framework_note/leetcode/note/medium/36_valid_sudoku.md

id, title, difficulty, tags, status, date_solved, leetcode_url, review_needed

id	title	difficulty	tags	status	date_solved	leetcode_url	review_needed
36	Valid Sudoku	Medium	array hash-table matrix	Solved	2026-06-05	https://leetcode.com/problems/valid-sudoku/	false

36. Valid Sudoku

[!info] Problem Link: LeetCode - Valid Sudoku

📝 Problem Description

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

1. Each row must contain the digits 1-9 without repetition.
2. Each column must contain the digits 1-9 without repetition.
3. 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 1

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

📥 Example 2

Input:

board = 
[["8","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: false Explanation: Same as Example 1, except with the 5 in the top left corner being modified to 8. Since there are two 8's in the top-left 3x3 sub-box, it is invalid.

---

💡 Approaches & Explanations

Approach 1: Hash Sets for Rows, Columns, and Boxes — Optimal

We use three collections of sets to track the numbers we've seen:

1. rows: 9 sets, one for each row.
2. cols: 9 sets, one for each column.
3. boxes: 9 sets, one for each 3x3 sub-grid.

We iterate through every cell (r, c) in the 9x9 board. If the cell is not empty (i.e., not .):

• Calculate the box index: box_idx = (r // 3) * 3 + (c // 3).
• Check if the digit already exists in rows[r], cols[c], or boxes[box_idx].
• If it exists, the board is invalid.
• If not, add the digit to all three sets and continue.

📊 Complexity Analysis

• Time Complexity: \mathcal{O}(1) or \mathcal{O}(N^2) where N=9. Since the board size is fixed at 9x9, we always perform 81 operations.
• Space Complexity: \mathcal{O}(1) or \mathcal{O}(N^2) to store the sets for rows, columns, and boxes. In the worst case, we store 81 entries.

---

💻 Code Implementations

Python3

class Solution:
    def isValidSudoku(self, board: List[List[str]]) -> bool:
        cols = collections.defaultdict(set)
        rows = collections.defaultdict(set)
        squares = collections.defaultdict(set)  # key = (r // 3, c // 3)

        for r in range(9):
            for c in range(9):
                if board[r][c] == ".":
                    continue
                if (
                    board[r][c] in rows[r]
                    or board[r][c] in cols[c]
                    or board[r][c] in squares[(r // 3, c // 3)]
                ):
                    return False
                cols[c].add(board[r][c])
                rows[r].add(board[r][c])
                squares[(r // 3, c // 3)].add(board[r][c])

        return True

---

🧠 Key Takeaways & Lessons

• Coordinate Mapping: Mapping a 2D coordinate (r, c) to a 1D sub-grid index or a tuple key (r // 3, c // 3) is a crucial technique for matrix problems.
• Trade-off: Using hash sets provides \mathcal{O}(1) lookup time, making the validation process very efficient.
• Constraints Matter: Since the board size is fixed (9x9), "optimal" here refers to the single-pass nature and clean logic rather than asymptotic growth beyond the constant size.