com.salesforce.jgrapht.alg.tour.HeldKarpTSP Maven / Gradle / Ivy
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/*
* (C) Copyright 2017-2018, 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 com.salesforce.jgrapht.alg.tour;
import com.salesforce.jgrapht.*;
import com.salesforce.jgrapht.alg.interfaces.*;
import com.salesforce.jgrapht.graph.*;
import com.salesforce.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
implements
HamiltonianCycleAlgorithm
{
/**
* Construct a new instance
*/
public HeldKarpTSP()
{
}
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)
{
final int n = graph.vertexSet().size(); // number of nodes
if (n == 0) {
throw new IllegalArgumentException("Graph contains no vertices");
}
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.");
}
if (n == 1) {
V startNode = graph.vertexSet().iterator().next();
return new GraphWalk<>(
graph, startNode, startNode, Collections.singletonList(startNode), null, 0);
}
// W[u, v] = the cost of the minimum weight between u and v
double[][] W = new double[n][n];
for (int i = 0; i < n; i++) {
Arrays.fill(W[i], Double.MAX_VALUE);
}
/*
* Normalize the graph by mapping each vertex to an integer.
*/
VertexToIntegerMapping vertexToIntegerMapping = Graphs.getVertexToIntegerMapping(graph);
Map vertexMap = vertexToIntegerMapping.getVertexMap();
List indexList = vertexToIntegerMapping.getIndexList();
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));
}
// 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];
for (int i = 0; i < n; i++) {
Arrays.fill(C[i], 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;
/*
* Reconstruct the tour
*/
List vertexList = new ArrayList<>(n);
List edgeList = 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));
edgeList.add(graph.getEdge(indexList.get(lastNode), indexList.get(nextNode)));
lastState ^= 1 << nextNode;
lastNode = nextNode;
}
// add start vertex
vertexList.add(indexList.get(0));
edgeList.add(graph.getEdge(indexList.get(lastNode), indexList.get(0)));
return new GraphWalk<>(
graph, indexList.get(0), indexList.get(0), vertexList, edgeList, tourWeight);
}
}