Problem Statement

We have to implement an isValidSudoku function that takes a 9x9 matrix (representing a Sudoku grid) as input and returns true if the grid is valid and false otherwise.

Sudoku Grid

Sudoku Grid

A Sudoku grid is valid if

  • Every row contains digits from 1 to 9 without repetition.

Rows in a Sudoku Grid

Rows in a Sudoku Grid

  • Every column contains digits from 1 to 9 without repetition.

Columns in a Sudoku Grid

Columns in a Sudoku Grid

  • Every 3x3 sub-matrix contains digits from 1 to 9 without repetition.

Sub Matrices in a Sudoku Grid

Sub-Matrices in a Sudoku Grid

The input Sudoku table could be incomplete where “.” will represent an empty cell.

Optimal Solution

The time complexity of validating a single row or column for repetition using a hashmap (like in containsDuplicate) will be $O(9)$. Repeating this process for every row and column in the input matrix will result in total time complexity of $9 \times O(9) + 9 \times O(9)$.

To validate all 3x3 sub-matrices we have to iterate over every 3rd row and column of the Sudoku grid. On each iteration, we will start from the top-left corner of the 3x3 grid and reach the bottom-right corner using nested loops. The total time complexity of validating each 3x3 submatrix will be $9 \times O(3) \times O(3)$.

Since the total time complexity of this solution will be constant for all inputs (because the Sudoku grid has a constant size of 9x9) this is the optimal solution.

Psuedo code for the Optimal Solution

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loop row_index from 0 to 9
  row_hashmap = Hashmap()
  sudoku_row = sudoku_grid[row_index]
  loop column_index from 0 to 9
    if row_hashmap[sudoku_row[column_index]]
      return false
    else
      row_hashmap[sudoku_row[column_index]] = 1

loop column_index from 0 to 9
  column_hashmap = Hashmap()
  loop row_index from 0 to 9
    value = sudoku_grid[row_index][column_index]
    if column_hashmap[value]
      return false
    else 
      column_hashmap[value] = 1

loop row_index from 0 to 6 with step 3
  loop column_index from 0 to 6 with step 3
    sub_matrix_hashmap = Hashmap()
    loop sm_row_index from 0 to 2
      loop sm_column_index from 0 to 2
        r_value = row_index+sm_row_index
        c_value = column_index+sm_column_index
        if sub_matrix_hashmap[r_value][c_value]
          return false
        else
          sub_matrix_hashmap[r_value][c_value] = 1

Time Complexity Analysis

Best Case Scenario

The best-case input for the optimal solution will be a completed Sudoku grid. The time taken to return the result will be $9 \times O(9) + 9 \times O(9) + O(81)$ which could be simplified to $O(1)$.

Worst Case Scenario

In the worst-case scenario, the input will be an invalid Sudoku grid. The time taken to produce the result will be the same as the best-case scenario i.e. O(1).

Space Complexity Analysis

The space complexity for the optimal solution will be the sum of memory space required by the row_hashmap, column_hashmap, and sub_matrix_hashmap i.e. $O(9)+O(9)+O(9)$ which could be simplified to $O(1)$.

Code for the Optimal Solution

First, we will implement the validCellValue function to check if the value in a cell is within 1 to 9.

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func validCellValue(inputValue byte)(bool){
    // Converting byte value to integer
    // to validate if inputValue lies between 1-len(board)
    
    intInputValue, _ := strconv.Atoi(string(inputValue))
    if ((intInputValue<10) && (intInputValue>0)){
        return true
    }
    return false
}

To validate each row we can create a function called checkRowValidity.

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func checkRowValidity(board [][]byte)(bool){
    // Iterating over each row
    // The length of the board could be hard coded to 9
    for rowIndex:=0;rowIndex<len(board);rowIndex++{
        
        // The rowHashMap would be used to check for
        // repeating values in the row
        rowHashMap:=make(map[byte]int)
        
        // Iterating over each column in the row
        for colIndex:=0;colIndex<len(board);colIndex++{
            cellValue := board[rowIndex][colIndex]
            
            // If the value of the cell isn't blank i.e. "."
            if(string(cellValue) != "."){
                
                // If the cell value is within 1-9
                if(validCellValue(cellValue)){
                    
                    // Check for value in rowHashMap
                    _, key_exists := rowHashMap[cellValue]
                    if key_exists{
                        
                        // Exit the function if we
                        // encountered a repeating value
                        // in the current row
                        return false
                    } else {
                        
                        // If value isn't present
                        // then add it the the hashmap
                        rowHashMap[cellValue] = 1
                    }
                } else {
                    
                    // Exit the function if we
                    // encountered a value that's not
                    // within range 1-9
                    return false
                }
            }
        }
    }
    
    return true
}

We can implement a similar function checkColumnValidity to validate columns.

