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Valid Sudoku

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Average time to solve is 40m

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Problem statement

You have been given a 9 X 9 2D matrix 'MATRIX' with some cells filled with digits(1 - 9), and some empty cells (denoted by 0).

You need to find whether there exists a way to fill all the empty cells with some digit(1 - 9) such that the final matrix is a valid Sudoku solution.

A Sudoku solution must satisfy all the following conditions-

1. Each of the digits 1 - 9 must occur exactly once in each row.
2. Each of the digits 1 - 9 must occur exactly once in each column.
3. Each of the digits 1 - 9 must occur exactly once in each of the 9, 3 x 3 sub-matrices of the matrix.

Note

1. There will always be a cell in the matrix which is empty.
2. The given initial matrix will always be consistent according to the rules mentioned in the problem statement.

Detailed explanation ( Input/output format, Notes, Images )

Input Format:

The first line contains a single integer 'T' denoting the number of test cases.
Then 'T' test cases follow.

Every test case contains 9 lines, with each line containing 9 single space-separated digits (0, if the cell is empty or a digit (1 - 9) otherwise).

Output Format:

For each test case, print a single line containing “yes”(without quotes), if there exists a Sudoku solution or “no” (without quotes) otherwise. Note the lowercase format of the output.

Note:

You do not need to print anything, it has already been taken care of. Just implement the given function.

Constraints:

1 <= 'T' <= 5
N = 9
0 <= MATRIX[i][j] <= 9

Where 'N' denotes the size of the given square matrix.

Time Limit: 1sec

Sample Input 1:

1
9 0 0 0 2 0 7 5 0 
6 0 0 0 5 0 0 4 0 
0 2 0 4 0 0 0 1 0 
2 0 8 0 0 0 0 0 0 
0 7 0 5 0 9 0 6 0 
0 0 0 0 0 0 4 0 1 
0 1 0 0 0 5 0 8 0 
0 9 0 0 7 0 0 0 4 
0 8 2 0 4 0 0 0 6

Sample Output 1:

yes

Explanation of the Sample Input1:

One of the possible solutions is:
9 4 1 3 2 6 7 5 8
6 3 7 1 5 8 2 4 9
8 2 5 4 9 7 6 1 3
2 6 8 7 1 4 3 9 5
1 7 4 5 3 9 8 6 2
3 5 9 6 8 2 4 7 1
4 1 3 2 6 5 9 8 7
5 9 6 8 7 3 1 2 4
7 8 2 9 4 1 5 3 6

Sample Input 2:

1
1 5 9 0 0 6 0 3 2
2 7 4 0 0 0 0 0 0
3 8 6 2 0 0 0 0 5
4 9 2 5 0 1 0 8 0
6 3 7 0 4 0 0 0 0
5 1 0 8 2 0 0 0 0
8 2 1 0 0 0 0 0 0
7 6 0 1 0 0 4 2 0
9 4 3 0 7 0 0 6 1

Sample Output 2:

no

Explanation of the Sample Input2:

In the third column from the left, there are two empty cells out of which one has to be filled with ‘8’, but we can’t put 8 in any of those two cells.

Hint 1

Hint 2

Use a brute force approach, try all configurations.

Approaches (2)

Approach 1

Approach 2

Brute Force

For all the empty cells of the matrix try all the arrangements of the digits (1 - 9) in those cells i.e filling all the empty cells with some digit.
For each arrangement, check whether the matrix now violates any of the given rules, if it does, then check for some other arrangement, otherwise, we found a solution.
If after checking all the arrangements/configurations, we can’t find a solution, then there doesn’t exist any solution to the given input.

Time Complexity

O(9 ^ K), where ‘K’ denotes the total number of empty cells in the given matrix.

The time complexity is exponential as for each cell we are trying all the digits from 1 to 9.

Space Complexity

O(K), where ‘K’ denotes the total number of empty cells in the given matrix.

We are checking the original matrix itself and the recursion stack can grow in the order of ‘K’ which can take max value as 9 * 9 = 81, the space complexity is approximately constant.

(100% EXP penalty)

Valid Sudoku

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