How to get better time complexity of N queens validator

Question

How can we lower the time complexity of this solution of NxN queens matrix validator? I have this solution while I check every row and every column and every diagonal of the matrix. If every row and column has exactly 1 queen, and the matrix has no more than 1 queen the output is true. This solution works but I think it's brute force.

   public static boolean solveMatrix(int[][] matrix) {

        int row, col;
        int rowCount = matrix.length;
        int columnCount = matrix[0].length;
        int counter = 0;

        // Checking if there is 1 queen per row
        for (int i = 0; i < 8; i++) {
            counter = 0;
            for (int j = 0; j < 8; j++) {

                if (matrix[i][j] == 1) {

                    counter++;
                }

            }
            if (counter != 1) {

                return false;
            }

        }
// Checking if there is 1 queen per column
        for (int i = 0; i < 8; i++) {
            counter = 0;
            for (int j = 0; j < 8; j++) {

                if (matrix[j][i] == 1) {

                    counter++;
                }

            }
            if (counter != 1) {

                return false;
            }

        }
        // Checking first side diagonals
        for (int k = 0; k < rowCount; k++) {
            counter = 0;
            for (row = k, col = 0; row >= 0 && col < columnCount; row--, col++) {
                if (matrix[row][col] == 1) {
                    counter++;
                }
            }
            if (counter > 1) {
                return false;
            }
        }
    // Checking first side diagonals
        for (int k = 1; k < columnCount; k++) {
            counter = 0;
            for (row = rowCount - 1, col = k; row >= 0 && col < columnCount; row--, col++) {
                if (matrix[row][col] == 1) {
                    counter++;
                }
            }
            if (counter > 1) {
                return false;
            }
        }
        // Checking second side diagonals
        for (int k = rowCount - 1; k >= 0; k--) {
            counter = 0;
            for (row = k, col = columnCount - 1; row >= 0 && col >= 0; row--, col--) {

                if (matrix[row][col] == 1) {
                    counter++;
                }

            }
            if (counter > 1) {
                return false;
            }
        }
// Checking second side diagonals
        for (int k = rowCount - 1; k >= 0; k--) {
            counter = 0;
            for (row = k, col = columnCount - 1; row >= 0 && col >= 0; row--, col--) {

                if (matrix[row][col] == 1) {
                    counter++;
                }

            }
            if (counter > 1) {
                return false;
            }
        }

        return true;

    }

You may find some ideas [here](https://stackoverflow.com/q/65610853/10553341). — Damien, Dec 01 '21 at 11:52
Your method's time complexity is `O(n^2)`. Since you are receiving your input as an NxN array, there's no way to improve that. You can get efficiency improvements, but the time complexity will still be `O(n^2)`. If you received the input as a list of N positions instead, then you could validate it in `O(n)`, but not with the current input form. — RBarryYoung, Dec 01 '21 at 18:43

score 0 · Accepted Answer · answered Dec 01 '21 at 11:56

You need to use 4 hashmaps, one for columns, one for rows, one for left to right diagonals, and one for right to left diagonals.

Now, do just one nested loop for rows and columns. When you find a queen do the following 4 steps:

Check if at that row index there is a queen in the row hashmap. If not, add the row index. If there is already one, return false.
Check if at that col index there is a queen in the col hashmap. If not, add the col index. If there is already one, return false.
Check if at that left to right diagonal, there is one queen in the left to right diagonal hashmap and act accordingly. Note that for each left to right diagonal rowIndex-columnIndex is always the same.
Check if at that right to left diagonal, there is one queen in the right to left diagonal hashmap and act accordingly. Note that for each right to left diagonal rowIndex+columnIndex is always the same.

If you successfully completed the above nested loop, it means that the board is valid. Return true.

This algorithm runs in O(n^2) where n is the length of one side of the square matrix.

It has a linear space complexity in the length of the side of the matrix n, since we are using 4 hashmaps with each n elements at most.

How to get better time complexity of N queens validator

1 Answers1