org.jgrapht.alg.tour.HeldKarpTSP Maven / Gradle / Ivy
/*
* (C) Copyright 2017-2023, by Alexandru Valeanu 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.tour;
import org.jgrapht.*;
import org.jgrapht.util.*;
import java.util.*;
/**
* A dynamic programming algorithm for the TSP problem.
*
*
* The travelling salesman problem (TSP) asks the following question: "Given a list of cities and
* the distances between each pair of cities, what is the shortest possible route that visits each
* city exactly once and returns to the origin city?".
*
*
* This is an implementation of the Held-Karp algorithm which returns a optimal, minimum-cost
* Hamiltonian tour. The implementation requires the input graph to contain at least one vertex. The
* running time is $O(2^{|V|} \times |V|^2)$ and it takes $O(2^{|V|} \times |V|)$ extra memory.
*
*
* See wikipedia for more
* details about TSP.
*
*
* See wikipedia for more
* details about the dynamic programming algorithm.
*
* @param the graph vertex type
* @param the graph edge type
*
* @author Alexandru Valeanu
*/
public class HeldKarpTSP
extends HamiltonianCycleAlgorithmBase
{
private double memo(int previousNode, int state, double[][] c, double[][] w)
{
// have we seen this state before?
if (c[previousNode][state] != Double.MIN_VALUE) {
return c[previousNode][state];
}
// no cycle has been found yet
double totalCost = Double.MAX_VALUE;
// check if all nodes have been visited (i.e. state + 1 == 2^n)
if (state == (1 << w.length) - 1) {
// check if there is a return edge we can use
if (w[previousNode][0] != Double.MAX_VALUE) {
totalCost = w[previousNode][0];
}
} else {
// try to find the 'best' next (i.e. unvisited and adjacent to previousNode) node in the
// tour
for (int i = 0; i < w.length; i++) {
if (((state >> i) & 1) == 0 && w[previousNode][i] != Double.MAX_VALUE) {
totalCost =
Math.min(totalCost, w[previousNode][i] + memo(i, state ^ (1 << i), c, w));
}
}
}
return c[previousNode][state] = totalCost;
}
/**
* Computes a minimum-cost Hamiltonian tour.
*
* @param graph the input graph
* @return a minimum-cost tour if one exists, null otherwise
* @throws IllegalArgumentException if the graph contains no vertices
* @throws IllegalArgumentException if the graph contains more than 31 vertices
*/
@Override
public GraphPath getTour(Graph graph)
{
requireNotEmpty(graph);
final int n = graph.vertexSet().size(); // number of nodes
if (n == 1) {
return getSingletonTour(graph);
}
if (n > 31) {
throw new IllegalArgumentException(
"The internal representation of the dynamic programming state "
+ "space cannot represent graphs containing more than 31 vertices. "
+ "The runtime complexity of this implementation, O(2^|V| x |V|^2), makes it unsuitable "
+ "for graphs with more than 31 vertices.");
}
// Normalize the graph by mapping each vertex to an integer.
VertexToIntegerMapping vertexToIntegerMapping = Graphs.getVertexToIntegerMapping(graph);
// W[u, v] = the cost of the minimum weight between u and v
double[][] w = computeMinimumWeights(vertexToIntegerMapping.getVertexMap(), graph);
// C[prevNode, state] = the minimum cost of a tour that ends in prevNode and contains all
// nodes in the bitmask state
double[][] c = new double[n][1 << n];
fill(c, Double.MIN_VALUE);
// start the tour from node 0 (because the tour is a cycle the start vertex does not matter)
double tourWeight = memo(0, 1, c, w);
// check if there is no tour
if (tourWeight == Double.MAX_VALUE) {
return null;
}
List vertexList = reconstructTour(vertexToIntegerMapping.getIndexList(), w, c);
return vertexListToTour(vertexList, graph);
}
private double[][] computeMinimumWeights(Map vertexMap, Graph graph)
{
final int n = vertexMap.size();
double[][] w = new double[n][n];
fill(w, Double.MAX_VALUE);
for (E e : graph.edgeSet()) {
V source = graph.getEdgeSource(e);
V target = graph.getEdgeTarget(e);
int u = vertexMap.get(source);
int v = vertexMap.get(target);
// use Math.min in case we deal with a multigraph
w[u][v] = Math.min(w[u][v], graph.getEdgeWeight(e));
// If the graph is undirected we need to also consider the reverse edge
if (graph.getType().isUndirected()) {
w[v][u] = Math.min(w[v][u], graph.getEdgeWeight(e));
}
}
return w;
}
private static void fill(double[][] array, double value)
{
for (double[] element : array) {
Arrays.fill(element, value);
}
}
private List reconstructTour(List indexList, double[][] w, double[][] c)
{
final int n = indexList.size();
List vertexList = new ArrayList<>(n);
int lastNode = 0;
int lastState = 1;
vertexList.add(indexList.get(lastNode));
for (int step = 1; step < n; step++) {
int nextNode = -1;
for (int node = 1; node < n; node++) {
if ((lastState & (1 << node)) == 0 && w[lastNode][node] != Double.MAX_VALUE
&& c[node][lastState ^ (1 << node)] != Double.MIN_VALUE
&& Double.compare(
c[node][lastState ^ (1 << node)] + w[lastNode][node],
c[lastNode][lastState]) == 0)
{
nextNode = node;
break;
}
}
assert nextNode != -1;
vertexList.add(indexList.get(nextNode));
lastState ^= 1 << nextNode;
lastNode = nextNode;
}
return vertexList;
}
}