org.jgrapht.alg.matching.SparseEdmondsMaximumCardinalityMatching Maven / Gradle / Ivy
/*
* (C) Copyright 2019-2021, by Dimitrios Michail 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.matching;
import org.jgrapht.*;
import org.jgrapht.alg.interfaces.*;
import org.jgrapht.alg.util.*;
import org.jgrapht.util.*;
import java.util.*;
/**
* Edmonds' blossom algorithm for maximum cardinality matching in general undirected graphs.
*
*
* A matching in a graph $G(V,E)$ is a subset of edges $M$ such that no two edges in $M$ have a
* vertex in common. A matching has at most $\frac{1}{2}|V|$ edges. A node $v$ in $G$ is matched by
* matching $M$ if $M$ contains an edge incident to $v$. A matching is perfect if all nodes are
* matched. By definition, a perfect matching consists of exactly $\frac{1}{2}|V|$ edges. This
* algorithm will return a perfect matching if one exists. If no perfect matching exists, then the
* largest (non-perfect) matching is returned instead. In the special case that the input graph is
* bipartite, consider using {@link HopcroftKarpMaximumCardinalityBipartiteMatching} instead.
*
*
* To compute a maximum cardinality matching, at most $n$ augmenting path computations are
* performed. Each augmenting path computation takes $O(m \alpha(m,n))$ time, where $\alpha(m,n)$ is
* the inverse of the Ackerman function, $n$ is the number of vertices, and $m$ the number of edges.
* This results in a total runtime complexity of O(n m alpha(m,n)). In practice, the number of
* augmenting path computations performed is far smaller than $n$, since an efficient heuristic is
* used to compute a near-optimal initial solution. The heuristic by default is the
* {@link GreedyMaximumCardinalityMatching} but can be changed using the appropriate constructor.
*
*
* The runtime complexity of this implementation could be improved to $O(n m)$ when the UnionFind
* data structure used in this implementation is replaced by the linear-time set union data
* structure proposed in: Gabow, H.N., Tarjan, R.E. A linear-time algorithm for a special case of
* disjoint set union. Proc. Fifteenth Annual ACM Symposium on Theory of Computing, 1982, pp.
* 246-251.
*
* Edmonds' original algorithm first appeared in Edmonds, J. Paths, trees, and flowers. Canadian
* Journal of Mathematics 17, 1965, pp. 449-467, and had a runtime complexity of $O(n^4)$.
*
*
* This is the algorithm and implementation described in the
* LEDA book. See the LEDA
* Platform of Combinatorial and Geometric Computing, Cambridge University Press, 1999.
*
*
* For future reference - A more efficient algorithm exists: S. Micali and V. Vazirani. An
* $O(\sqrt{n}m)$ algorithm for finding maximum matching in general graphs. Proc. 21st Ann. Symp. on
* Foundations of Computer Science, IEEE, 1980, pp. 17–27. This is the most efficient algorithm
* known for computing maximum cardinality matchings in general graphs. More details on this
* algorithm can be found in:
*
* - Presentation from
* Vazirani 'Dispelling an Old Myth about an Ancient Algorithm'
* - Vazirani, V. A Simplification of the MV Matching
* Algorithm and its Proof, 2013
*
*
* @param the graph vertex type
* @param the graph edge type
*
* @author Dimitrios Michail
* @author Joris Kinable
*/
public class SparseEdmondsMaximumCardinalityMatching
implements
MatchingAlgorithm
{
private final Graph graph;
private MatchingAlgorithm initializer;
private Matching result;
private Map oddSetCover;
/**
* Constructs a new instance of the algorithm. {@link GreedyMaximumCardinalityMatching} is used
* to quickly generate a near optimal initial solution.
*
* @param graph the input graph
*/
public SparseEdmondsMaximumCardinalityMatching(Graph graph)
{
this(graph, new GreedyMaximumCardinalityMatching<>(graph, false));
}
/**
* Constructs a new instance of the algorithm.
*
* @param graph undirected graph (graph does not have to be simple)
* @param initializer heuristic matching algorithm used to quickly generate a (near optimal)
* initial feasible solution
*/
public SparseEdmondsMaximumCardinalityMatching(
Graph graph, MatchingAlgorithm initializer)
{
this.graph = GraphTests.requireUndirected(graph);
this.initializer = initializer;
}
/**
* The actual implementation as an inner class. We use this pattern in order to free the work
* memory after computation. The outer class can cache the result but can also release all the
* auxiliary memory.
