N Queens Backtracking in Java

// N Queens Backtracking in Java
//
// 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
// Java 17+: primitive int arrays, Arrays.binarySearch for LIS, and Math exact arithmetic.
// Invalid input throws IllegalArgumentException; arithmetic overflow throws
// ArithmeticException.
//
// EXAMPLE
// Expected return: 1.
// Run: javac AlgorithmExample.java && java AlgorithmExample
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

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

    static int n_queens(int n) {
        alg_require(((n >= 0) && (n <= 1000000)));
        alg_require(((n >= 0) && (n <= 12)));
        int[] columns = new int[alg_size(Math.addExact(n, 1))];
        int[] rising = new int[alg_size(Math.addExact(Math.multiplyExact(2, n), 1))];
        int[] falling = new int[alg_size(Math.addExact(Math.multiplyExact(2, n), 1))];
        int[] placement = new int[alg_size(Math.addExact(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[Math.subtractExact(col, 1)] = 0;
                rising[Math.subtractExact(Math.addExact(row, col), 1)] = 0;
                falling[Math.addExact(Math.subtractExact(row, col), n)] = 0;
            }
            while (((col < n) && ((columns[col] != 0) || (rising[Math.addExact(row, col)] != 0) || (falling[Math.subtractExact(Math.addExact(Math.subtractExact(row, col), n), 1)] != 0)))) {
                col = Math.addExact(col, 1);
            }
            if ((col == n)) {
                placement[row] = 0;
                row = Math.subtractExact(row, 1);
            } else {
                placement[row] = Math.addExact(col, 1);
                columns[col] = 1;
                rising[Math.addExact(row, col)] = 1;
                falling[Math.subtractExact(Math.addExact(Math.subtractExact(row, col), n), 1)] = 1;
                if ((row == Math.subtractExact(n, 1))) {
                    count = Math.addExact(count, 1);
                } else {
                    row = Math.addExact(row, 1);
                    placement[row] = 0;
                }
            }
        }
        return count;
    }

    public static void main(String[] args) {
        int n = 0;
        int result = n_queens(n);
        System.out.println(result);
    }
}