Optimize Fibonacci and Making Change functions with caching for improved performance

Author: Emilianouz

Sprint-2/improve_with_caches/fibonacci/fibonacci.py

@@ -1,4 +1,12 @@
-def fibonacci(n):
+def fibonacci(n, cache={}): 
+    '''Adding a dictionary to store a copy of what we have already calculated, the complexity becomes linear.
+    With the cache, each value is computed once, bringing it down to O(n)
+    '''
     if n <= 1:
         return n
-    return fibonacci(n - 1) + fibonacci(n - 2)
+
+    if n in cache:
+        return cache[n]
+
+    cache[n] = fibonacci(n - 1, cache) + fibonacci(n - 2, cache)
+    return cache[n]

Sprint-2/improve_with_caches/making_change/making_change.py

@@ -7,26 +7,35 @@ 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])
+    cache = {}
+    return ways_to_make_change_helper(total, [200, 100, 50, 20, 10, 5, 2, 1], 0, cache)
 
 
-def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
+def ways_to_make_change_helper(
+    total: int, coins: List[int], coin_index: int, cache: dict
+) -> 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:
+        return 1
+    if coin_index == len(coins):
         return 0
 
+    key = (total, coin_index)
+    if key in cache:
+        return cache[key]
+
+    coin = coins[coin_index]
     ways = 0
-    for coin_index in range(len(coins)):
-        coin = coins[coin_index]
-        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:])
-                ways += intermediate
-            count_of_coin += 1
+
+    # the remaining coins can complete whatever total is left over
+    while coin * count <= total:
+        ways += ways_to_make_change_helper(total - coin * count, coins, coin_index + 1, cache)
+        count += 1
+
+    # counting combinations that don't use the current coin 
+    ways += ways_to_make_change_helper(total, coins, coin_index + 1, cache)
+
+    cache[key] = ways
     return ways