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/*
* (C) Copyright 2011-2021, by Robby McKilliam, Ernst de Ridder and Contributors.
*
* JGraphT : a free Java graph-theory library
*
* See the CONTRIBUTORS.md file distributed with this work for additional
* information regarding copyright ownership.
*
* This program and the accompanying materials are made available under the
* terms of the Eclipse Public License 2.0 which is available at
* http://www.eclipse.org/legal/epl-2.0, or the
* GNU Lesser General Public License v2.1 or later
* which is available at
* http://www.gnu.org/licenses/old-licenses/lgpl-2.1-standalone.html.
*
* SPDX-License-Identifier: EPL-2.0 OR LGPL-2.1-or-later
*/
package org.jgrapht.alg;
import org.jgrapht.*;
import org.jgrapht.graph.*;
import java.util.*;
/**
* Implements the Stoer and Wagner minimum cut
* algorithm. Deterministically computes the minimum cut in $O(|V||E| + |V| \log |V|)$ time.
* This implementation uses Java's PriorityQueue and requires $O(|V||E| \log |E|)$ time. M. Stoer
* and F. Wagner, "A Simple Min-Cut Algorithm", Journal of the ACM, volume 44, number 4. pp 585-591,
* 1997.
*
* @param the graph vertex type
* @param the graph edge type
*
* @author Robby McKilliam
* @author Ernst de Ridder
*/
public class StoerWagnerMinimumCut
{
final Graph, DefaultWeightedEdge> workingGraph;
protected double bestCutWeight = Double.POSITIVE_INFINITY;
protected Set bestCut;
/**
* Will compute the minimum cut in graph.
*
* @param graph graph over which to run algorithm
*
* @throws IllegalArgumentException if a negative weight edge is found
* @throws IllegalArgumentException if graph has less than 2 vertices
*/
public StoerWagnerMinimumCut(Graph graph)
{
GraphTests.requireUndirected(graph, "Graph must be undirected");
if (graph.vertexSet().size() < 2) {
throw new IllegalArgumentException("Graph has less than 2 vertices");
}
// get a version of this graph where each vertex is wrapped with a list
workingGraph = new SimpleWeightedGraph<>(DefaultWeightedEdge.class);
Map> vertexMap = new HashMap<>();
for (V v : graph.vertexSet()) {
Set list = new HashSet<>();
list.add(v);
vertexMap.put(v, list);
workingGraph.addVertex(list);
}
for (E e : graph.edgeSet()) {
if (graph.getEdgeWeight(e) < 0.0) {
throw new IllegalArgumentException("Negative edge weights not allowed");
}
V s = graph.getEdgeSource(e);
Set sNew = vertexMap.get(s);
V t = graph.getEdgeTarget(e);
Set tNew = vertexMap.get(t);
// For multigraphs, we sum the edge weights (either all are
// contained in a cut, or none)
DefaultWeightedEdge eNew = workingGraph.getEdge(sNew, tNew);
if (eNew == null) {
eNew = workingGraph.addEdge(sNew, tNew);
workingGraph.setEdgeWeight(eNew, graph.getEdgeWeight(e));
} else {
workingGraph
.setEdgeWeight(eNew, workingGraph.getEdgeWeight(eNew) + graph.getEdgeWeight(e));
}
}
// arbitrary vertex used to seed the algorithm.
Set a = workingGraph.vertexSet().iterator().next();
while (workingGraph.vertexSet().size() > 1) {
minimumCutPhase(a);
}
}
/**
* Implements the MinimumCutPhase function of Stoer and Wagner.
*
* @param a the vertex
*/
protected void minimumCutPhase(Set a)
{
// The last and before last vertices added to A.
Set last = a, beforelast = null;
// queue contains vertices not in A ordered by max weight of edges to A.