*
* @param the vertex type
* @param the edge type
*/
private static class Algorithm
{
private static final int NULL = -1;
/**
* Even, odd and unlabeled labels.
*/
private enum Label
{
EVEN,
ODD,
UNLABELED
}
private final Graph graph;
private MatchingAlgorithm initializer;
private int nodes;
private Map vertexIndexMap;
private V[] vertexMap;
private int[] mate;
private Label[] label;
private int[] pred;
double strue;
private double[] path1;
private double[] path2;
private int[] sourceBridge;
private int[] targetBridge;
private VertexPartition base;
private FixedSizeIntegerQueue queue;
private List labeledNodes;
public Algorithm(Graph graph, MatchingAlgorithm initializer)
{
this.graph = graph;
this.initializer = initializer;
}
@SuppressWarnings("unchecked")
private void initialize()
{
// index graph
this.nodes = graph.vertexSet().size();
this.vertexIndexMap = CollectionUtil.newHashMapWithExpectedSize(nodes);
this.vertexMap = (V[]) new Object[nodes];
int vIndex = 0;
for (V vertex : graph.vertexSet()) {
vertexIndexMap.put(vertex, vIndex);
vertexMap[vIndex] = vertex;
vIndex++;
}
this.mate = new int[nodes];
this.base = new VertexPartition(nodes);
this.label = new Label[nodes];
this.pred = new int[nodes];
this.path1 = new double[nodes];
this.path2 = new double[nodes];
this.sourceBridge = new int[nodes];
this.targetBridge = new int[nodes];
for (int i = 0; i < nodes; i++) {
this.mate[i] = NULL;
this.label[i] = Label.EVEN;
this.pred[i] = NULL;
this.path1[i] = 0d;
this.path2[i] = 0d;
this.sourceBridge[i] = NULL;
this.targetBridge[i] = NULL;
}
this.strue = 0d;
this.queue = new FixedSizeIntegerQueue(nodes);
this.labeledNodes = new ArrayList<>();
}
private void runInitializer()
{
if (initializer == null) {
return;
}
for (E e : initializer.getMatching()) {
V u = graph.getEdgeSource(e);
V v = graph.getEdgeTarget(e);
int uIndex = vertexIndexMap.get(u);
int vIndex = vertexIndexMap.get(v);
mate[uIndex] = vIndex;
label[uIndex] = Label.UNLABELED;
mate[vIndex] = uIndex;
label[vIndex] = Label.UNLABELED;
}
}
private void findPath(Deque p, int x, int y)
{
if (x == y) {
p.add(x);
return;
}
if (label[x] == Label.EVEN) {
p.add(x);
p.add(mate[x]);
findPath(p, pred[mate[x]], y);
return;
}
// x is ODD
p.add(x);
Deque p2 = new ArrayDeque<>();
findPath(p2, sourceBridge[x], mate[x]);
while (!p2.isEmpty()) {
p.add(p2.removeLast());
}
findPath(p, targetBridge[x], y);
}
private void shrinkPath(int b, int v, int w)
{
int x = base.find(v);
while (x != b) {
base.union(x, b);
x = mate[x];
base.union(x, b);
base.name(b); // make sure b is called the same
queue.enqueue(x);
sourceBridge[x] = v;
targetBridge[x] = w;
x = base.find(pred[x]);
}
}
public Set computeMatching()
{
initialize();
runInitializer();
for (int i = 0; i < nodes; i++) {
if (mate[i] != NULL) {
continue;
}
queue.clear();
queue.enqueue(i);
labeledNodes.clear();
labeledNodes.add(i);
boolean breakThrough = false;
while (!breakThrough && !queue.isEmpty()) { // grow tree
int v = queue.poll();
V vAsVertex = vertexMap[v];
for (E e : graph.edgesOf(vAsVertex)) {
V wAsVertex = Graphs.getOppositeVertex(graph, e, vAsVertex);
int w = vertexIndexMap.get(wAsVertex);
if (base.find(v) == base.find(w) || label[base.find(w)] == Label.ODD) {
continue;
}
if (label[w] == Label.UNLABELED) { // grow tree
