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

Backtracking/held_karp.js

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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 };

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));