PriorityQueue queue = new PriorityQueue<>();
// Maps vertices to elements of queue
Map, VertexAndWeight> dmap = new HashMap<>();
// Initialize queue
for (Set v : workingGraph.vertexSet()) {
if (v == a) {
continue;
}
DefaultWeightedEdge e = workingGraph.getEdge(v, a);
Double w = (e == null) ? 0.0 : workingGraph.getEdgeWeight(e);
VertexAndWeight vandw = new VertexAndWeight(v, w, e != null);
queue.add(vandw);
dmap.put(v, vandw);
}
// Now iteratively update the queue to get the required vertex ordering
while (!queue.isEmpty()) {
Set v = queue.poll().vertex;
dmap.remove(v);
beforelast = last;
last = v;
for (DefaultWeightedEdge e : workingGraph.edgesOf(v)) {
Set vc = Graphs.getOppositeVertex(workingGraph, e, v);
VertexAndWeight vcandw = dmap.get(vc);
if (vcandw != null) {
queue.remove(vcandw); // this is O(log n) but could be O(1)?
vcandw.active = true;
vcandw.weight += workingGraph.getEdgeWeight(e);
queue.add(vcandw); // this is O(log n) but could be O(1)?
}
}
}
// Update the best cut
double w = vertexWeight(last);
if (w < bestCutWeight) {
bestCutWeight = w;
bestCut = last;
}
// merge the last added vertices
mergeVertices(beforelast, last);
}
/**
* Return the weight of the minimum cut
*
* @return the weight of the minimum cut
*/
public double minCutWeight()
{
return bestCutWeight;
}
/**
* Return a set of vertices on one side of the cut
*
* @return a set of vertices on one side of the cut
*/
public Set minCut()
{
return bestCut;
}
/**
* Merges vertex $t$ into vertex $s$, summing the weights as required. Returns the merged vertex
* and the sum of its weights
*
* @param s the first vertex
* @param t the second vertex
*
* @return the merged vertex and its weight
*/
protected VertexAndWeight mergeVertices(Set s, Set t)
{
// construct the new combinedvertex
Set set = new HashSet<>();
set.addAll(s);
set.addAll(t);
workingGraph.addVertex(set);
// add edges and weights to the combined vertex
double wsum = 0.0;
for (Set v : workingGraph.vertexSet()) {
if ((s != v) && (t != v)) {
double neww = 0.0;
DefaultWeightedEdge etv = workingGraph.getEdge(t, v);
DefaultWeightedEdge esv = workingGraph.getEdge(s, v);
if (etv != null) {
neww += workingGraph.getEdgeWeight(etv);
}
if (esv != null) {
neww += workingGraph.getEdgeWeight(esv);
}
if ((etv != null) || (esv != null)) {
wsum += neww;
workingGraph.setEdgeWeight(workingGraph.addEdge(set, v), neww);
}
}
}
// remove original vertices
workingGraph.removeVertex(t);
workingGraph.removeVertex(s);
return new VertexAndWeight(set, wsum, false);
}
/**
* Compute the sum of the weights entering a vertex
*
* @param v the vertex
* @return the sum of the weights entering a vertex
*/
public double vertexWeight(Set v)
{
double wsum = 0.0;
for (DefaultWeightedEdge e : workingGraph.edgesOf(v)) {
wsum += workingGraph.getEdgeWeight(e);
}
return wsum;
}
/**
* Class for weighted vertices
*/
protected class VertexAndWeight
implements
Comparable
{
public Set vertex;
public Double weight;
public boolean active; // active == neighbour in A
/**
* Construct a new weighted vertex.
*
* @param v the vertex
* @param w the weight of the vertex
* @param active whether it is active
*/
public VertexAndWeight(Set v, double w, boolean active)
{
this.vertex = v;
this.weight = w;
this.active = active;
}
/**
* compareTo that sorts in reverse order because we need extract-max and queue provides
* extract-min.
*/
@Override
public int compareTo(VertexAndWeight that)
{
if (this.active && that.active) {
return -Double.compare(weight, that.weight);
}
if (this.active && !that.active) {
return -1;
}
if (!this.active && that.active) {
return +1;
}
// both inactive
return 0;
}
@Override
public String toString()
{
return "(" + vertex + ", " + weight + ")";
}
}
}
