package edu.princeton.cs.algs4;
import edu.princeton.cs.algs4.ch13.Queue;
import edu.princeton.cs.introcs.*;
/*************************************************************************
* Compilation: javac FordFulkerson.java
* Execution: java FordFulkerson V E
* Dependencies: FlowNetwork.java FlowEdge.java Queue.java
*
* Ford-Fulkerson algorithm for computing a max flow and
* a min cut using shortest augmenting path rule.
*
*********************************************************************/
/**
* The <tt>FordFulkerson</tt> class represents a data type for computing a
* <em>maximum st-flow</em> and <em>minimum st-cut</em> in a flow
* network.
* <p>
* This implementation uses the <em>Ford-Fulkerson</em> algorithm with
* the <em>shortest augmenting path</em> heuristic.
* The constructor takes time proportional to <em>E V</em> (<em>E</em> + <em>V</em>)
* in the worst case and extra space (not including the network)
* proportional to <em>V</em>, where <em>V</em> is the number of vertices
* and <em>E</em> is the number of edges. In practice, the algorithm will
* run much faster.
* Afterwards, the <tt>inCut()</tt> and <tt>value()</tt> methods take
* constant time.
* <p>
* If the capacities and initial flow values are all integers, then this
* implementation guarantees to compute an integer-valued maximum flow.
* If the capacities and floating-point numbers, then floating-point
* roundoff error can accumulate.
* <p>
* For additional documentation, see <a href="http://algs4.cs.princeton.edu/64maxflow">Section 6.4</a>
* <i>Algorithms, 4th Edition</i> by Robert Sedgewick and Kevin Wayne.
*
* @author Robert Sedgewick
* @author Kevin Wayne
*/
public class FordFulkerson {
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 <tt>G</tt>
* from vertex <tt>s</tt> to vertex <tt>t</tt>.
* @param G the flow network
* @param s the source vertex
* @param t the sink vertex
* @throws IndexOutOfBoundsException unless 0 <= s < V
* @throws IndexOutOfBoundsException unless 0 <= t < V
* @throws IllegalArgumentException if s = t
* @throws IllegalArgumentException if initial flow is infeasible
*/
public FordFulkerson(FlowNetwork G, int s, int t) {
validate(s, G.V());
validate(t, G.V());
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;
}
// is v in the s side of the min s-t cut?
/**
* Is vertex <tt>v</tt> on the <tt>s</tt> side of the minimum st-cut?
* @return <tt>true</tt> if vertex <tt>v</tt> is on the <tt>s</tt> side of the micut,
* and <tt>false</tt> if vertex <tt>v</tt> is on the <tt>t</tt> side.
* @throws IndexOutOfBoundsException unless 0 <= v < V
*/
public boolean inCut(int v) {
validate(v, marked.length);
return marked[v];
}
// throw an exception if v is outside prescibed range
private void validate(int v, int V) {
if (v < 0 || v >= V)
throw new IndexOutOfBoundsException("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
private boolean hasAugmentingPath(FlowNetwork G, int s, int t) {
edgeTo = new FlowEdge[G.V()];
marked = new boolean[G.V()];
// breadth-first search
Queue<Integer> queue = new Queue<Integer>();
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) {
double EPSILON = 1E-11;
// check that capacity constraints are satisfied
for (int v = 0; v < G.V(); v++) {
for (FlowEdge e : G.adj(v)) {
if (e.flow() < -EPSILON || e.flow() > e.capacity() + 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)) > 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)) > 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)) > 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();
}
}
double EPSILON = 1E-11;
if (Math.abs(mincutValue - value) > EPSILON) {
System.err.println("Max flow value = " + value + ", min cut value = " + mincutValue);
return false;
}
return true;
}
/**
* Unit tests the <tt>FordFulkerson</tt> data type.
*/
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());
}
}
/*************************************************************************
* Copyright 2002-2012, Robert Sedgewick and Kevin Wayne.
*
* This file is part of algs4-package.jar, which accompanies the textbook
*
* Algorithms, 4th edition by Robert Sedgewick and Kevin Wayne,
* Addison-Wesley Professional, 2011, ISBN 0-321-57351-X.
* http://algs4.cs.princeton.edu
*
*
* algs4-package.jar is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* algs4-package.jar is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
* You should have received a copy of the GNU General Public License
* along with algs4-package.jar. If not, see http://www.gnu.org/licenses.
*************************************************************************/