Valid Sudoku

Problem Statement

Determine if a 9 x 9 Sudoku board is valid. Only the filled cells need to be validated according to the following rules:

• Each row must contain the digits 1-9 without repetition.
• Each column must contain the digits 1-9 without repetition.
• Each of the nine 3 x 3 sub-boxes of the grid must contain the digits 1-9 without repetition.

Note: A Sudoku board (partially filled) could be valid but is not necessarily solvable. Only the filled cells need to be validated according to the mentioned rules.

Example:

Input: board =
[
 ["5","3",".",".","7",".",".",".","."],
 ["6",".",".","1","9","5",".",".","."],
 [".","9","8",".",".",".",".","6","."],
 ["8",".",".",".","6",".",".",".","3"],
 ["4",".",".","8",".","3",".",".","1"],
 ["7",".",".",".","2",".",".",".","6"],
 [".","6",".",".",".",".","2","8","."],
 [".",".",".","4","1","9",".",".","5"],
 [".",".",".",".","8",".",".","7","9"]
]

Output: true

Solution: Hashing

• To ensure the elements in each row, column, square are unique, maintain a set for each
• To track the elements in squares, maintain a 3x3 grid where each cell contains the digits in a square. The square grid cell that an element at [x][y] index position in input board will go to will be given by calculating [x/3][y/3]

bool isValidSudoku (vector<vector<char>> &board) {
    // Type alias for set of uniqie digits
    using uniqueDigits = unordered_set<int>;

    // To track digits in each row, column, square respectively
    vector<uniqueDigits> rowVals(9);
    vector<uniqueDigits> colVals(9);
    vector<vector<uniqueDigits>> squareVals(3, vector<uniqueDigits>(3));

    // Traverse each cell of board
    for (int r = 0; r < 9; r++) {
        for (int c = 0; c < 9; c++) {
            // Ignore cell if it's not a number
            if (board[r][c] == '.') continue;

            // '1'-'9' charcter digit converted to 0-8 index
            int curr = board[r][c] - '0' - 1;
            // Calcuate the position in 3*3 square grid
            int squareX = r / 3, squareY = c / 3;

            // If current digit already present in either of it's
            // row or column or square, the sudoku is INVALID
            if (
                rowVals[r].count(curr) || colVals[c].count(curr)
                || squareVals[squareX][squareY].count(curr)
            ) {
                return false;
            }

            // Add current cell to tracked entries
            rowVals[r].insert(curr);
            colVals[c].insert(curr);
            squareVals[squareX][squareY].insert(curr);
        }
    }

    // No conficts. So valid sudoku
    return true;
}

Time taken:

Traversing each cell of board takes \( O(N^2) \) time and each access or insert operation into the hash-set takes \( O(1) \) time on average. Thus, overall time complexity is \( O(N^2) \)

Space used:

• The row and column sets would have at-most \( N \) elements filled at each of the \( N \) total rows and columns i.e. \( O(N^2) \) space over all elements
• The grid for each square would have dimensions \( \sqrt{N} \ast \sqrt{N} \) and will contain at-most \( N \) elements at each cell of grid i.e. \( O(N^2) \) space over all elements