Algorithms: Backtracking

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// N Queens Backtracking in C
//
// 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
// C99: checked calloc/free workspace, const input pointers, explicit size_t lengths. Errors
// return INT_MIN with errno=EINVAL, ERANGE, or ENOMEM. Check errno before using a result.
// Input/output arrays must not alias. No variable-length stack arrays.
//
// EXAMPLE
// Expected return: 1.
// Run: gcc -std=c99 -O2 example.c -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static int alg_checked(int64_t v, int *error) {
    if (v < INT_MIN || v > INT_MAX) { *error = 1; return 0; }
    return (int)v;
}
static int alg_add(int a, int b, int *e) { return alg_checked((int64_t)a+b,e); }
static int alg_sub(int a, int b, int *e) { return alg_checked((int64_t)a-b,e); }
static int alg_mul(int a, int b, int *e) { return alg_checked((int64_t)a*b,e); }

int n_queens(int n) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *columns = NULL;
    int *rising = NULL;
    int *falling = NULL;
    int *placement = NULL;
    if (!(((n >= 0) && (n <= 1000000)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((n >= 0) && (n <= 12)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    {
        int alg_length = alg_add(n, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        columns = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*columns));
        if (!columns) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = alg_add(alg_mul(2, n, &alg_error), 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        rising = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*rising));
        if (!rising) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = alg_add(alg_mul(2, n, &alg_error), 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        falling = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*falling));
        if (!falling) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = alg_add(n, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        placement = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*placement));
        if (!placement) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    int row = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int col = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int count = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int alg_condition_1 = (n == 0);
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_condition_1) {
        alg_result = 1;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        goto alg_cleanup;
    }
    while (1) {
        int alg_condition = (row >= 0);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        col = placement[row];
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        int alg_condition_2 = (col > 0);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_condition_2) {
            columns[alg_sub(col, 1, &alg_error)] = 0;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            rising[alg_sub(alg_add(row, col, &alg_error), 1, &alg_error)] = 0;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            falling[alg_add(alg_sub(row, col, &alg_error), n, &alg_error)] = 0;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        while (1) {
            int alg_condition = ((col < n) && ((columns[col] != 0) || (rising[alg_add(row, col, &alg_error)] != 0) || (falling[alg_sub(alg_add(alg_sub(row, col, &alg_error), n, &alg_error), 1, &alg_error)] != 0)));
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            col = alg_add(col, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        int alg_condition_3 = (col == n);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_condition_3) {
            placement[row] = 0;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            row = alg_sub(row, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        } else {
            placement[row] = alg_add(col, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            columns[col] = 1;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            rising[alg_add(row, col, &alg_error)] = 1;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            falling[alg_sub(alg_add(alg_sub(row, col, &alg_error), n, &alg_error), 1, &alg_error)] = 1;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            int alg_condition_4 = (row == alg_sub(n, 1, &alg_error));
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (alg_condition_4) {
                count = alg_add(count, 1, &alg_error);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            } else {
                row = alg_add(row, 1, &alg_error);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                placement[row] = 0;
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            }
        }
    }
    alg_result = count;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    free(columns);
    free(rising);
    free(falling);
    free(placement);
    return alg_result;
}

int main(void) {
    int n = 0;
    int result = n_queens(n);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// N Queens Backtracking in C++
//
// 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
// C++17: const vector references for inputs, mutable references for outputs, RAII workspace,
// standard lower_bound for LIS. Invalid inputs throw invalid_argument; overflow throws
// overflow_error; allocation failure propagates bad_alloc.
//
// EXAMPLE
// Expected return: 1.
// Run: g++ -std=c++17 -O2 example.cpp -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static void alg_require(bool valid) { if (!valid) throw std::invalid_argument("invalid algorithm input"); }
static int alg_checked(int64_t v) { if (v < INT32_MIN || v > INT32_MAX) throw std::overflow_error("32-bit result overflow"); return (int)v; }
static int alg_add(int a,int b) { return alg_checked((int64_t)a+b); }
static int alg_sub(int a,int b) { return alg_checked((int64_t)a-b); }
static int alg_mul(int a,int b) { return alg_checked((int64_t)a*b); }
static size_t alg_size(int n) { alg_require(n>=0 && n<=4000000); return (size_t)n; }

int n_queens(int n) {
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((n >= 0) && (n <= 12)));
    std::vector<int> columns(alg_size(alg_add(n, 1)), 0);
    std::vector<int> rising(alg_size(alg_add(alg_mul(2, n), 1)), 0);
    std::vector<int> falling(alg_size(alg_add(alg_mul(2, n), 1)), 0);
    std::vector<int> placement(alg_size(alg_add(n, 1)), 0);
    int row = 0;
    int col = 0;
    int count = 0;
    if ((n == 0)) {
        return 1;
    }
    while ((row >= 0)) {
        col = placement[row];
        if ((col > 0)) {
            columns[alg_sub(col, 1)] = 0;
            rising[alg_sub(alg_add(row, col), 1)] = 0;
            falling[alg_add(alg_sub(row, col), n)] = 0;
        }
        while (((col < n) && ((columns[col] != 0) || (rising[alg_add(row, col)] != 0) || (falling[alg_sub(alg_add(alg_sub(row, col), n), 1)] != 0)))) {
            col = alg_add(col, 1);
        }
        if ((col == n)) {
            placement[row] = 0;
            row = alg_sub(row, 1);
        } else {
            placement[row] = alg_add(col, 1);
            columns[col] = 1;
            rising[alg_add(row, col)] = 1;
            falling[alg_sub(alg_add(alg_sub(row, col), n), 1)] = 1;
            if ((row == alg_sub(n, 1))) {
                count = alg_add(count, 1);
            } else {
                row = alg_add(row, 1);
                placement[row] = 0;
            }
        }
    }
    return count;
}