label[w] = Label.ODD;
labeledNodes.add(w);
pred[w] = v;
label[mate[w]] = Label.EVEN;
labeledNodes.add(mate[w]);
queue.enqueue(mate[w]);
} else { // augment or shrink blossom
int hv = base.find(v);
int hw = base.find(w);
strue += 1d;
path1[hv] = strue;
path2[hw] = strue;
while ((path1[hw] != strue && path2[hv] != strue)
&& (mate[hv] != NULL || mate[hw] != NULL))
{
if (mate[hv] != NULL) {
hv = base.find(pred[mate[hv]]);
path1[hv] = strue;
}
if (mate[hw] != NULL) {
hw = base.find(pred[mate[hw]]);
path2[hw] = strue;
}
}
if (path1[hw] == strue || path2[hv] == strue) {
// shrink blossom
int b = (path1[hw] == strue) ? hw : hv; // base
shrinkPath(b, v, w);
shrinkPath(b, w, v);
} else {
// augment
Deque p = new ArrayDeque<>();
findPath(p, v, hv);
p.addFirst(w);
while (!p.isEmpty()) {
int a = p.pop();
int b = p.pop();
mate[a] = b;
mate[b] = a;
}
labeledNodes.add(w);
for (Integer k : labeledNodes) {
label[k] = Label.UNLABELED;
}
base.split(labeledNodes);
breakThrough = true;
break;
}
}
}
}
}
// compute resulting matching
Set matching = new HashSet<>();
for (E e : graph.edgeSet()) {
V u = graph.getEdgeSource(e);
V v = graph.getEdgeTarget(e);
if (u.equals(v)) {
continue;
}
int uIndex = vertexIndexMap.get(u);
int vIndex = vertexIndexMap.get(v);
if (uIndex != vIndex && mate[uIndex] == vIndex) {
matching.add(e);
// cleanup
mate[uIndex] = uIndex;
mate[vIndex] = vIndex;
}
}
return matching;
}
public Map computeOddSetCover()
{
int[] osc = new int[nodes];
Arrays.fill(osc, -1);
int numberOfUnlabeled = 0;
int arbUNode = -1;
for (int v = 0; v < nodes; v++) {
if (label[v] == Label.UNLABELED) {
numberOfUnlabeled++;
arbUNode = v;
}
}
if (numberOfUnlabeled > 0) {
osc[arbUNode] = 1;
int lambda = (numberOfUnlabeled == 2 ? 0 : 2);
for (int v = 0; v < nodes; v++) {
if (label[v] == Label.UNLABELED && v != arbUNode) {
osc[v] = lambda;
}
}
}
int kappa = (numberOfUnlabeled <= 2 ? 2 : 3);
for (int v = 0; v < nodes; v++) {
if (base.find(v) != v && osc[base.find(v)] == -1) {
osc[base.find(v)] = kappa++;
}
}
for (int v = 0; v < nodes; v++) {
if (base.find(v) == v && osc[v] == -1) {
if (label[v] == Label.EVEN) {
osc[v] = 0;
}
if (label[v] == Label.ODD) {
osc[v] = 1;
}
}
if (base.find(v) != v) {
osc[v] = osc[base.find(v)];
}
}
Map oddSetCover = new HashMap<>();
for (int v = 0; v < nodes; v++) {
oddSetCover.put(vertexMap[v], osc[v]);
}
return oddSetCover;
}
}
@Override
public Matching getMatching()
{
if (result == null) {
Algorithm alg = new Algorithm<>(graph, initializer);
Set matchingEdges = alg.computeMatching();
int cardinality = matchingEdges.size();
result = new MatchingImpl<>(graph, matchingEdges, cardinality);
oddSetCover = alg.computeOddSetCover();
}
return result;
}
/**
* Get an odd set cover which proves the optimality of the computed matching.
*
*
* In order to check for optimality one needs to check that the odd-set-cover is a node labeling
* that (a) covers the graph and (b) whose capacity is equal to the cardinality of the matching.
* For (a) we check that every edge is either incident to a node with label 1 or connects two
* nodes labeled $i$ for some $i \ge 2$. For (b) we count for each $i$ the number $n_i$ of nodes
* with label $i$ and compute $S = n_1 + \sum_{i \ge 2} \floor{n_i/2}$.
