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TspDynamicProgrammingIterative.java
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TspDynamicProgrammingIterative.java
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/**
* An implementation of the traveling salesman problem in Java using dynamic programming to improve
* the time complexity from O(n!) to O(n^2 * 2^n).
*
* <p>Time Complexity: O(n^2 * 2^n) Space Complexity: O(n * 2^n)
*
* @author William Fiset, william.alexandre.fiset@gmail.com
*/
package com.williamfiset.algorithms.graphtheory;
import java.util.ArrayList;
import java.util.Collections;
import java.util.List;
public class TspDynamicProgrammingIterative {
private final int N, start;
private final double[][] distance;
private List<Integer> tour = new ArrayList<>();
private double minTourCost = Double.POSITIVE_INFINITY;
private boolean ranSolver = false;
public TspDynamicProgrammingIterative(double[][] distance) {
this(0, distance);
}
public TspDynamicProgrammingIterative(int start, double[][] distance) {
N = distance.length;
if (N <= 2) throw new IllegalStateException("N <= 2 not yet supported.");
if (N != distance[0].length) throw new IllegalStateException("Matrix must be square (n x n)");
if (start < 0 || start >= N) throw new IllegalArgumentException("Invalid start node.");
if (N > 32)
throw new IllegalArgumentException(
"Matrix too large! A matrix that size for the DP TSP problem with a time complexity of"
+ "O(n^2*2^n) requires way too much computation for any modern home computer to handle");
this.start = start;
this.distance = distance;
}
// Returns the optimal tour for the traveling salesman problem.
public List<Integer> getTour() {
if (!ranSolver) solve();
return tour;
}
// Returns the minimal tour cost.
public double getTourCost() {
if (!ranSolver) solve();
return minTourCost;
}
// Solves the traveling salesman problem and caches solution.
public void solve() {
if (ranSolver) return;
final int END_STATE = (1 << N) - 1;
Double[][] memo = new Double[N][1 << N];
// Add all outgoing edges from the starting node to memo table.
for (int end = 0; end < N; end++) {
if (end == start) continue;
memo[end][(1 << start) | (1 << end)] = distance[start][end];
}
for (int r = 3; r <= N; r++) {
for (int subset : combinations(r, N)) {
if (notIn(start, subset)) continue;
for (int next = 0; next < N; next++) {
if (next == start || notIn(next, subset)) continue;
int subsetWithoutNext = subset ^ (1 << next);
double minDist = Double.POSITIVE_INFINITY;
for (int end = 0; end < N; end++) {
if (end == start || end == next || notIn(end, subset)) continue;
double newDistance = memo[end][subsetWithoutNext] + distance[end][next];
if (newDistance < minDist) {
minDist = newDistance;
}
}
memo[next][subset] = minDist;
}
}
}
// Connect tour back to starting node and minimize cost.
for (int i = 0; i < N; i++) {
if (i == start) continue;
double tourCost = memo[i][END_STATE] + distance[i][start];
if (tourCost < minTourCost) {
minTourCost = tourCost;
}
}
int lastIndex = start;
int state = END_STATE;
tour.add(start);
// Reconstruct TSP path from memo table.
for (int i = 1; i < N; i++) {
int bestIndex = -1;
double bestDist = Double.POSITIVE_INFINITY;
for (int j = 0; j < N; j++) {
if (j == start || notIn(j, state)) continue;
double newDist = memo[j][state] + distance[j][lastIndex];
if (newDist < bestDist) {
bestIndex = j;
bestDist = newDist;
}
}
tour.add(bestIndex);
state = state ^ (1 << bestIndex);
lastIndex = bestIndex;
}
tour.add(start);
Collections.reverse(tour);
ranSolver = true;
}
private static boolean notIn(int elem, int subset) {
return ((1 << elem) & subset) == 0;
}
// This method generates all bit sets of size n where r bits
// are set to one. The result is returned as a list of integer masks.
public static List<Integer> combinations(int r, int n) {
List<Integer> subsets = new ArrayList<>();
combinations(0, 0, r, n, subsets);
return subsets;
}
// To find all the combinations of size r we need to recurse until we have
// selected r elements (aka r = 0), otherwise if r != 0 then we still need to select
// an element which is found after the position of our last selected element
private static void combinations(int set, int at, int r, int n, List<Integer> subsets) {
// Return early if there are more elements left to select than what is available.
int elementsLeftToPick = n - at;
if (elementsLeftToPick < r) return;
// We selected 'r' elements so we found a valid subset!
if (r == 0) {
subsets.add(set);
} else {
for (int i = at; i < n; i++) {
// Try including this element
set ^= (1 << i);
combinations(set, i + 1, r - 1, n, subsets);
// Backtrack and try the instance where we did not include this element
set ^= (1 << i);
}
}
}
public static void main(String[] args) {
// Create adjacency matrix
int n = 6;
double[][] distanceMatrix = new double[n][n];
for (double[] row : distanceMatrix) java.util.Arrays.fill(row, 10000);
distanceMatrix[5][0] = 10;
distanceMatrix[1][5] = 12;
distanceMatrix[4][1] = 2;
distanceMatrix[2][4] = 4;
distanceMatrix[3][2] = 6;
distanceMatrix[0][3] = 8;
int startNode = 0;
TspDynamicProgrammingIterative solver =
new TspDynamicProgrammingIterative(startNode, distanceMatrix);
// Prints: [0, 3, 2, 4, 1, 5, 0]
System.out.println("Tour: " + solver.getTour());
// Print: 42.0
System.out.println("Tour cost: " + solver.getTourCost());
}
}