(s);
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marked[s] = true;
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while (!q.isEmpty()) {
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int v = q.dequeue();
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for (FlowEdge e : G.adj(v)) {
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int w = e.other(v);
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// if residual capacity from v to w
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if (e.residualCapacityTo(w) > 0) {
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if (!marked[w]) {
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edgeTo[w] = e;
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marked[w] = true;
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q.enqueue(w);
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}
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}
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}
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}
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// is there an augmenting path?
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return marked[t];
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}
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// return excess flow at vertex v
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private double excess(FlowNetwork G, int v) {
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double excess = 0.0;
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for (FlowEdge e : G.adj(v)) {
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if (v == e.from()) excess -= e.flow();
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else excess += e.flow();
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}
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return excess;
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}
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// return excess flow at vertex v
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private boolean isFeasible(FlowNetwork G, int s, int t) {
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double EPSILON = 1E-11;
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// check that capacity constraints are satisfied
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for (int v = 0; v < G.V(); v++) {
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for (FlowEdge e : G.adj(v)) {
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if (e.flow() < -EPSILON || e.flow() > e.capacity() + EPSILON) {
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System.err.println("Edge does not satisfy capacity constraints: " + e);
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return false;
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}
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}
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}
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// check that net flow into a vertex equals zero, except at source and sink
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if (Math.abs(value + excess(G, s)) > EPSILON) {
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System.err.println("Excess at source = " + excess(G, s));
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System.err.println("Max flow = " + value);
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return false;
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}
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if (Math.abs(value - excess(G, t)) > EPSILON) {
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System.err.println("Excess at sink = " + excess(G, t));
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System.err.println("Max flow = " + value);
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return false;
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}
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for (int v = 0; v < G.V(); v++) {
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if (v == s || v == t) continue;
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else if (Math.abs(excess(G, v)) > EPSILON) {
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System.err.println("Net flow out of " + v + " doesn't equal zero");
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return false;
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}
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}
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return true;
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}
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// check optimality conditions
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private boolean check(FlowNetwork G, int s, int t) {
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// check that flow is feasible
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if (!isFeasible(G, s, t)) {
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System.err.println("Flow is infeasible");
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return false;
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}
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// check that s is on the source side of min cut and that t is not on source side
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if (!inCut(s)) {
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System.err.println("source " + s + " is not on source side of min cut");
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return false;
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}
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if (inCut(t)) {
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System.err.println("sink " + t + " is on source side of min cut");
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return false;
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}
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// check that value of min cut = value of max flow
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double mincutValue = 0.0;
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for (int v = 0; v < G.V(); v++) {
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for (FlowEdge e : G.adj(v)) {
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if ((v == e.from()) && inCut(e.from()) && !inCut(e.to()))
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mincutValue += e.capacity();
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}
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}
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double EPSILON = 1E-11;
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if (Math.abs(mincutValue - value) > EPSILON) {
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System.err.println("Max flow value = " + value + ", min cut value = " + mincutValue);
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return false;
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}
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return true;
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}
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/**
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* Unit tests the FordFulkerson data type.
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*/
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public static void main(String[] args) {
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// create flow network with V vertices and E edges
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int V = Integer.parseInt(args[0]);
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int E = Integer.parseInt(args[1]);
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int s = 0, t = V-1;
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FlowNetwork G = new FlowNetwork(V, E);
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StdOut.println(G);
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// compute maximum flow and minimum cut
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FordFulkerson maxflow = new FordFulkerson(G, s, t);
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StdOut.println("Max flow from " + s + " to " + t);
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for (int v = 0; v < G.V(); v++) {
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for (FlowEdge e : G.adj(v)) {
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if ((v == e.from()) && e.flow() > 0)
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StdOut.println(" " + e);
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}
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}
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// print min-cut
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StdOut.print("Min cut: ");
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for (int v = 0; v < G.V(); v++) {
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if (maxflow.inCut(v)) StdOut.print(v + " ");
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}
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StdOut.println();
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StdOut.println("Max flow value = " + maxflow.value());
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}
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}
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