Sudoku Solver
Problem Description
Write a program to solve a Sudoku puzzle by filling the empty cells.
A sudoku solution must satisfy all of the following rules:
- Each of the digits 1-9 must occur exactly once in each row.
- Each of the digits 1-9 must occur exactly once in each column.
- Each of the digits 1-9 must occur exactly once in each of the 9 sub-boxes of the grid.
The '.' character indicates empty cells.
Examples
Example 1:
Input: board =
[["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"]]
Output:
[["5","3","4","6","7","8","9","1","2"],["6","7","2","1","9","5","3","4","8"],["1","9","8","3","4","2","5","6","7"],["8","5","9","7","6","1","4","2","3"],["4","2","6","8","5","3","7","9","1"],["7","1","3","9","2","4","8","5","6"],["9","6","1","5","3","7","2","8","4"],["2","8","7","4","1","9","6","3","5"],["3","4","5","2","8","6","1","7","9"]]
Explanation: The input board is shown above and the only valid solution is shown below:
Constraints
- board.length == 9
- board[i].length == 9
- board[i][j] is a digit 1-9 or '.'.
- It is guaranteed that the input board has only one solution.
Approach
👉 Iterate Through Each Cell: The algorithm iterates through each cell of the Sudoku board
👉 Empty Cell Check: If the current cell is empty (contains ‘.’), the algorithm proceeds to try different numbers (from ‘1’ to ‘9’) in that cell
👉 Number Placement: For each number, the algorithm checks if placing that number in the current cell is valid according to the Sudoku rules (no repetition in the same row, column, or sub-box).
👉 Recursive Exploration: If placing a number is valid, the algorithm sets that number in the cell and recursively explores the next empty cell.
👉 Backtracking: If the recursive exploration leads to an invalid solution, the algorithm backtracks by undoing the choice (resetting the cell to ‘.’) and trying the next number.
👉 Solution Check: The algorithm continues exploring possibilities until a valid solution is found, or it exhaustively searches all possibilities.
Solution Code
Python
class Solution:
def solveSudoku(self, board: List[List[str]]) -> None:
def isValid(row: int, col: int, c: str) -> bool:
for i in range(9):
if board[i][col] == c or \
board[row][i] == c or \
board[3 * (row // 3) + i // 3][3 * (col // 3) + i % 3] == c:
return False
return True
def solve(s: int) -> bool:
if s == 81:
return True
i = s // 9
j = s % 9
if board[i][j] != '.':
return solve(s + 1)
for c in string.digits[1:]:
if isValid(i, j, c):
board[i][j] = c
if solve(s + 1):
return True
board[i][j] = '.'
return False
solve(0)
Java
class Solution {
public void solveSudoku(char[][] board) {
int[][] f=new int[9][10];
int[][] row=new int[9][10];
int[][] col=new int[9][10];
int cell=0;
for(int i=0;i<9;i++){
for(int j=0;j<9;j++){
cell=((i/3)*3)+j/3;
if(board[i][j]!='.'){
f[cell][(int)(board[i][j]-'0')]=1;
row[i][(int)(board[i][j]-'0')]=1;
col[j][(int)(board[i][j]-'0')]=1;
}}
}
get(f,board,0,0,row,col);
return;
}
static boolean get(int[][] f,char[][] board, int i, int j, int[][] row, int[][] col ){
if(i==9)return true;
if(board[i][j]!='.'){
if(j==8){
return get(f,board,i+1,0,row,col);
}else{
return get(f,board,i,j+1,row,col);
}
}else{
boolean t=false;
int cell=((i/3)*3) + j/3;
for(int k=1;k<=9;k++){
if(f[cell][k]==0 && row[i][k]==0 && col[j][k]==0){
f[cell][k]=1;
row[i][k]=1;
col[j][k]=1;
board[i][j]=(char) (k + '0');
if(j==8){
t=get(f,board,i+1,0,row,col);
}else{
t=get(f,board,i,j+1,row,col);
}
if(t)return true;
f[cell][k]=0;
row[i][k]=0;
col[j][k]=0;
board[i][j]='.';
}
}
}
return false;
}
}
C++
class Solution {
public:
void solveSudoku(vector<vector<char>>& board) {
solve(board);
}
bool solve(vector<vector<char>>& board){
for(int i=0;i<board.size();i++)
{
for(int j=0;j<board[0].size();j++)
{
if(board[i][j]=='.')
{
for(char c='1';c<='9';c++)
{
if(isValid(board,i,j,c))
{
board[i][j]=c;
if(solve(board)==true)
return true;
else
board[i][j]='.';
}
}
return false;
}
}
}
return true;
}
bool isValid(vector<vector<char>>& board,int row,int col, char c)
{
for(int i=0;i<9;i++)
{
if((board[row][i]==c) || (board[i][col]==c) || (board[3*(row/3) + i/3][3*(col/3) + i%3] ==c))
return false;
}
return true;
}
};
Conclusion
✅ Time Complexity:
✔ The solve method uses a backtracking approach to explore the solution space.
✔ In the worst case, the algorithm tries out all possibilities for each empty cell until a valid solution is found or all possibilities are exhausted.
✔ The time complexity of the backtracking algorithm is typically exponential, but in practice, it tends to be much less than the worst case.
✔ The algorithm explores 9 possibilities for each of the 81 cells, resulting in a time complexity of , where m represents the number of empty cells.
✅ Space Complexity:
✔ The algorithm uses a recursive approach to explore the solution space.
✔ The recursive stack depth is proportional to the number of empty cells in the Sudoku board.
✔ The space complexity of the backtracking algorithm is typically linear, but in practice, it tends to be much less.
✔ The algorithm explores 9 possibilities for each of the 81 cells, resulting in a space complexity of , where m represents the number of empty cells.
✅ The algorithm solves the Sudoku puzzle by backtracking through the solution space, exploring possibilities until a valid solution is found or all possibilities are exhausted.
✅ The algorithm uses a recursive approach to explore the solution space, setting numbers in empty cells and checking if the placement is valid according to the Sudoku rules.
✅ The algorithm backtracks when an invalid solution is found, undoing the choice and trying the next number.