Optimize cache by using start_index instead of coin sub-arrays

Author: aydaeslami

Sprint-2/improve_with_caches/making_change/making_change.py

@@ -1,26 +1,27 @@
 from typing import List
+COINS = [200, 100, 50, 20, 10, 5, 2, 1]
 
 cache = {}
 
 def ways_to_make_change(total: int) -> int:
-    return ways_to_make_change_helper(total, [200, 100, 50, 20, 10, 5, 2, 1])
+    return ways_to_make_change_helper(total, 0)
 
 
-def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
-    key = (total, tuple(coins))
+def ways_to_make_change_helper(total: int, start_index: int) -> int:
+    key = (total, start_index)
 
     if key in cache:
         return cache[key]
 
     if total == 0:
         return 1
 
-    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]
+    for coin_index in range(start_index, len(COINS)):
+        coin = COINS[coin_index]
         count_of_coin = 1
         while coin * count_of_coin <= total:
             total_from_coins = coin * count_of_coin
@@ -29,7 +30,7 @@ def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
             else:
                 ways += ways_to_make_change_helper(
                     total - total_from_coins,
-                    coins=coins[coin_index + 1:]
+                    coin_index + 1
                 )
             count_of_coin += 1