Source code

001package algs64; // section 6.4
002import stdlib.*;
003import algs13.Queue;
004/* ***********************************************************************
005 *  Compilation:  javac FordFulkerson.java
006 *  Execution:    java FordFulkerson V E
007 *  Dependencies: FlowNetwork.java FlowEdge.java Queue.java
008 *
009 *  Ford-Fulkerson algorithm for computing a max flow and
010 *  a min cut using shortest augmenthing path rule.
011 *
012 *********************************************************************/
013
014public class FordFulkerson {
015        private boolean[] marked;     // marked[v] = true iff s->v path in residual graph
016        private FlowEdge[] edgeTo;    // edgeTo[v] = last edge on shortest residual s->v path
017        private double value;         // current value of max flow
018
019        // max flow in flow network G from s to t
020        public FordFulkerson(FlowNetwork G, int s, int t) {
021                if (s < 0 || s >= G.V()) {
022                        throw new IndexOutOfBoundsException("Source s is invalid: " + s);
023                }
024                if (t < 0 || t >= G.V()) {
025                        throw new IndexOutOfBoundsException("Sink t is invalid: " + t);
026                }
027                if (s == t) {
028                        throw new IllegalArgumentException("Source equals sink");
029                }
030                value = excess(G, t);
031                if (!isFeasible(G, s, t)) {
032                        throw new Error("Initial flow is infeasible");
033                }
034
035                // while there exists an augmenting path, use it
036                while (hasAugmentingPath(G, s, t)) {
037
038                        // compute bottleneck capacity
039                        double bottle = Double.POSITIVE_INFINITY;
040                        for (int v = t; v != s; v = edgeTo[v].other(v)) {
041                                bottle = Math.min(bottle, edgeTo[v].residualCapacityTo(v));
042                        }
043
044                        // augment flow
045                        for (int v = t; v != s; v = edgeTo[v].other(v)) {
046                                edgeTo[v].addResidualFlowTo(v, bottle);
047                        }
048
049                        value += bottle;
050                }
051
052                // check optimality conditions
053                assert check(G, s, t);
054        }
055
056        // return value of max flow
057        public double value()  {
058                return value;
059        }
060
061        // is v in the s side of the min s-t cut?
062        public boolean inCut(int v)  {
063                return marked[v];
064        }
065
066
067        // return an augmenting path if one exists, otherwise return null
068        private boolean hasAugmentingPath(FlowNetwork G, int s, int t) {
069                edgeTo = new FlowEdge[G.V()];
070                marked = new boolean[G.V()];
071
072                // breadth-first search
073                Queue<Integer> q = new Queue<>();
074                q.enqueue(s);
075                marked[s] = true;
076                while (!q.isEmpty()) {
077                        int v = q.dequeue();
078
079                        for (FlowEdge e : G.adj(v)) {
080                                int w = e.other(v);
081
082                                // if residual capacity from v to w
083                                if (e.residualCapacityTo(w) > 0) {
084                                        if (!marked[w]) {
085                                                edgeTo[w] = e;
086                                                marked[w] = true;
087                                                q.enqueue(w);
088                                        }
089                                }
090                        }
091                }
092
093                // is there an augmenting path?
094                return marked[t];
095        }
096
097
098
099        // return excess flow at vertex v
100        private double excess(FlowNetwork G, int v) {
101                double excess = 0.0;
102                for (FlowEdge e : G.adj(v)) {
103                        if (v == e.from()) excess -= e.flow();
104                        else               excess += e.flow();
105                }
106                return excess;
107        }
108
109        // return excess flow at vertex v
110        private boolean isFeasible(FlowNetwork G, int s, int t) {
111                double EPSILON = 1E-11;
112
113                // check that capacity constraints are satisfied
114                for (int v = 0; v < G.V(); v++) {
115                        for (FlowEdge e : G.adj(v)) {
116                                if (e.flow() < -EPSILON || e.flow() > e.capacity() + EPSILON) {
117                                        System.err.println("Edge does not satisfy capacity constraints: " + e);
118                                        return false;
119                                }
120                        }
121                }
122
123                // check that net flow into a vertex equals zero, except at source and sink
124                if (Math.abs(value + excess(G, s)) > EPSILON) {
125                        System.err.println("Excess at source = " + excess(G, s));
126                        System.err.println("Max flow         = " + value);
127                        return false;
128                }
129                if (Math.abs(value - excess(G, t)) > EPSILON) {
130                        System.err.println("Excess at sink   = " + excess(G, t));
131                        System.err.println("Max flow         = " + value);
132                        return false;
133                }
134                for (int v = 0; v < G.V(); v++) {
135                        if (v == s || v == t) continue;
136                        else if (Math.abs(excess(G, v)) > EPSILON) {
137                                System.err.println("Net flow out of " + v + " doesn't equal zero");
138                                return false;
139                        }
140                }
141                return true;
142        }
143
144
145
146        // check optimality conditions
147        private boolean check(FlowNetwork G, int s, int t) {
148
149                // check that flow is feasible
150                if (!isFeasible(G, s, t)) {
151                        System.err.println("Flow is infeasible");
152                        return false;
153                }
154
155                // check that s is on the source side of min cut and that t is not on source side
156                if (!inCut(s)) {
157                        System.err.println("source " + s + " is not on source side of min cut");
158                        return false;
159                }
160                if (inCut(t)) {
161                        System.err.println("sink " + t + " is on source side of min cut");
162                        return false;
163                }
164
165                // check that value of min cut = value of max flow
166                double mincutValue = 0.0;
167                for (int v = 0; v < G.V(); v++) {
168                        for (FlowEdge e : G.adj(v)) {
169                                if ((v == e.from()) && inCut(e.from()) && !inCut(e.to()))
170                                        mincutValue += e.capacity();
171                        }
172                }
173
174                double EPSILON = 1E-11;
175                if (Math.abs(mincutValue - value) > EPSILON) {
176                        System.err.println("Max flow value = " + value + ", min cut value = " + mincutValue);
177                        return false;
178                }
179
180                return true;
181        }
182
183
184        // test client that creates random network, solves max flow, and prints results
185        public static void main(String[] args) {
186
187                // create flow network with V vertices and E edges
188                int V = Integer.parseInt(args[0]);
189                int E = Integer.parseInt(args[1]);
190                int s = 0, t = V-1;
191                FlowNetwork G = new FlowNetwork(V, E);
192                StdOut.println(G);
193
194                // compute maximum flow and minimum cut
195                FordFulkerson maxflow = new FordFulkerson(G, s, t);
196                StdOut.println("Max flow from " + s + " to " + t);
197                for (int v = 0; v < G.V(); v++) {
198                        for (FlowEdge e : G.adj(v)) {
199                                if ((v == e.from()) && e.flow() > 0)
200                                        StdOut.println("   " + e);
201                        }
202                }
203
204                // print min-cut
205                StdOut.print("Min cut: ");
206                for (int v = 0; v < G.V(); v++) {
207                        if (maxflow.inCut(v)) StdOut.print(v + " ");
208                }
209                StdOut.println();
210
211                StdOut.println("Max flow value = " +  maxflow.value());
212        }
213
214}