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/******************************************************************************
* Compilation: javac FordFulkerson.java
* Execution: java FordFulkerson V E
* Dependencies: FlowNetwork.java FlowEdge.java Queue.java
* Data files: https://algs4.cs.princeton.edu/65maxflow/tinyFN.txt
*
* Ford-Fulkerson algorithm for computing a max flow and
* a min cut using shortest augmenting path rule.
*
******************************************************************************/
package edu.princeton.cs.algs4;
/**
* The {@code FordFulkerson} class represents a data type for computing a
* maximum st-flow and minimum st-cut in a flow
* network.
*
* This implementation uses the Ford-Fulkerson algorithm with
* the shortest augmenting path heuristic.
* The constructor takes time proportional to E V (E + V)
* in the worst case and extra space (not including the network)
* proportional to V, where V is the number of vertices
* and E is the number of edges. In practice, the algorithm will
* run much faster.
* Afterwards, the {@code inCut()} and {@code value()} methods take
* constant time.
*
* 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.
*
* For additional documentation,
* see Section 6.4 of
* Algorithms, 4th Edition by Robert Sedgewick and Kevin Wayne.
*
* @author Robert Sedgewick
* @author Kevin Wayne
*/
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
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());
}
}
/******************************************************************************
* Copyright 2002-2018, Robert Sedgewick and Kevin Wayne.
*
* This file is part of algs4.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.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.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.jar. If not, see http://www.gnu.org/licenses.
******************************************************************************/