N Queens Backtracking in Python

from bisect import bisect_left

# N Queens Backtracking in Python
#
# PURPOSE
# N-queens places one queen in each row with no shared columns or diagonals.
#
# METHOD
# Try columns row by row. Keep occupied-column and diagonal marker arrays for constant-time
# safety checks. Backtrack when no column remains and count complete placements.
#
# COMPLEXITY
# Exponential search, bounded by O(n!) candidate placements with additional column-scan
# overhead; O(n) auxiliary space.
#
# CONTRACT
# 0<=n<=12. Returns solution count. The empty board has one placement. This implementation
# avoids recursion.
# Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
# counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
# Sorted order, correct tree topology, and caller-owned data-structure invariants are
# preconditions; checking them on every query would change query complexity.
# Scalar and shape checks are included. Value-range checks are included where needed to
# prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
# workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
# uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
# partly updated after arithmetic failure. These are single-threaded, in-memory
# implementations.
#
# LANGUAGE DETAILS
# Python 3.9+: automatic lengths, native lists, arbitrary-precision arithmetic, and
# bisect_left for LIS. Invalid scalar/shape inputs raise ValueError. Integer element types
# and sorted/tree invariants remain caller preconditions.
#
# EXAMPLE
# Expected return: 1.
# Run: python example.py
# Tests for this exact source artifact run before publication. The accompanying source
# archive includes unit cases, runtime results, and source hashes.

def alg_require(condition):
    if not condition:
        raise ValueError("invalid algorithm input; see contract")
def alg_size(n):
    alg_require(0 <= n <= 4000000)
    return n

def n_queens(n: int) -> int:
    alg_require(type(n) is int and -2147483648 <= n <= 2147483647)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(n >= 0 and n <= 12)
    columns: list[int] = [0] * alg_size(n + 1)
    rising: list[int] = [0] * alg_size(2 * n + 1)
    falling: list[int] = [0] * alg_size(2 * n + 1)
    placement: list[int] = [0] * alg_size(n + 1)
    row: int = 0
    col: int = 0
    count: int = 0
    if n == 0:
        return 1
    while row >= 0:
        col = placement[row]
        if col > 0:
            columns[col - 1] = 0
            rising[row + col - 1] = 0
            falling[row - col + n] = 0
        while col < n and (columns[col] != 0 or rising[row + col] != 0 or falling[row - col + n - 1] != 0):
            col += 1
        if col == n:
            placement[row] = 0
            row -= 1
        else:
            placement[row] = col + 1
            columns[col] = 1
            rising[row + col] = 1
            falling[row - col + n - 1] = 1
            if row == n - 1:
                count += 1
            else:
                row += 1
                placement[row] = 0
    return count

n = 0
result = n_queens(n)
print(result)