refactor(ways_to_make_change): optimize cache keys with integer start_index

Author: RomanSanaye

Sprint-2/improve_with_caches/making_change/making_change.py

@@ -1,5 +1,9 @@
 from typing import List
 
+# Defined once globally to avoid recreation overhead
+COINS = [200, 100, 50, 20, 10, 5, 2, 1]
+
+
 # ===============> change function <============
 def ways_to_make_change(total: int) -> int:
     """
@@ -9,36 +13,41 @@ def ways_to_make_change(total: int) -> int:
     """
     # Initialize a cache dictionary
     cache = {}
-    return ways_to_make_change_helper(total, [200, 100, 50, 20, 10, 5, 2, 1], cache)
+    # Passing 0 (the starting index) instead of the full list
+    return ways_to_make_change_helper(total, 0, cache)
+
 
 # =================> change helper function <==============
-def ways_to_make_change_helper(total: int, coins: List[int], cache: dict) -> int:
+def ways_to_make_change_helper(total: int, start_index: int, cache: dict) -> int:
     """
     Helper function for ways_to_make_change to avoid exposing the coins parameter to callers.
     """
     # Create unique key for this subproblem
-    cache_key = (total, tuple(coins))
+    cache_key = (total, start_index)
 
     # Return cached result if already calculated
     if cache_key in cache:
         return cache[cache_key]
 
-    if total == 0 or len(coins) == 0:
+    if total == 0 or start_index >= len(COINS):
         return 0
 
     ways = 0
-    for coin_index in range(len(coins)):
-        coin = coins[coin_index]
+    # Iterate through the global COINS list using the integer index
+    for i in range(start_index, len(COINS)):
+        coin = COINS[i]
         count_of_coin = 1
         while coin * count_of_coin <= total:
             total_from_coins = coin * count_of_coin
             if total_from_coins == total:
                 ways += 1
             else:
-                intermediate = ways_to_make_change_helper(total - total_from_coins, coins=coins[coin_index+1:], cache=cache)
+                # Just pass an integer (i + 1)
+                intermediate = ways_to_make_change_helper(
+                    total - total_from_coins, start_index=i + 1, cache=cache
+                )
                 ways += intermediate
             count_of_coin += 1
 
-    # Save result before returning
     cache[cache_key] = ways
     return ways