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This approach involves using backtracking to place queens row by row, exploring valid positions recursively. Each queen placement is validated based on column and diagonal restrictions. If a valid placement is found, the algorithm progresses to the next row, otherwise it backtracks and attempts alternative placements.
Time Complexity: O(N!), because we are trying N possible positions for each row for N queens.
Space Complexity: O(N^2), due to the storage of the board states and recursion stack.
1function solveNQueens(n) {
2 const solutions = [];
3
4 function isSafe(board, row, col) {
5 for (let i = 0; i < row; i++) {
6 if (board[i] === col ||
7 board[i] - i === col - row ||
8 board[i] + i === col + row) {
9 return false;
10 }
11 }
12 return true;
13 }
14
15 function placeQueens(row, board) {
16 if (row === n) {
17 solutions.push(board.map(c => '.'.repeat(c) + 'Q' + '.'.repeat(n - c - 1)));
18 return;
19 }
20 for (let col = 0; col < n; col++) {
21 if (isSafe(board, row, col)) {
22 board[row] = col;
23 placeQueens(row + 1, board);
24 }
25 }
26 }
27
28 placeQueens(0, []);
29 return solutions;
30}
31
32const solutions = solveNQueens(4);
33console.log(solutions);This JavaScript solution leans on an array storage for queens, using indices as rows and values as columns. The function isSafe checks for column and diagonal threats. Recursive exploration in placeQueens accumulates legit solutions in solutions.
This approach optimizes the placement verification using bit manipulation, reducing the auxiliary space by managing columns and diagonals in integer bitmasks. This permits efficient bitwise operations for placing queens and checking potential attacks.
Time Complexity: O(N!), each row considers N placements, simplified by binary check.
Space Complexity: O(N), due to condensed state tracking via differential lists.
1import java.util.*;
2
3
This solution in Java embodies bit manipulation to track column and diagonal availability via bitmasks, ensuring low overhead while solving N-Queens recursively. Solutions are compiled as lists of strings to represent the board state efficiently.