Backtracking | Set 7 (Sudoku)

Given a partially filled 9×9 2D array ‘grid[9][9]’, the goal is to assign digits (from 1 to 9) to the empty cells so that every row, column, and subgrid of size 3×3 contains exactly one instance of the digits from 1 to 9.

Naive Algorithm
The Naive Algorithm is to generate all possible configurations of numbers from 1 to 9 to fill the empty cells. Try every configuration one by one until the correct configuration is found.

Backtracking Algorithm
Like all other Backtracking problems, we can solve Sudoku by one by one assigning numbers to empty cells. Before assigning a number, we check whether it is safe to assign. We basically check that the same number is not present in current row, current column and current 3X3 subgrid. After checking for safety, we assign the number, and recursively check whether this assignment leads to a solution or not. If the assignment doesn’t lead to a solution, then we try next number for current empty cell. And if none of number (1 to 9) lead to solution, we return false.

  Find row, col of an unassigned cell
  If there is none, return true
  For digits from 1 to 9
    a) If there is no conflict for digit at row,col
        assign digit to row,col and recursively try fill in rest of grid
    b) If recursion successful, return true
    c) Else, remove digit and try another
  If all digits have been tried and nothing worked, return false

Following are C++ and Python implementation for Sudoku problem. It prints the completely filled grid as output.

C/C++

// A Backtracking program  in C++ to solve Sudoku problem
#include <stdio.h>

// UNASSIGNED is used for empty cells in sudoku grid
#define UNASSIGNED 0

// N is used for size of Sudoku grid. Size will be NxN
#define N 9

// This function finds an entry in grid that is still unassigned
bool FindUnassignedLocation(int grid[N][N], int &row, int &col);

// Checks whether it will be legal to assign num to the given row,col
bool isSafe(int grid[N][N], int row, int col, int num);

/* Takes a partially filled-in grid and attempts to assign values to
  all unassigned locations in such a way to meet the requirements
  for Sudoku solution (non-duplication across rows, columns, and boxes) */
bool SolveSudoku(int grid[N][N])
{
    int row, col;

    // If there is no unassigned location, we are done
    if (!FindUnassignedLocation(grid, row, col))
       return true; // success!

    // consider digits 1 to 9
    for (int num = 1; num <= 9; num++)
    {
        // if looks promising
        if (isSafe(grid, row, col, num))
        {
            // make tentative assignment
            grid[row][col] = num;

            // return, if success, yay!
            if (SolveSudoku(grid))
                return true;

            // failure, unmake & try again
            grid[row][col] = UNASSIGNED;
        }
    }
    return false; // this triggers backtracking
}

/* Searches the grid to find an entry that is still unassigned. If
   found, the reference parameters row, col will be set the location
   that is unassigned, and true is returned. If no unassigned entries
   remain, false is returned. */
bool FindUnassignedLocation(int grid[N][N], int &row, int &col)
{
    for (row = 0; row < N; row++)
        for (col = 0; col < N; col++)
            if (grid[row][col] == UNASSIGNED)
                return true;
    return false;
}

/* Returns a boolean which indicates whether any assigned entry
   in the specified row matches the given number. */
bool UsedInRow(int grid[N][N], int row, int num)
{
    for (int col = 0; col < N; col++)
        if (grid[row][col] == num)
            return true;
    return false;
}

/* Returns a boolean which indicates whether any assigned entry
   in the specified column matches the given number. */
bool UsedInCol(int grid[N][N], int col, int num)
{
    for (int row = 0; row < N; row++)
        if (grid[row][col] == num)
            return true;
    return false;
}

/* Returns a boolean which indicates whether any assigned entry
   within the specified 3x3 box matches the given number. */
bool UsedInBox(int grid[N][N], int boxStartRow, int boxStartCol, int num)
{
    for (int row = 0; row < 3; row++)
        for (int col = 0; col < 3; col++)
            if (grid[row+boxStartRow][col+boxStartCol] == num)
                return true;
    return false;
}

/* Returns a boolean which indicates whether it will be legal to assign
   num to the given row,col location. */
bool isSafe(int grid[N][N], int row, int col, int num)
{
    /* Check if 'num' is not already placed in current row,
       current column and current 3x3 box */
    return !UsedInRow(grid, row, num) &&
           !UsedInCol(grid, col, num) &&
           !UsedInBox(grid, row - row%3 , col - col%3, num);
}

/* A utility function to print grid  */
void printGrid(int grid[N][N])
{
    for (int row = 0; row < N; row++)
    {
       for (int col = 0; col < N; col++)
             printf("%2d", grid[row][col]);
        printf("\n");
    }
}