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func checkColValidity(board [][]byte)(bool){
    // Iterating over each column
    for colIndex:=0;colIndex<len(board);colIndex++{

        // The colHashMap would be used to check for
        // repeating values in the column
        colHashMap:=make(map[byte]int)

        // Iterating over each row in the column
        for rowIndex:=0;rowIndex<len(board);rowIndex++{
            cellValue := board[rowIndex][colIndex]

            // If the value of the cell isn't blank i.e. "."
            if(string(cellValue)!="."){

                // If the cell value is within 1-9
                if(validCellValue(cellValue)){

                    // Check for value in colHashMap
                    _, key_exists := colHashMap[cellValue]
                    if key_exists{

                        // Exit the function if we
                        // encountered a repeating value
                        // in the current column
                        return false
                    } else {

                        // If value isn't present
                        // then add it the the hashmap
                        colHashMap[cellValue] = 1
                    }
                } else {

                    // Exit the function if we
                    // encountered a value that's not
                    // within range 1-9
                    return false
                }
            }
        }
    }
    
    return true
}

For validating every 3x3 sub-matrix we can implement checkSubMatrixValidity.

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func checkSubMatrixValidity(board [][]byte)(bool){
    // Iterating over every 3rd row
    for rowIndex:=0;rowIndex<len(board);rowIndex+=3{
        
        // Iterating over every 3rd column
        for colIndex:=0;colIndex<len(board);colIndex+=3{
            
            // The subMatrixHashMap would be used to check for
            // repeating values in the 3x3 sub-matrix
            subMatrixHashMap := make(map[byte]int)
            
            // Iterate over every row in the sub-matrix
            for i:=0;i<3;i++{

                // Iterate over every column in the sub-matrix
                for j:=0;j<3;j++{
                    cellValue := board[rowIndex+i][colIndex+j]
                    
                    // If the value of the cell isn't blank i.e. "."
                    if(string(cellValue)!="."){

                        // If the cell value is within 1-9
                        if(validCellValue(cellValue)){
                            
                            // Check for value in subMatrixHashMap
                            _, key_exists := subMatrixHashMap[cellValue]
                            if key_exists{

                                // Exit the function if we
                                // encountered a repeating value
                                // in the current sub-matrix
                                return false
                            } else {

                                // If value isn't present
                                // then add it the the hashmap
                                subMatrixHashMap[cellValue] = 1 
                            }
                        } else {

                            // Exit the function if we
                            // encountered a value that's not
                            // within range 1-9
                            return false
                        }
                    }
                }
            }
        }
    }
    
    return true
}

The isvalidSudoku function will validate the board by calling checkRowValidity, checkColumnValidity, and checkSubMatrixValidity and if all three functions return true then the input Sudoku board is valid.

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func isValidSudoku(board [][]byte)(bool){
    if(checkRowValidity(board) && 
        checkColValidity(board) && 
        checkSubMatrixValidity(board)){
        return true
    }
    return false
}

Complete Code for the Optimal Solution

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package main

import (
    "fmt"
    "strconv"
)

func validCellValue(inputValue byte)(bool){
    // Converting byte value to integer
    // to validate if inputValue lies between 1-len(board)
    
    intInputValue, _ := strconv.Atoi(string(inputValue))
    if ((intInputValue<10) && (intInputValue>0)){
        return true
    }
    return false
}

func checkRowValidity(board [][]byte)(bool){
    // Iterating over each row
    // The length of board the could be hard coded to 9
    for rowIndex:=0;rowIndex<len(board);rowIndex++{
        
        // The rowHashMap would be used to check for
        // repeating values in the row
        rowHashMap:=make(map[byte]int)
        
        // Iterating over each column in the row
        for colIndex:=0;colIndex<len(board);colIndex++{
            cellValue := board[rowIndex][colIndex]
            
            // If the value of the cell isn't blank i.e. "."
            if(string(cellValue) != "."){
                
                // If the cell value is within 1-9
                if(validCellValue(cellValue)){
                    
                    // Check for value in rowHashMap
                    _, key_exists := rowHashMap[cellValue]
                    if key_exists{
                        
                        // Exit the function if we
                        // encountered a repeating value
                        // in the current row
                        return false
                    } else {
                        
                        // If value isn't present
                        // then add it the the hashmap
                        rowHashMap[cellValue] = 1
                    }
                } else {
                    
                    // Exit the function if we
                    // encountered a value that's not
                    // within range 1-9
                    return false
                }
            }
        }
    }
    
    return true
}

func checkColValidity(board [][]byte)(bool){
    // Iterating over each column
    for colIndex:=0;colIndex<len(board);colIndex++{

        // The colHashMap would be used to check for
        // repeating values in the column
        colHashMap:=make(map[byte]int)

        // Iterating over each row in the column
        for rowIndex:=0;rowIndex<len(board);rowIndex++{
            cellValue := board[rowIndex][colIndex]

