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

@@ -1,32 +1,59 @@
 from typing import List
-
+cache = {}
 
 def ways_to_make_change(total: int) -> int:
     """
     Given access to coins with the values 1, 2, 5, 10, 20, 50, 100, 200, returns a count of all of the ways to make the passed total value.
 
     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] 
+    return ways_to_make_change_helper(total, coins, 0)
 
 
-def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
+def ways_to_make_change_helper(total: int, coins: List[int],start_index: 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
+# No coins left to try as we moved past the end of the list
+    if start_index >= len(coins): 
         return 0
 
+# Cache key now uses (total, start_index) instead of hashing the whole coin list
+    key = (total, start_index)
+    if key in cache:
+        return cache[key]
+
     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
+
+ # Try using the current coin 1,2,3... times as long as we don't exceed the total.
+    coin = coins[start_index]
+    count_of_coin = 1
+
+    while coin * count_of_coin <= total:
+        total_from_coins = coin * count_of_coin
+
+        if total_from_coins == total:
+# if exact match then one valid way
+
+            ways += 1
+        else:
+# Remaining total, and move to the next coin (no slicing!)
+            intermediate = ways_to_make_change_helper(
+                total - total_from_coins,
+                coins,
+                start_index + 1
+            )
+            ways += intermediate
+
+        count_of_coin += 1
+
+# Also consider the case where we skip this coin entirely
+    ways += ways_to_make_change_helper(total, coins, start_index + 1)
+
+    cache[key] = ways
     return ways