Refactor memoisation to avoid repeated sub-array creation

Author: sheida-shab

Sprint-2/improve_with_caches/making_change/making_change.py

@@ -7,21 +7,29 @@ def ways_to_make_change(total: int) -> int:
 
     For instance, there are two ways to make a value of 3: with 3x 1 coins, or with 1x 1 coin and 1x 2 coin.
     """
-    return ways_to_make_change_helper(total, [200, 100, 50, 20, 10, 5, 2, 1])
 
+    coins=[200, 100, 50, 20, 10, 5, 2, 1]
+    all_subarrays=[]
+    for i  in range(len(coins)) :
+        all_subarrays.append( coins[i:] )
 
-def ways_to_make_change_helper(total: int, coins: List[int],cache=None) -> int:
+    cache={}    
+         
+    return ways_to_make_change_helper(total, all_subarrays,0,cache)
+
+
+def ways_to_make_change_helper(total: int, all_subarrays: List[List[int]],subarray_id: int,cache) -> int:
     """
     Helper function for ways_to_make_change to avoid exposing the coins parameter to callers.
     """
-    if total == 0 or len(coins) == 0:
+    if total == 0 or subarray_id >= len(all_subarrays) :
         return 0
-    if cache is None:
-        cache={}
-    key=(total,tuple(coins))    
+    
+    key = (total, subarray_id)
     if key in cache:
         return cache[key]
 
+    coins = all_subarrays[subarray_id]    
     ways = 0
     for coin_index in range(len(coins)):
         coin = coins[coin_index]
@@ -31,8 +39,10 @@ def ways_to_make_change_helper(total: int, coins: List[int],cache=None) -> int:
             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)
-                ways += intermediate
+                ways += ways_to_make_change_helper(total - total_from_coins, all_subarrays, subarray_id + coin_index + 1, cache)
+               
             count_of_coin += 1
-    cache[key]=ways        
+
+    cache[key]=ways   
+
     return ways