int main(void) {
    int n = 0;
    int result = n_queens(n);
    printf("%d\n", result);
    return 0;
}

using System;
// N Queens Backtracking in C#
//
// 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
// C# / .NET 10: ReadOnlySpan<int> inputs, Span<int> outputs, primitive workspace arrays,
// Array.BinarySearch for LIS, and checked integer arithmetic. Invalid arguments and overflow
// throw exceptions.
//
// EXAMPLE
// Expected return: 1.
// Run: Use Program.cs in a net10.0 console project; dotnet run.
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class Program {
    static void alg_require(bool valid) { if (!valid) throw new ArgumentException("invalid algorithm input"); }
    static int alg_size(int n) { alg_require(n>=0 && n<=4000000); return n; }

    static int n_queens(int n) {
        checked {
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require(((n >= 0) && (n <= 12)));
            int[] columns = new int[alg_size((n + 1))];
            int[] rising = new int[alg_size(((2 * n) + 1))];
            int[] falling = new int[alg_size(((2 * n) + 1))];
            int[] placement = new int[alg_size((n + 1))];
            int row = 0;
            int col = 0;
            int count = 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) && ((columns[col] != 0) || (rising[(row + col)] != 0) || (falling[(((row - col) + n) - 1)] != 0)))) {
                    col = (col + 1);
                }
                if ((col == n)) {
                    placement[row] = 0;
                    row = (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 = (count + 1);
                    } else {
                        row = (row + 1);
                        placement[row] = 0;
                    }
                }
            }
            return count;
        }
    }

    static void Main() {
        int n = 0;
        int result = n_queens(n);
        Console.WriteLine(result);
    }
}

<?php
declare(strict_types=1);
// N Queens Backtracking in PHP
//
// 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
// PHP 8.1+ on a 64-bit runtime: packed arrays, strict scalar types, read-only inputs by
// value using copy-on-write, references only for outputs, and checked 32-bit arithmetic.
// Invalid input and overflow throw exceptions. PHP arrays have higher overhead than native
// packed buffers in C/Java.
//
// EXAMPLE
// Expected return: 1.
// Run: php example.php
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

function alg_require(bool $valid): void { if (!$valid) throw new InvalidArgumentException("invalid algorithm input"); }
function alg_checked(int|float $v): int { if ($v < -2147483648 || $v > 2147483647 || floor($v) != $v) throw new OverflowException("32-bit result overflow"); return (int)$v; }
function alg_add(int $a,int $b): int { return alg_checked($a+$b); }
function alg_sub(int $a,int $b): int { return alg_checked($a-$b); }
function alg_mul(int $a,int $b): int { return alg_checked($a*$b); }
function alg_size(int $n): int { alg_require($n>=0 && $n<=4000000); return $n; }

function n_queens(int $n): int {
    alg_require((($n >= 0) && ($n <= 1000000)));
    alg_require((($n >= 0) && ($n <= 12)));
    $columns = array_fill(0, alg_size(alg_add($n, 1)), 0);
    $rising = array_fill(0, alg_size(alg_add(alg_mul(2, $n), 1)), 0);
    $falling = array_fill(0, alg_size(alg_add(alg_mul(2, $n), 1)), 0);
    $placement = array_fill(0, alg_size(alg_add($n, 1)), 0);
    $row = 0;
    $col = 0;
    $count = 0;
    if (($n == 0)) {
        return 1;
    }
    while (($row >= 0)) {
        $col = $placement[$row];
        if (($col > 0)) {
            $columns[alg_sub($col, 1)] = 0;
            $rising[alg_sub(alg_add($row, $col), 1)] = 0;
            $falling[alg_add(alg_sub($row, $col), $n)] = 0;
        }
        while ((($col < $n) && (($columns[$col] != 0) || ($rising[alg_add($row, $col)] != 0) || ($falling[alg_sub(alg_add(alg_sub($row, $col), $n), 1)] != 0)))) {
            $col = alg_add($col, 1);
        }
        if (($col == $n)) {
            $placement[$row] = 0;
            $row = alg_sub($row, 1);
        } else {
            $placement[$row] = alg_add($col, 1);
            $columns[$col] = 1;
            $rising[alg_add($row, $col)] = 1;
            $falling[alg_sub(alg_add(alg_sub($row, $col), $n), 1)] = 1;
            if (($row == alg_sub($n, 1))) {
                $count = alg_add($count, 1);
            } else {
                $row = alg_add($row, 1);
                $placement[$row] = 0;
            }
        }
    }
    return $count;
}

$n = 0;
$result = n_queens($n);
echo $result, PHP_EOL;