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"""

In this problem, we want to determine all possible combinations of k

numbers out of 1 ... n. We use backtracking to solve this problem.

Time complexity: O(C(n,k)) which is O(n choose k) = O((n!/(k! * (n - k)!))),

"""

from __future__ import annotations

from itertools import combinations

def combination_lists(n: int, k: int) -> list[list[int]]:

"""

>>> combination_lists(n=4, k=2)

[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]

"""

return [list(x) for x in combinations(range(1, n + 1), k)]

def generate_all_combinations(n: int, k: int) -> list[list[int]]:

"""

>>> generate_all_combinations(n=4, k=2)

[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]

>>> generate_all_combinations(n=0, k=0)

[[]]

>>> generate_all_combinations(n=10, k=-1)

Traceback (most recent call last):

...

ValueError: k must not be negative

>>> generate_all_combinations(n=-1, k=10)

Traceback (most recent call last):

...

ValueError: n must not be negative

>>> generate_all_combinations(n=5, k=4)

[[1, 2, 3, 4], [1, 2, 3, 5], [1, 2, 4, 5], [1, 3, 4, 5], [2, 3, 4, 5]]

>>> from itertools import combinations

>>> all(generate_all_combinations(n, k) == combination_lists(n, k)

... for n in range(1, 6) for k in range(1, 6))

True

"""

if k < 0:

raise ValueError("k must not be negative")

if n < 0:

raise ValueError("n must not be negative")

result: list[list[int]] = []

create_all_state(1, n, k, [], result)

return result

def create_all_state(

increment: int,

total_number: int,

level: int,

current_list: list[int],

total_list: list[list[int]],

) -> None:

if level == 0:

total_list.append(current_list[:])

return

for i in range(increment, total_number - level + 2):

current_list.append(i)

create_all_state(i + 1, total_number, level - 1, current_list, total_list)

current_list.pop()

if __name__ == "__main__":

from doctest import testmod

testmod()

print(generate_all_combinations(n=4, k=2))

tests = ((n, k) for n in range(1, 5) for k in range(1, 5))

for n, k in tests:

print(n, k, generate_all_combinations(n, k) == combination_lists(n, k))

print("Benchmark:")

from timeit import timeit

for func in ("combination_lists", "generate_all_combinations"):

print(f"{func:>25}(): {timeit(f'{func}(n=4, k = 2)', globals=globals())}")

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