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Glasgow | 25-SDC-Nov | Nataliia Volkova | Sprint 2 | improve_with_caches #106

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Glasgow | 25-SDC-Nov | Nataliia Volkova | Sprint 2 | improve_with_caches #106

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fibonacci
Nataliia74 Feb 15, 2026
62e1e71
making_change
Nataliia74 Feb 15, 2026
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12 changes: 10 additions & 2 deletions Sprint-2/improve_with_caches/fibonacci/fibonacci.py
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Original file line number Diff line number Diff line change
@@ -1,4 +1,12 @@
cache = {}

def fibonacci(n):
if n in cache:
return cache[n]
if n <= 1:
return n
return fibonacci(n - 1) + fibonacci(n - 2)
result = n
else:
result = fibonacci(n - 1) + fibonacci(n - 2)

cache[n] = result
return result
18 changes: 16 additions & 2 deletions Sprint-2/improve_with_caches/making_change/making_change.py
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@cjyuan cjyuan Feb 22, 2026

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Array creation is a relatively costly operation.

From line 41, we know coins can only be one of the following 9 arrays:

[200, 100, 50, 20, 10, 5, 2, 1]
[100, 50, 20, 10, 5, 2, 1]
[50, 20, 10, 5, 2, 1]
...
[1]
[]

We could further improve the performance if we can

  • avoid repeatedly creating the same sub-arrays at line 41 (e.g. use another cache), and
  • create key as (total, a_unique_integer_identifying_the_subarray) instead of as (total, tuple of coins)
    • There are only a small number of different subarrays. We can easily assign each subarray a unique integer.

Original file line number Diff line number Diff line change
@@ -1,5 +1,6 @@
from typing import List

cache = {}

def ways_to_make_change(total: int) -> int:
"""
Expand All @@ -8,14 +9,25 @@ 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])



def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
key = (total, tuple(coins))
if key in cache:
return cache[key]
"""
Helper function for ways_to_make_change to avoid exposing the coins parameter to callers.
"""
if total == 0 or len(coins) == 0:
return 0
result = 0
cache[key] = result
return result

if total == 1 or len(coins) == 1:
result = 1
cache[key] = result
return result
Comment on lines +27 to +30
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@cjyuan cjyuan Feb 22, 2026

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When total == 1 or len(coins) == 1 is true, the result may not be 1 if the last coin is not 1 (e.g. the initial list of coins given is different)

On the other hand, when len(coins) == 1, there is simple way to determine if the current total (any value) can be made by the coin value.


ways = 0
for coin_index in range(len(coins)):
Expand All @@ -29,4 +41,6 @@ def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
intermediate = ways_to_make_change_helper(total - total_from_coins, coins=coins[coin_index+1:])
ways += intermediate
count_of_coin += 1
return ways

cache[key] = ways
return ways