Feature: Add-on of Held-karp algorithm in backtracking folder #1744

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	/*
	Held-karp algorithm (https://en.wikipedia.org/wiki/Held-Karp_algorithm)

	- Held-karp algorithm solve TSP problem using dynamic programming paradigm.
	- It computes the minimum cost to visit each city exactly once & return to start point.
	considering all subsets of cities & using memoization.
	- Comparing it with hamiltonian algo. vs Held-karp algo. ( n! vs ~2^n ) this is more efficient.

	*/

	function heldKarp(dist) {
	const n = dist.length;
	const memo = Array.from({ length: n }, () => Array(1 << n).fill(null));

	// Base case: distance from the starting point to itself is 0
	for (let i = 0; i < n; i++) {
	memo[i][1 << i] = dist[i][0];
	}

	// Iterate through all subsets of vertices
	for (let mask = 0; mask < (1 << n); mask++) {
	for (let u = 0; u < n; u++) {
	if (!(mask & (1 << u))) continue; // u must be in the subset
	// Iterate through all vertices to find the minimum cost
	for (let v = 0; v < n; v++) {
	if (mask & (1 << v) \|\| u === v) continue; // v must not be in the subset
	const newMask = mask \| (1 << v);
	const newCost = memo[u][mask] + dist[u][v];

	if (memo[v][newMask] === null \|\| newCost < memo[v][newMask]) {
	memo[v][newMask] = newCost;
	}
	}
	}
	}

	// Get the minimum cost to complete the tour
	let minCost = Infinity;
	for (let u = 1; u < n; u++) {
	const cost = memo[u][(1 << n) - 1] + dist[u][0];
	minCost = Math.min(minCost, cost);
	}

	return minCost;
	}

	module.exports = { heldKarp };

62 changes: 62 additions & 0 deletions Backtracking/tests/held_karp.test.js

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	const { heldKarp } = require('../held_karp');

	const distanceMatrix1 = [
	[0, 10, 15, 20],
	[10, 0, 35, 25],
	[15, 35, 0, 30],
	[20, 25, 30, 0]
	];
	console.log("Test Case 1 - Minimum cost of visiting all cities:", heldKarp(distanceMatrix1));

	const distanceMatrix2 = [
	[0, 5, 10],
	[5, 0, 15],
	[10, 15, 0]
	];
	console.log("Test Case 2 - Minimum cost of visiting all cities:", heldKarp(distanceMatrix2));

	const distanceMatrix3 = [
	[0, 10, 15, 20, 25],
	[10, 0, 35, 25, 30],
	[15, 35, 0, 30, 20],
	[20, 25, 30, 0, 10],
	[25, 30, 20, 10, 0]
	];
	console.log("Test Case 3 - Minimum cost of visiting all cities:", heldKarp(distanceMatrix3));

	const distanceMatrix4 = [
	[0]
	];
	console.log("Test Case 4 - Minimum cost of visiting all cities:", heldKarp(distanceMatrix4));

	const distanceMatrix5 = [
	[0, 5],
	[5, 0]
	];
	console.log("Test Case 5 - Minimum cost of visiting all cities:", heldKarp(distanceMatrix5));

	const distanceMatrix6 = [
	[0, 1, 1, 1],
	[1, 0, 1, 1],
	[1, 1, 0, 1],
	[1, 1, 1, 0]
	];
	console.log("Test Case 6 - Minimum cost of visiting all cities:", heldKarp(distanceMatrix6));

	const distanceMatrix7 = [
	[0, 10, 20, 30],
	[5, 0, 15, 25],
	[10, 5, 0, 20],
	[20, 15, 10, 0]
	];
	console.log("Test Case 7 - Minimum cost of visiting all cities:", heldKarp(distanceMatrix7));

	const distanceMatrix8 = [
	[0, 1000, 2000, 3000],
	[1000, 0, 1500, 2500],
	[2000, 1500, 0, 3500],
	[3000, 2500, 3500, 0]
	];
	console.log("Test Case 8 - Minimum cost of visiting all cities:", heldKarp(distanceMatrix8));

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