Added Combination Sum Problem in backtracking

backtracking/combination_sum.cpp

@@ -0,0 +1,86 @@
+/**
+ * Combination Sum - Backtracking Problem
+ *
+ * Given a set of candidate numbers (distinct integers) and a target number,
+ * find all unique combinations in candidates where the candidate numbers sum to the target.
+ * You may use the same number multiple times.
+ *
+ * Example:
+ * Input: candidates = [2, 3, 6, 7], target = 7
+ * Output: [[2,2,3],[7]]
+ *
+ * ---------------------------------------------
+ * Brute Force Approach:
+ * Try every possible subset and check if the sum equals the target.
+ * Time Complexity: O(2^n * k) where k is average length of subset
+ * Space Complexity: O(k * x) where x is number of valid combinations
+ *
+ * ---------------------------------------------
+ * Optimized Approach - Backtracking:
+ * Use recursion and backtrack when the current sum exceeds the target.
+ * Time Complexity: Better than brute force, but still exponential in worst-case.
+ */
+
+#include <iostream>
+#include <vector>
+using namespace std;
+
+// ------------------- Brute Force Style (for explanation) -------------------
+void findAllSubsets(int index, int target, vector<int>& candidates, vector<int>& current, vector<vector<int>>& result) {
+    if (index == candidates.size()) {
+        if (target == 0) result.push_back(current);
+        return;
+    }
+
+    // Pick the current element
+    if (candidates[index] <= target) {
+        current.push_back(candidates[index]);
+        findAllSubsets(index, target - candidates[index], candidates, current, result);
+        current.pop_back();
+    }
+
+    // Do not pick the current element
+    findAllSubsets(index + 1, target, candidates, current, result);
+}
+
+// ------------------- Optimized Backtracking -------------------
+void backtrack(int index, int target, vector<int>& candidates, vector<int>& current, vector<vector<int>>& result) {
+    if (target == 0) {
+        result.push_back(current);
+        return;
+    }
+
+    for (int i = index; i < candidates.size(); i++) {
+        if (candidates[i] <= target) {
+            current.push_back(candidates[i]);
+            backtrack(i, target - candidates[i], candidates, current, result); // not i+1 because we can reuse
+            current.pop_back(); // backtrack
+        }
+    }
+}
+
+int main() {
+    vector<int> candidates = {2, 3, 6, 7};
+    int target = 7;
+
+    vector<vector<int>> bruteForceResult, backtrackingResult;
+    vector<int> temp;
+
+    // Brute Force Result
+    findAllSubsets(0, target, candidates, temp, bruteForceResult);
+    cout << "Brute Force Result:\n";
+    for (auto& comb : bruteForceResult) {
+        for (int num : comb) cout << num << " ";
+        cout << endl;
+    }
+
+    // Backtracking Result
+    backtrack(0, target, candidates, temp, backtrackingResult);
+    cout << "\nOptimized Backtracking Result:\n";
+    for (auto& comb : backtrackingResult) {
+        for (int num : comb) cout << num << " ";
+        cout << endl;
+    }
+
+    return 0;
+}