What is the brute force approach and complexity for Sudoku Solver?

Go through the empty cells in order; each try scans its row, column and box (27 reads) with time complexity O(9^E) and space complexity O(1).

What is the optimal approach and complexity for Sudoku Solver?

Bit masks for O(1) checks, and always fill the empty cell with the fewest digits that fit with optimal time complexity O(9^E) and space complexity O(1).

How do you identify the pattern for Sudoku Solver?

Look for key signals in the prompt: "fill in the empty cells" indicates A search over choices: each empty cell takes one digit, and a wrong digit must be undone.; "each row, column and box holds 1 to 9" indicates Three overlapping groups of nine cells. Which digits each group already holds is the state, so bit masks fit.; "exactly one solution" indicates The first full board is the answer, so each caller must stop as soon as the recursion succeeds..

Sudoku Solver

Hard·LeetCode #37 Pro OnlySlight Variation

LeetCode

Problem Statement

You are given a 9 × 9 Sudoku board. Digits 1 to 9 are clues. A . marks an empty cell.

Fill in every empty cell so that each row, each column and each of the nine 3 × 3 boxes holds the digits 1 to 9 exactly once. Change the board in place. The function returns nothing.

The puzzle has exactly one solution.

Input

board: a 9 × 9 grid of characters. Each one is a digit 1 to 9 or ..

Output

The same grid, filled in. Nothing is returned.

Example 1

Input: board = the 9 × 9 grid from LeetCode's example 1. Its first row is 5 3 . . 7 . . . .

Output: The same grid, filled in. Its first row is 5 3 4 6 7 8 9 1 2.

The puzzle has one solution. The blanks in the first row get 4, 6, 8, 9, 1 and 2.

Constraints

• board.length == 9
• board[i].length == 9
• board[i][j] is a digit or '.'
• It is guaranteed that the input board has only one solution.

Pattern Recognition Clues

Recognising these key signals in the problem description immediately identifies the pattern:

“fill in the empty cells”

→ A search over choices: each empty cell takes one digit, and a wrong digit must be undone.

“each row, column and box holds 1 to 9”

→ Three overlapping groups of nine cells. Which digits each group already holds is the state, so bit masks fit.

“exactly one solution”

→ The first full board is the answer, so each caller must stop as soon as the recursion succeeds.

Approach Complexity Ladder

Brute Force — fill in order, check by scanningO(9^E)O(1)

Go through the empty cells in order; each try scans its row, column and box (27 reads)

Better — bit masks for O(1) checksO(9^E)O(1)

The same cell order as the brute force, but each digit check is one AND on three bit masks

Optimal — bit masks plus the fewest-options cell first (MRV)O(9^E)O(1)

Bit masks for O(1) checks, and always fill the empty cell with the fewest digits that fit

Brute Force Intuition

Walk the empty cells in row-by-row order. For each one, try the digits 1 to 9. A digit is allowed when the row, the column and the box do not already hold it, which is found by scanning them. If a cell has no digit that fits, go back to the cell before it. The first full board is the answer.

Recognition rule for this sub-pattern:

Place items one by one, checking the rules before each placement. Prune as early as possible.

Remember this · Backtracking

Trigger

"All subsets / permutations / combinations", place items under rules (N-Queens, Sudoku), find a path in a grid.

Think

At each step, what are my choices, and how do I undo a choice?

Key question

What is my state, when is it complete, and when do I prune?

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