Update __init__.py

backtracking/__init__.py

@@ -0,0 +1,116 @@
+"""
+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]]:
+    """
+    Generates all possible combinations of k numbers out of 1 ... n using itertools.
+
+    >>> 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]]:
+    """
+    Generates all possible combinations of k numbers out of 1 ... n using backtracking.
+
+    >>> 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]]
+    >>> generate_all_combinations(n=3, k=3)
+    [[1, 2, 3]]
+    >>> generate_all_combinations(n=3, k=1)
+    [[1], [2], [3]]
+    >>> generate_all_combinations(n=1, k=0)
+    [[]]
+    >>> generate_all_combinations(n=1, k=1)
+    [[1]]
+    >>> 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:
+    """
+    Helper function to recursively build all combinations.
+
+    >>> create_all_state(1, 4, 2, [], result := [])
+    >>> result
+    [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]
+    >>> create_all_state(1, 3, 3, [], result := [])
+    >>> result
+    [[1, 2, 3]]
+    >>> create_all_state(2, 2, 1, [1], result := [])
+    >>> result
+    [[1, 2]]
+    >>> create_all_state(1, 0, 0, [], result := [])
+    >>> result
+    [[]]
+    >>> create_all_state(1, 4, 0, [1, 2], result := [])
+    >>> result
+    [[1, 2]]
+    >>> create_all_state(5, 4, 2, [1, 2], result := [])
+    >>> result
+    []
+    """
+    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())}")