*
*
* Method {{@link #isOptimalMatching(Graph, Set, Map)} performs this check given a matching and
* an odd-set-cover.
*
* @return an odd set cover whose capacity is the same as the matching's cardinality
*/
public Map getOddSetCover()
{
getMatching();
return oddSetCover;
}
/**
* Check whether a matching is optimal.
*
* The method first checks whether the matching is indeed a matching. Then it checks whether the
* odd-set-cover provided is a node labeling that covers the graph and whose capacity is equal
* to the cardinality of the matching.
*
* First, we count for each $i$ the number $n_i$ of nodes with label $i$, and then compute $S =
* n_1 + \sum_{i \ge 2} \floor{n_i/2}$. $S$ should be equal to the size of the matching. Then,
* we check that every edge is incident to a node label one or connects two nodes labeled $i$
* for some $i \ge 2$.
*
* This method runs in linear time.
*
* @param graph the graph
* @param matching a matching
* @param oddSetCover an odd set cover
* @return true if the matching is optimal, false otherwise
*
* @param graph vertex type
* @param graph edge type
*/
public static boolean isOptimalMatching(
Graph graph, Set matching, Map oddSetCover)
{
// check matching
Set matched = new HashSet<>();
for (E e : matching) {
V s = graph.getEdgeSource(e);
if (!matched.add(s)) {
return false;
}
V t = graph.getEdgeTarget(e);
if (!matched.add(t)) {
return false;
}
}
// check optimality
int n = Math.max(2, graph.vertexSet().size());
int kappa = 1;
int[] count = new int[n];
for (int i = 0; i < n; i++) {
count[i] = 0;
}
for (V v : graph.vertexSet()) {
Integer osc = oddSetCover.get(v);
if (osc < 0 || osc >= n) {
return false;
}
count[osc]++;
if (osc > kappa) {
kappa = osc;
}
}
int s = count[1];
for (int i = 2; i <= kappa; i++) {
s += count[i] / 2;
}
if (s != matching.size()) {
return false;
}
for (E e : graph.edgeSet()) {
V v = graph.getEdgeSource(e);
V w = graph.getEdgeTarget(e);
int oscv = oddSetCover.get(v);
int oscw = oddSetCover.get(w);
if (v.equals(w) || oscv == 1 || oscw == 1 || (oscv == oscw && oscv >= 2)) {
continue;
}
return false;
}
return true;
}
/**
* Special integer vertex union-find.
*
* @author Dimitrios Michail
*/
private static class VertexPartition
{
private Item[] items;
public VertexPartition(int n)
{
this.items = new Item[n];
for (int i = 0; i < n; i++) {
items[i] = new Item(i);
}
}
public int find(int e)
{
return findItem(e).rep;
}
public void union(int a, int b)
{
assert a >= 0 && a < items.length;
assert b >= 0 && b < items.length;
Item ia = findItem(a);
Item ib = findItem(b);
// check if the elements are already in the same set
if (ia == ib) {
return;
}
// union by rank
if (ia.rank > ib.rank) {
ib.parent = ia;
} else if (ia.rank < ib.rank) {
ia.parent = ib;
} else {
ib.parent = ia;
ia.rank += 1;
}
}
/**
* Name the representative of the group where e belongs as e.
*
* @param e a vertex
*/
public void name(int e)
{
Item ie = findItem(e);
ie.rep = e;
}
/**
* Split a partition. Assumes that it contains all members, otherwise bad things may happen.
*
* @param toSplit all members of a partition
*/
public void split(List toSplit)
{
for (int i : toSplit) {
Item item = items[i];
item.parent = item;
item.rep = i;
item.rank = 0;
}
}
private Item findItem(int e)
{
assert e >= 0 && e < items.length;
// lookup
Item current = items[e];
while (true) {
Item parent = current.parent;
if (parent.equals(current)) {
break;
}
current = parent;
}
// path compression
final Item root = current;
current = items[e];
while (!current.equals(root)) {
Item parent = current.parent;
current.parent = root;
current = parent;
}
return root;
}
// the item class
private static class Item
{
public int rep;
public int rank;
Item parent;
public Item(int rep)
{
this.rep = rep;
this.rank = 0;
this.parent = this;
}
}
}
}