Added inverse_coin_change.py

backtracking/inverse_coin_change.py

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+"""
+This module finds all possible ways to combine given coin denominations
+to sum up to a specified target amount using a backtracking method.
+
+The time complexity grows exponentially based on the amount and the set of coins.
+"""
+
+from __future__ import annotations
+from typing import List
+
+
+def inverse_coin_change(amount: int, coins: List[int]) -> List[List[int]]:
+    """
+    Computes every possible combination of coins that add up exactly to the target amount.
+    Coins can be chosen multiple times.
+
+    :param amount: The total amount to form with the coins.
+    :param coins: Available coin denominations.
+
+    Example 1:
+    >>> inverse_coin_change(4, [1, 2])
+    [[1, 1, 1, 1], [1, 1, 2], [2, 2]]
+
+    Example 2:
+    >>> inverse_coin_change(3, [2])
+    []
+
+    Example 3:
+    >>> inverse_coin_change(0, [1, 2])
+    [[]]
+    """
+    results: List[List[int]] = []
+
+    def backtrack(remaining: int, current_combo: List[int], start_index: int) -> None:
+        if remaining == 0:
+            results.append(current_combo.copy())
+            return
+        for i in range(start_index, len(coins)):
+            coin = coins[i]
+            if coin <= remaining:
+                current_combo.append(coin)
+                backtrack(remaining - coin, current_combo, i)  # Allow coin reuse
+                current_combo.pop()
+
+    coins.sort()
+    backtrack(amount, [], 0)
+    return results
+
+
+# Uncomment the lines below to accept user input
+# print("Please enter the target amount:")
+# target_amount = int(input())
+# print("Enter the coin denominations separated by spaces:")
+# coin_values = list(map(int, input().split()))
+# combinations = inverse_coin_change(target_amount, coin_values)
+# print(combinations)
+
+
+if __name__ == "__main__":
+    print("Possible combinations for amount = 4 with coins [1, 2]:")
+    print(inverse_coin_change(4, [1, 2]))
+
+    print("\nPossible combinations for amount = 5 with coins [1, 3, 5]:")
+    print(inverse_coin_change(5, [1, 3, 5]))