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83 lines
2.4 KiB
Python
83 lines
2.4 KiB
Python
"""
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In this problem, we want to determine all possible combinations of k
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numbers out of 1 ... n. We use backtracking to solve this problem.
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Time complexity: O(C(n,k)) which is O(n choose k) = O((n!/(k! * (n - k)!))),
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"""
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from __future__ import annotations
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from itertools import combinations
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def combination_lists(n: int, k: int) -> list[list[int]]:
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"""
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>>> combination_lists(n=4, k=2)
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[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]
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"""
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return [list(x) for x in combinations(range(1, n + 1), k)]
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def generate_all_combinations(n: int, k: int) -> list[list[int]]:
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"""
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>>> generate_all_combinations(n=4, k=2)
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[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]
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>>> generate_all_combinations(n=0, k=0)
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[[]]
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>>> generate_all_combinations(n=10, k=-1)
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Traceback (most recent call last):
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...
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ValueError: k must not be negative
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>>> generate_all_combinations(n=-1, k=10)
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Traceback (most recent call last):
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...
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ValueError: n must not be negative
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>>> generate_all_combinations(n=5, k=4)
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[[1, 2, 3, 4], [1, 2, 3, 5], [1, 2, 4, 5], [1, 3, 4, 5], [2, 3, 4, 5]]
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>>> from itertools import combinations
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>>> all(generate_all_combinations(n, k) == combination_lists(n, k)
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... for n in range(1, 6) for k in range(1, 6))
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True
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"""
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if k < 0:
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raise ValueError("k must not be negative")
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if n < 0:
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raise ValueError("n must not be negative")
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result: list[list[int]] = []
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create_all_state(1, n, k, [], result)
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return result
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def create_all_state(
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increment: int,
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total_number: int,
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level: int,
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current_list: list[int],
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total_list: list[list[int]],
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) -> None:
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if level == 0:
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total_list.append(current_list[:])
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return
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for i in range(increment, total_number - level + 2):
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current_list.append(i)
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create_all_state(i + 1, total_number, level - 1, current_list, total_list)
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current_list.pop()
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if __name__ == "__main__":
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from doctest import testmod
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testmod()
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print(generate_all_combinations(n=4, k=2))
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tests = ((n, k) for n in range(1, 5) for k in range(1, 5))
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for n, k in tests:
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print(n, k, generate_all_combinations(n, k) == combination_lists(n, k))
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print("Benchmark:")
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from timeit import timeit
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for func in ("combination_lists", "generate_all_combinations"):
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print(f"{func:>25}(): {timeit(f'{func}(n=4, k = 2)', globals=globals())}")
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