            // If the value of the cell isn't blank i.e. "."
            if(string(cellValue)!="."){

                // If the cell value is within 1-9
                if(validCellValue(cellValue)){

                    // Check for value in colHashMap
                    _, key_exists := colHashMap[cellValue]
                    if key_exists{

                        // Exit the function if we
                        // encountered a repeating value
                        // in the current column
                        return false
                    } else {

                        // If value isn't present
                        // then add it the the hashmap
                        colHashMap[cellValue] = 1
                    }
                } else {

                    // Exit the function if we
                    // encountered a value that's not
                    // within range 1-9
                    return false
                }
            }
        }
    }
    
    return true
}

func checkSubMatrixValidity(board [][]byte)(bool){
    // Iterating over every 3rd row
    for rowIndex:=0;rowIndex<len(board);rowIndex+=3{
        
        // Iterating over every 3rd column
        for colIndex:=0;colIndex<len(board);colIndex+=3{
            
            // The subMatrixHashMap would be used to check for
            // repeating values in the 3x3 sub-matrix
            subMatrixHashMap := make(map[byte]int)
            
            // Iterate over every row in the sub-matrix
            for i:=0;i<3;i++{

                // Iterate over every column in the sub-matrix
                for j:=0;j<3;j++{
                    cellValue := board[rowIndex+i][colIndex+j]
                    
                    // If the value of the cell isn't blank i.e. "."
                    if(string(cellValue)!="."){

                        // If the cell value is within 1-9
                        if(validCellValue(cellValue)){
                            
                            // Check for value in subMatrixHashMap
                            _, key_exists := subMatrixHashMap[cellValue]
                            if key_exists{

                                // Exit the function if we
                                // encountered a repeating value
                                // in the current sub-matrix
                                return false
                            } else {

                                // If value isn't present
                                // then add it the the hashmap
                                subMatrixHashMap[cellValue] = 1 
                            }
                        } else {

                            // Exit the function if we
                            // encountered a value that's not
                            // within range 1-9
                            return false
                        }
                    }
                }
            }
        }
    }
    
    return true
}

func isValidSudoku(board [][]byte)(bool){
    if(checkRowValidity(board) && 
        checkColValidity(board) && 
        checkSubMatrixValidity(board)){
        return true
    }
    return false
}

func main(){
    // A valid sudoku grid
    // inputBoard := [][]string{
    //     {"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"},
    // }
    inputBoard := [][]byte{
        {53,51,46,46,55,46,46,46,46}, 
        {54,46,46,49,57,53,46,46,46}, 
        {46,57,56,46,46,46,46,54,46},
        {56,46,46,46,54,46,46,46,51}, 
        {52,46,46,56,46,51,46,46,49}, 
        {55,46,46,46,50,46,46,46,54},
        {46,54,46,46,46,46,50,56,46},
        {46,46,46,52,49,57,46,46,53},
        {46,46,46,46,56,46,46,55,57}}
    fmt.Println("Validitity of sudoku grid:", isValidSudoku(inputBoard))

    // An invalid sudoku grid (Repeating 8s in first column)
    // inputBoard := [][]string{
    //     {"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"},
    // }
    inputBoard = [][]byte{
        {56,51,46,46,55,46,46,46,46}, 
        {54,46,46,49,57,53,46,46,46}, 
        {46,57,56,46,46,46,46,54,46},
        {56,46,46,46,54,46,46,46,51}, 
        {52,46,46,56,46,51,46,46,49}, 
        {55,46,46,46,50,46,46,46,54},
        {46,54,46,46,46,46,50,56,46},
        {46,46,46,52,49,57,46,46,53},
        {46,46,46,46,56,46,46,55,57}}
    fmt.Println("Validitity of sudoku grid:", isValidSudoku(inputBoard))

    // An invalid sudoku grid (Repeating 8s in the first 3x3 submatrix)
    // inputBoard := [][]string{
    //     {"5", "3", ".", ".", "7", ".", ".", ".", "."},
    //     {"6", "8", ".", "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"},
    // }
    inputBoard = [][]byte{
        {53,51,46,46,55,46,46,46,46}, 
        {54,56,46,49,57,53,46,46,46}, 
        {46,57,56,46,46,46,46,54,46},
        {56,46,46,46,54,46,46,46,51}, 
        {52,46,46,56,46,51,46,46,49}, 
        {55,46,46,46,50,46,46,46,54},
        {46,54,46,46,46,46,50,56,46},
        {46,46,46,52,49,57,46,46,53},
        {46,46,46,46,56,46,46,55,57}}
    fmt.Println("Validitity of sudoku grid:", isValidSudoku(inputBoard))
}

// Output
// Validitity of sudoku grid: true
// Validitity of sudoku grid: false
// Validitity of sudoku grid: false

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Resources

How to Play Sudoku?
36. Valid Sudoku
Valid Sudoku - Amazon Interview Question - Leetcode 36 - Python