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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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2da2744
fibonacci
Nataliia74 Feb 15, 2026
62e1e71
making_change
Nataliia74 Feb 15, 2026
403c901
simplify complexity, math instead recursion on 0 total, and 0 number…
Nataliia74 Apr 21, 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
36 changes: 21 additions & 15 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.

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@Nataliia74 Nataliia74 Apr 21, 2026

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Eliminated slicing, which is costly and creates a new array every recursion, and also many duplicates; the tuple conversion was costly as well. Changed by exchanging the tuple at the index an array.

Original file line number Diff line number Diff line change
@@ -1,32 +1,38 @@
from typing import List

cache = {}
coins = [1, 2, 5, 10, 20, 50, 100, 200]
Comment on lines +3 to +4
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@cjyuan cjyuan Apr 23, 2026 •
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Should you need to avoid introducing variables in the global scope, you can consider:

  • Declare them in ways_to_make_change()
  • Define ways_to_make_change_helper() as an inner function of ways_to_make_change(). Inner function can access the variables in the outer scope.


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])
return ways_to_make_change_helper(total, 0)



def ways_to_make_change_helper(total: int, coins: List[int]) -> int:
def ways_to_make_change_helper(total: int, index: int) -> int:
key = (total, index)
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:
if total == 0:
cache[key] = 1
return 1

if index == len(coins):
cache[key] = 0
return 0

coin=coins[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
return ways
count = 0
while coin * count <= total:
ways += ways_to_make_change_helper(total - coin * count, index + 1)
count +=1
cache[key] = ways
return ways
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