WestMidlands| SDC NOV 2025 |Sara Tahir| Sprint 2 | Improve with caches

Sprint-2/improve_with_caches/fibonacci/fibonacci.py

@@ -1,4 +1,17 @@
+#initialising a dictionary to store a copy of what we already know or have already computed
+#caching reduces the fibonacci time complexity from exponential to linear.
+cache = {}
+
 def fibonacci(n):
-    if n <= 1:
+    if n in cache: 
+        return cache[n]
+
+    if n <= 1: #when n is 0 or 1, we don’t need to calculate anything,just return th known value
         return n
-    return fibonacci(n - 1) + fibonacci(n - 2)
+
+    result = fibonacci(n - 1) + fibonacci(n - 2)
+
+    #saving result in our dict before returning 
+    cache[n] = result
+    return result
+

Sprint-2/improve_with_caches/making_change/making_change.py

@@ -1,5 +1,5 @@
 from typing import List
-
+cache = {}
 
 def ways_to_make_change(total: int) -> int:
     """
@@ -11,22 +11,38 @@ def ways_to_make_change(total: int) -> int:
 
 
 def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
+
     """
     Helper function for ways_to_make_change to avoid exposing the coins parameter to callers.
     """
-    if total == 0 or len(coins) == 0:
+#We found one valid way if its an exact match.
+    if total == 0:
+        return 1
+#base case no coins left but still have remaining total but no way to form it
+    if not coins:
         return 0
+#creating a cache key from our current prob, converting list to a tuple to make it hashable
+    key = (total, tuple(coins)) 
+    if key in cache:
+        return cache[key]
+
 
     ways = 0
+# Try using the current coin 1,2,3...times as long as we don't exceed the total.
     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:
+            if total_from_coins == total:   #if we get an exact match ++ on ways
                 ways += 1
+
+#remaining total and remaining coins after using the current one
             else:
                 intermediate = ways_to_make_change_helper(total - total_from_coins, coins=coins[coin_index+1:])
                 ways += intermediate
+
             count_of_coin += 1
+    cache[key] = ways 
     return ways
+