From c8d5bd5b5d66461da2e77cf2e0aecc598589d986 Mon Sep 17 00:00:00 2001
From: Sanjoli Singh <sanjolisingh90@gmail.com>
Date: Wed, 22 Jul 2026 13:14:51 -0500
Subject: [PATCH] Completed Backtracking 1

---
 CombinationSum.java        | 34 ++++++++++++++++++++++++++++
 ExpressionAddOperator.java | 46 ++++++++++++++++++++++++++++++++++++++
 2 files changed, 80 insertions(+)
 create mode 100644 CombinationSum.java
 create mode 100644 ExpressionAddOperator.java

diff --git a/CombinationSum.java b/CombinationSum.java
new file mode 100644
index 00000000..98210b09
--- /dev/null
+++ b/CombinationSum.java
@@ -0,0 +1,34 @@
+class Solution {
+    /*
+        TC = O(N^M), where M is the target/smallest number and N is the total numbers 
+        SC - O(M) - recursive stack, where M is the target/smallest number
+        Maintain a path representing the current combination.
+        If target == 0, add the current path to the result.
+        If target < 0, stop exploring that branch.
+        Iterate from the current pivot index to avoid duplicate combinations.
+        Recursively call with the same index (i) so the current number can be reused unlimited times.
+        After recursion, remove the last element (backtrack) and try the next candidate.
+    */
+    List<List<Integer>> result;
+    public List<List<Integer>> combinationSum(int[] candidates, int target) {
+        this.result = new ArrayList<>();
+        helper(candidates, target, new ArrayList<>(), 0);
+        return result;
+    }
+
+    private void helper(int[] candidates, int target, List<Integer> path, int pivot) {
+        if (target < 0) {
+            return;
+        }
+        if (target == 0) {
+            result.add(new ArrayList<>(path));
+            return;
+        }
+
+        for (int i = pivot; i < candidates.length; i++) {
+            path.add(candidates[i]);
+            helper(candidates, target - candidates[i], path, i);
+            path.remove(path.size() - 1);
+        }
+    }
+}
\ No newline at end of file
diff --git a/ExpressionAddOperator.java b/ExpressionAddOperator.java
new file mode 100644
index 00000000..a3a611d8
--- /dev/null
+++ b/ExpressionAddOperator.java
@@ -0,0 +1,46 @@
+class Solution {
+    /*
+        TC - O(4^N), N is the input length and 4 is the possible number of combinations we can choose from
+        SC - O(N), N is the input length
+
+        This solution uses backtracking (DFS) to generate all possible expressions by inserting +, -, and * operators between digits of the input string.
+        The helper function recursively chooses the next number from the remaining digits.
+        calc keeps the current evaluated value of the expression.
+        tail stores the last operand value, which helps correctly evaluate multiplication due to operator precedence.
+        When adding multiplication: calc - tail + tail * curr removes the previous operand and replaces it with the multiplied result.
+        We skip numbers with leading zeros (like "05") to avoid invalid expressions.
+        When all digits are processed, if calc == target, the expression is added to the result list.
+    */
+    List<String> result;
+    public List<String> addOperators(String num, int target) {
+        this.result = new ArrayList<>();
+        helper(num, target, 0, 0l, 0l, new StringBuilder());
+        return result;
+    }
+
+    private void helper(String num, int target, int pivot, long calc, long tail, StringBuilder path) {
+        if (pivot == num.length()) {
+            if (calc == target) {
+                result.add(path.toString());
+            }
+        }
+
+        int len = path.length();
+
+        for (int  i = pivot; i < num.length(); i++) {
+            if (pivot != i && num.charAt(pivot) == '0') continue;
+            long curr = Long.parseLong(num.substring(pivot, i + 1));
+            if (pivot == 0) {
+                helper(num, target, i + 1, curr, curr, path.append(curr));
+                path.setLength(len);
+            } else {
+                helper(num, target, i + 1, calc + curr, curr, path.append("+" + curr));
+                path.setLength(len);
+                helper(num, target, i + 1, calc - curr, -curr, path.append("-" + curr));
+                path.setLength(len);
+                helper(num, target, i + 1, calc - tail + tail * curr, tail * curr, path.append("*" + curr));
+                path.setLength(len);
+            }
+        }
+    }
+}
\ No newline at end of file