Sudoku Solver Using Backtracking in DSA Javascript - Time & Space Complexity

Time Complexity: Sudoku Solver Using Backtracking

O(9^n)

Understanding Time Complexity

We want to understand how the time needed to solve a Sudoku puzzle grows as the puzzle size or difficulty changes.

Specifically, how does the backtracking method's work increase with more empty cells?

Scenario Under Consideration

Analyze the time complexity of the following Sudoku solver code using backtracking.

function solveSudoku(board) {
  for (let row = 0; row < 9; row++) {
    for (let col = 0; col < 9; col++) {
      if (board[row][col] === 0) {
        for (let num = 1; num <= 9; num++) {
          if (isValid(board, row, col, num)) {
            board[row][col] = num;
            if (solveSudoku(board)) return true;
            board[row][col] = 0;
          }
        }
        return false;
      }
    }
  }
  return true;
}

This code tries numbers 1 to 9 in empty cells, checks if valid, and uses recursion to fill the board.

Identify Repeating Operations

Look at the loops and recursion that repeat work.

• Primary operation: Trying numbers 1 to 9 in each empty cell and recursively solving the rest.
• How many times: Up to 9 tries per empty cell, repeated recursively for each empty cell.

How Execution Grows With Input

Each empty cell can try up to 9 numbers, and for each try, the solver moves to the next empty cell, repeating this process.

Empty Cells (n)	Approx. Operations
10	Up to 9^10 = 3,486,784,401 tries
20	Up to 9^20 = very large (over 10^19) tries
30	Up to 9^30 = extremely huge number of tries

Pattern observation: The number of tries grows exponentially with the number of empty cells.

Final Time Complexity

Time Complexity: O(9^n)

This means the time needed can grow very fast as the number of empty cells increases, because each cell tries many possibilities recursively.

Common Mistake

[X] Wrong: "The solver always tries all 9 numbers for every empty cell."

[OK] Correct: The solver stops trying numbers once it finds a valid number that leads to a solution, so it often tries fewer than 9 per cell.

Interview Connect

Understanding this exponential growth helps you explain why backtracking can be slow and why pruning or smarter methods matter, a skill valued in problem solving.

Self-Check

"What if we used a heuristic to pick the next empty cell with the fewest valid options? How would that affect the time complexity?"

Practice

1. What is the main idea behind using backtracking to solve a Sudoku puzzle?

easy

A. Try placing numbers one by one and backtrack if a conflict occurs

B. Fill all empty cells randomly and check if the puzzle is solved

C. Use a sorting algorithm to arrange numbers in order

D. Replace all zeros with the number 9 directly

Solution

1. Step 1: Understand backtracking approach

   Backtracking tries numbers in empty cells one by one and checks if they fit the Sudoku rules.

2. Step 2: Handle conflicts by backtracking

   If a number causes a conflict, the algorithm removes it and tries the next number, going back if needed.

3. Final Answer:

   Try placing numbers one by one and backtrack if a conflict occurs -> Option A

4. Quick Check:

   Backtracking = Try and undo [OK]

Hint: Backtracking means try, check, undo if wrong [OK]

Common Mistakes:

• Thinking backtracking fills cells randomly
• Assuming sorting solves Sudoku
• Replacing zeros without checks

2. Which of the following is the correct way to check if a number can be placed in a Sudoku cell in JavaScript?

easy

A. if isSafe(board, row, col, num) { placeNumber(); }

B. if isSafe(board, row, col, num) then placeNumber();

C. if (isSafe board, row, col, num) { placeNumber(); }

D. if (isSafe(board, row, col, num)) { placeNumber(); }

Solution

1. Step 1: Understand JavaScript if syntax

   JavaScript requires parentheses around the condition in an if statement.

2. Step 2: Check function call syntax

   Function calls need parentheses with arguments inside, like isSafe(board, row, col, num).

3. Final Answer:

   if (isSafe(board, row, col, num)) { placeNumber(); } -> Option D

4. Quick Check:

   if(condition) with parentheses [OK]

Hint: Use parentheses around conditions in if statements [OK]

Common Mistakes:

• Missing parentheses in if condition
• Using 'then' keyword in JavaScript
• Incorrect function call syntax

3. What will be the output of the following code snippet when solving a Sudoku board?

const board = [
  [5,3,0,0,7,0,0,0,0],
  [6,0,0,1,9,5,0,0,0],
  [0,9,8,0,0,0,0,6,0],
  [8,0,0,0,6,0,0,0,3],
  [4,0,0,8,0,3,0,0,1],
  [7,0,0,0,2,0,0,0,6],
  [0,6,0,0,0,0,2,8,0],
  [0,0,0,4,1,9,0,0,5],
  [0,0,0,0,8,0,0,7,9]
];
solveSudoku(board);
console.log(board);

medium

A. An error because solveSudoku is not defined

B. The original board unchanged with zeros intact

C. A fully solved Sudoku board with no zeros

D. A partially solved board with some zeros replaced

Solution

1. Step 1: Understand solveSudoku function purpose

   solveSudoku uses backtracking to fill all empty cells correctly, solving the puzzle.

2. Step 2: Check console.log output after solving

   After solveSudoku runs, the board is fully solved with no zeros remaining.

3. Final Answer:

   A fully solved Sudoku board with no zeros -> Option C

4. Quick Check:

   Backtracking solves all cells [OK]

Hint: Backtracking fills all empty cells correctly [OK]

Common Mistakes:

• Expecting original board unchanged
• Thinking partial fill is final
• Assuming function is undefined

4. Consider this buggy code snippet for checking if a number can be placed in a Sudoku cell:

function isSafe(board, row, col, num) {
  for (let x = 0; x < 9; x++) {
    if (board[row][x] === num || board[x][col] === num) {
      return false;
    }
  }
  // Missing 3x3 box check
  return true;
}

What is the main problem here?

medium

A. It returns false too early

B. It does not check the 3x3 subgrid for duplicates

C. It uses wrong loop variable x

D. It modifies the board inside the function

Solution

1. Step 1: Analyze row and column checks

   The code correctly checks if num exists in the same row or column.

2. Step 2: Identify missing 3x3 subgrid check

   Sudoku rules require checking the 3x3 box; this code does not do that, causing errors.

3. Final Answer:

   It does not check the 3x3 subgrid for duplicates -> Option B

4. Quick Check:

   3x3 box check missing = bug [OK]

Hint: Always check row, column, and 3x3 box [OK]

Common Mistakes:

• Ignoring 3x3 box validation
• Thinking loop variable causes error
• Assuming early return is wrong

5. You want to optimize the Sudoku solver by choosing the next empty cell with the fewest possible valid numbers to try. Which technique does this describe?

hard

A. Heuristic with Minimum Remaining Value (MRV) strategy

B. Random guessing of empty cells

C. Depth-first search without pruning

D. Sorting the board rows before solving

Solution

1. Step 1: Understand MRV heuristic

   MRV chooses the empty cell with the fewest valid options to reduce branching and speed up solving.

2. Step 2: Compare with other options

   Random guessing and depth-first search without pruning are less efficient; sorting rows does not help solving.

3. Final Answer:

   Heuristic with Minimum Remaining Value (MRV) strategy -> Option A

4. Quick Check:

   MRV = choose cell with fewest options [OK]

Hint: Pick cell with fewest choices first [OK]

Common Mistakes:

• Confusing random guessing with heuristics
• Ignoring pruning benefits
• Thinking sorting helps solving