Implement A Sudoku Solver - Sudoku Solving Backtracking Algorithm ("Sudoku Solver" on LeetCode)

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Question: Implement a Sudoku solver. (a classic Sudoku board is 9x9)

The Rules of Sudoku:

We are working with numbers 1 thorugh 9.

Each number can only appear once in a:

Row
Column
And one of the 3 x 3 subboxes
Approach 1 (Brute Force)

Try every possible assignment to empty entries and then at the end validate if the sudoku board is valid or not.

With no constraints, given a 9x9 board we have at max 81 spaces where in each space we can place the number 1-9 (but on a normal board many spaces will already have a number before we even start solving)

This means that there are 9^81 total arrangements of the board (as a loose upper bound). This is 1.9662705 * 10 ^ 77 boards just for a 9 x 9 board.

This would take too long.
Approach 2 (Apply The Constraints)

The backtracking approach.

We can traverse and place an entry in empty cells one by one.

We then check the whole board to see if it is still valid.

If it is we continue our recursion.

If it is not we do not even follow that path.

When all entries have been filled, we are finished.
The 3 Keys To Backtracking:

Our Choice

What we place in an empty cell that we see.

Our Constraints

Standard Sudoku constraints. We cannot make a placement that will break the board.

Our Goal

Place all empty spaces on the board. We will know we have done this when we have placed a valid value in the last cell in our search which programmatically will be the bottom right cell in the 2D matrix.
Validating Placements Before We Place Them

INSIGHT: We could traverse every row, column, and 3x3 subgrid to perform validation but at every decision point we know that we are adding onto a grid that is already valid.

So we just check the row, column, and subgrid of the item that has been added.

Whenever backtracking we know that every decision that we made stayed within the constraint that we started with being that the board was valid.
Complexities

Since Sudoku is traditionally 9x9 we can't really discuss complexities because nothing can scale.

Solving Sudoku generalized to n x n boards is a NP-Complete problem so we know for sure that our time complexity is exponential at the very least.
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HackerRank: https://www.youtube.com/channel/UCOf7UPMHBjAavgD0Qw5q5ww

Tuschar Roy: https://www.youtube.com/user/tusharroy2525

GeeksForGeeks: https://www.youtube.com/channel/UC0RhatS1pyxInC00YKjjBqQ

Jarvis Johnson: https://www.youtube.com/user/VSympathyV

Success In Tech: https://www.youtube.com/channel/UC-vYrOAmtrx9sBzJAf3x_xw

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This question is number 16.9 in "Elements of Programming Interviews" by Adnan Aziz, Tsung-Hsien Lee, and Amit Prakash.

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