Sudoku Backtracking Solver in Python

from bisect import bisect_left

# Sudoku Backtracking Solver in Python
#
# PURPOSE
# A Sudoku solver fills a 9 by 9 grid while respecting row, column, and box uniqueness.
#
# METHOD
# Validate givens into three occupancy tables, collect empty cells, then try candidate
# digits with iterative backtracking. On exhaustion restore each original empty cell to
# zero.
#
# COMPLEXITY
# Exponential worst-case search in the number of empty cells; O(81) bounded workspace.
#
# CONTRACT
# board has 81 values in [0,9], where 0 means empty. Returns 1 after solving, 0 if
# unsatisfiable, -1 for conflicting givens. On failure leaves board unchanged.
# 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.
# Expected board after call: [5, 3, 4, 6, 7, 8, 9, 1, 2, 6, 7, 2, 1, 9, 5, 3, 4, 8, 1, 9, 8,
# 3, 4, 2, 5, 6, 7, 8, 5, 9, 7, 6, 1, 4, 2, 3, 4, 2, 6, 8, 5, 3, 7, 9, 1, 7, 1, 3, 9, 2, 4,
# 8, 5, 6, 9, 6, 1, 5, 3, 7, 2, 8, 4, 2, 8, 7, 4, 1, 9, 6, 3, 5, 3, 4, 5, 2, 8, 6, 1, 7, 9].
# 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 sudoku_solve(board: list[int]) -> int:
    alg_require(isinstance(board, list) and len(board) <= 1000000)
    alg_require(len(board) == 81)
    for value in board:
        alg_require(value >= 0 and value <= 9)
    rows: list[int] = [0] * alg_size(90)
    cols: list[int] = [0] * alg_size(90)
    boxes: list[int] = [0] * alg_size(90)
    empty: list[int] = [0] * alg_size(81)
    count: int = 0
    position: int = 0
    cell: int = 0
    r: int = 0
    c: int = 0
    box: int = 0
    digit: int = 0
    while cell < 81:
        r = cell // 9
        c = cell % 9
        box = r // 3 * 3 + c // 3
        digit = board[cell]
        if digit == 0:
            empty[count] = cell
            count += 1
        else:
            if rows[r * 10 + digit] != 0 or cols[c * 10 + digit] != 0 or boxes[box * 10 + digit] != 0:
                return -1
            rows[r * 10 + digit] = 1
            cols[c * 10 + digit] = 1
            boxes[box * 10 + digit] = 1
        cell += 1
    while position >= 0 and position < count:
        cell = empty[position]
        r = cell // 9
        c = cell % 9
        box = r // 3 * 3 + c // 3
        digit = board[cell]
        if digit != 0:
            rows[r * 10 + digit] = 0
            cols[c * 10 + digit] = 0
            boxes[box * 10 + digit] = 0
        digit += 1
        while digit <= 9 and (rows[r * 10 + digit] != 0 or cols[c * 10 + digit] != 0 or boxes[box * 10 + digit] != 0):
            digit += 1
        if digit == 10:
            board[cell] = 0
            position -= 1
        else:
            board[cell] = digit
            rows[r * 10 + digit] = 1
            cols[c * 10 + digit] = 1
            boxes[box * 10 + digit] = 1
            position += 1
    if position == count:
        return 1
    return 0

board = [5, 3, 4, 6, 7, 8, 9, 1, 2, 6, 7, 2, 1, 9, 5, 3, 4, 8, 1, 9, 8, 3, 4, 2, 5, 6, 7, 8, 5, 9, 7, 6, 1, 4, 2, 3, 4, 2, 6, 8, 5, 3, 7, 9, 1, 7, 1, 3, 9, 2, 4, 8, 5, 6, 9, 6, 1, 5, 3, 7, 2, 8, 4, 2, 8, 7, 4, 1, 9, 6, 3, 5, 3, 4, 5, 2, 8, 6, 1, 7, 0]
result = sudoku_solve(board)
print(result)
# Expected board: [5, 3, 4, 6, 7, 8, 9, 1, 2, 6, 7, 2, 1, 9, 5, 3, 4, 8, 1, 9, 8, 3, 4, 2, 5, 6, 7, 8, 5, 9, 7, 6, 1, 4, 2, 3, 4, 2, 6, 8, 5, 3, 7, 9, 1, 7, 1, 3, 9, 2, 4, 8, 5, 6, 9, 6, 1, 5, 3, 7, 2, 8, 4, 2, 8, 7, 4, 1, 9, 6, 3, 5, 3, 4, 5, 2, 8, 6, 1, 7, 9]
print(board)