/* Driver Program to test above functions */
int main()
{
    // 0 means unassigned cells
    int grid[N][N] = {{3, 0, 6, 5, 0, 8, 4, 0, 0},
                      {5, 2, 0, 0, 0, 0, 0, 0, 0},
                      {0, 8, 7, 0, 0, 0, 0, 3, 1},
                      {0, 0, 3, 0, 1, 0, 0, 8, 0},
                      {9, 0, 0, 8, 6, 3, 0, 0, 5},
                      {0, 5, 0, 0, 9, 0, 6, 0, 0},
                      {1, 3, 0, 0, 0, 0, 2, 5, 0},
                      {0, 0, 0, 0, 0, 0, 0, 7, 4},
                      {0, 0, 5, 2, 0, 6, 3, 0, 0}};
    if (SolveSudoku(grid) == true)
          printGrid(grid);
    else
         printf("No solution exists");

    return 0;
}

Python

# A Backtracking program  in Pyhton to solve Sudoku problem


# A Utility Function to print the Grid
def print_grid(arr):
    for i in range(9):
        for j in range(9):
            print arr[i][j],
        print ('\n')

        
# Function to Find the entry in the Grid that is still  not used
# Searches the grid to find an entry that is still unassigned. If
# found, the reference parameters row, col will be set the location
# that is unassigned, and true is returned. If no unassigned entries
# remain, false is returned.
# 'l' is a list  variable that has been passed from the solve_sudoku function
# to keep track of incrementation of Rows and Columns
def find_empty_location(arr,l):
    for row in range(9):
        for col in range(9):
            if(arr[row][col]==0):
                l[0]=row
                l[1]=col
                return True
    return False

# Returns a boolean which indicates whether any assigned entry
# in the specified row matches the given number.
def used_in_row(arr,row,num):
    for i in range(9):
        if(arr[row][i] == num):
            return True
    return False

# Returns a boolean which indicates whether any assigned entry
# in the specified column matches the given number.
def used_in_col(arr,col,num):
    for i in range(9):
        if(arr[i][col] == num):
            return True
    return False

# Returns a boolean which indicates whether any assigned entry
# within the specified 3x3 box matches the given number
def used_in_box(arr,row,col,num):
    for i in range(3):
        for j in range(3):
            if(arr[i+row][j+col] == num):
                return True
    return False

# Checks whether it will be legal to assign num to the given row,col
#  Returns a boolean which indicates whether it will be legal to assign
#  num to the given row,col location.
def check_location_is_safe(arr,row,col,num):
    
    # Check if 'num' is not already placed in current row,
    # current column and current 3x3 box
    return not used_in_row(arr,row,num) and not used_in_col(arr,col,num) and not used_in_box(arr,row - row%3,col - col%3,num)

# Takes a partially filled-in grid and attempts to assign values to
# all unassigned locations in such a way to meet the requirements
# for Sudoku solution (non-duplication across rows, columns, and boxes)
def solve_sudoku(arr):
    
    # 'l' is a list variable that keeps the record of row and col in find_empty_location Function    
    l=[0,0]
    
    # If there is no unassigned location, we are done    
    if(not find_empty_location(arr,l)):
        return True
    
    # Assigning list values to row and col that we got from the above Function 
    row=l[0]
    col=l[1]
    
    # consider digits 1 to 9
    for num in range(1,10):
        
        # if looks promising
        if(check_location_is_safe(arr,row,col,num)):
            
            # make tentative assignment
            arr[row][col]=num

            # return, if sucess, ya!
            if(solve_sudoku(arr)):
                return True

            # failure, unmake & try again
            arr[row][col] = 0
            
    # this triggers backtracking        
    return False 

# Driver main function to test above functions
if __name__=="__main__":
    
    # creating a 2D array for the grid
    grid=[[0 for x in range(9)]for y in range(9)]
    
    # assigning values to the grid
    grid=[[3,0,6,5,0,8,4,0,0],
          [5,2,0,0,0,0,0,0,0],
          [0,8,7,0,0,0,0,3,1],
          [0,0,3,0,1,0,0,8,0],
          [9,0,0,8,6,3,0,0,5],
          [0,5,0,0,9,0,6,0,0],
          [1,3,0,0,0,0,2,5,0],
          [0,0,0,0,0,0,0,7,4],
          [0,0,5,2,0,6,3,0,0]]
    
    # if sucess print the grid
    if(solve_sudoku(grid)):
        print_grid(grid)
    else:
        print "No solution exists"

# The above code has been contributed by Harshit Sidhwa.

Output:

  3 1 6 5 7 8 4 9 2
  5 2 9 1 3 4 7 6 8
  4 8 7 6 2 9 5 3 1
  2 6 3 4 1 5 9 8 7
  9 7 4 8 6 3 1 2 5
  8 5 1 7 9 2 6 4 3
  1 3 8 9 4 7 2 5 6
  6 9 2 3 5 1 8 7 4
  7 4 5 2 8 6 3 1 9

References:
http://see.stanford.edu/materials/icspacs106b/H19-RecBacktrackExamples.pdf

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