Sudoku Solver Using Backtracking in DSA C++ - Time & Space Complexity

Time Complexity: Sudoku Solver Using Backtracking

O(9^m)

Understanding Time Complexity

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

Specifically, how does the backtracking method scale when trying to fill the board?

Scenario Under Consideration

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

bool solveSudoku(int board[9][9]) {
  for (int row = 0; row < 9; row++) {
    for (int col = 0; col < 9; col++) {
      if (board[row][col] == 0) {
        for (int num = 1; num <= 9; num++) {
          if (isSafe(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 and uses recursion to fill the board step-by-step.

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 attempts per empty cell, and recursion explores many board states.

How Execution Grows With Input

The number of empty cells affects how many guesses and recursive calls happen.

Input Size (empty cells)	Approx. Operations
10	Up to 9^10 = 3,486,784,401 tries
20	Up to 9^20 = 1.21e+19 tries (very large)
40	Up to 9^40 = 1.21e+38 tries (astronomical)

Pattern observation: The operations grow exponentially with the number of empty cells, making the problem very costly as empties increase.

Final Time Complexity

Time Complexity: O(9^m) where m is the number of empty cells

This means the time grows very fast as more empty cells need to be filled, because each cell can try up to 9 numbers recursively.

Common Mistake

[X] Wrong: "The solver always takes the same time regardless of the puzzle."

[OK] Correct: The time depends heavily on how many empty cells and how constrained the puzzle is; some puzzles solve quickly, others take much longer.

Interview Connect

Understanding this exponential growth helps you explain why backtracking can be slow and why pruning or heuristics matter in real problems.

Self-Check

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

Practice

1. What is the main idea behind the backtracking technique used in a Sudoku solver?

easy

A. Use a fixed pattern to fill the board

B. Fill the board randomly without checking rules

C. Try placing numbers and undo if they lead to conflicts

D. Only fill cells with a single possible number

Solution

1. Step 1: Understand backtracking in Sudoku

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

2. Step 2: Undo if conflict arises

   If a number causes a conflict later, it is removed (backtracked) and another number is tried.

3. Final Answer:

   Try placing numbers and undo if they lead to conflicts -> Option C

4. Quick Check:

   Backtracking = Try and undo [OK]

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

Common Mistakes:

• Thinking backtracking fills randomly
• Assuming backtracking fills only once without undo
• Believing backtracking uses fixed patterns

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

easy

A. if (isSafe(board, row, col, num)) { board[row][col] = num; }

B. if (board[row][col] == num) { placeNumber(); }

C. if (num > 9) { board[row][col] = num; }

D. if (board[row][col] != 0) { board[row][col] = num; }

Solution

1. Step 1: Understand the safety check function

   The function isSafe checks if placing num at board[row][col] follows Sudoku rules.

2. Step 2: Place number only if safe

   Only if isSafe returns true, the number is placed in the cell.

3. Final Answer:

   if (isSafe(board, row, col, num)) { board[row][col] = num; } -> Option A

4. Quick Check:

   Use isSafe before placing number [OK]

Hint: Always check safety before placing number [OK]

Common Mistakes:

• Placing number without checking rules
• Checking if cell already has the number incorrectly
• Using invalid conditions for placement

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

int board[9][9] = {
  {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);

Assuming solveSudoku correctly solves the puzzle, what will be the value at board[0][2] after solving?

medium

A. 9

B. 4

C. 7

D. 1

Solution

1. Step 1: Identify the empty cell at (0,2)

   The cell at row 0, column 2 is initially 0 (empty).

2. Step 2: Solve Sudoku and find correct number

   After solving, the number 4 fits at (0,2) without conflicts in row, column, or box.

3. Final Answer:

   4 -> Option B

4. Quick Check:

   board[0][2] = 4 after solve [OK]

Hint: Check solved board cell for correct number [OK]

Common Mistakes:

• Guessing without checking Sudoku rules
• Confusing row and column indices
• Assuming initial zeros remain zeros

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

bool isSafe(int board[9][9], int row, int col, int num) {
  for (int x = 0; x < 9; x++) {
    if (board[row][x] == num || board[x][col] == num)
      return false;
  }
  int startRow = row - row % 3;
  int startCol = col - col % 3;
  for (int i = 0; i < 3; i++) {
    for (int j = 0; j < 3; j++) {
      if (board[i][j] == num)
        return false;
    }
  }
  return true;
}

What is the bug in this code?

medium

A. The 3x3 box check uses wrong indices; should use startRow + i and startCol + j

B. The row and column checks are missing

C. The function does not return false when number is found

D. The modulo operation is incorrect

Solution

1. Step 1: Analyze 3x3 box checking loop

   The inner loop checks board[i][j] instead of board[startRow + i][startCol + j], so it checks wrong cells.

2. Step 2: Correct indices for box check

   To check the correct 3x3 box, add startRow and startCol offsets to i and j.

3. Final Answer:

   The 3x3 box check uses wrong indices; should use startRow + i and startCol + j -> Option A

4. Quick Check:

   Box check indices must include startRow/startCol [OK]

Hint: Add startRow/startCol offsets when checking 3x3 box [OK]

Common Mistakes:

• Checking wrong cells in 3x3 box
• Ignoring row or column checks
• Incorrect modulo usage

5. You want to optimize a Sudoku solver using backtracking by adding a heuristic. Which of the following strategies best reduces the search space?

hard

A. Fill cells row-wise ignoring constraints until stuck

B. Try numbers in random order for each empty cell

C. Always fill the first empty cell found without checking possibilities

D. Choose the empty cell with the fewest possible valid numbers first

Solution

1. Step 1: Understand heuristic to reduce search space

   Choosing the cell with the fewest valid options first reduces branching and speeds up solving.

2. Step 2: Compare other strategies

   Filling first empty cell or random tries increase backtracking steps; ignoring constraints causes errors.

3. Final Answer:

   Choose the empty cell with the fewest possible valid numbers first -> Option D

4. Quick Check:

   Minimum remaining values heuristic = choose cell with fewest options [OK]

Hint: Pick cell with least options first to speed solving [OK]

Common Mistakes:

• Filling cells without checking possibilities
• Trying numbers randomly wastes time
• Ignoring constraints leads to wrong solutions