Abstract
We describe a way to transfer efficiently Sudoku grids through the Internet. This is done by using linearization together with compression and decompression that use the information structure present in all sudoku grids. The compression and the corresponding decompression are based on the fact that in each Sudoku grid there are information dependencies and so some of the information is redundant.
Highlights
In this study, we use the established terminology (Delahaye, 2006) - a Sudoku grid is a square 9 9 table with 81 cells
There is a single digit from 1 to 9 and each Sudoku grid fulfills three types of constraints: (1) Each row has each of the digits from 1 to 9 exactly once; (2) each column has each of the digits from 1 to 9 exactly once; and (3) each of the small 3 3 squares has each of the digits from 1 to 9 exactly once
The figure on the left is an example of a Sudoku grid and on the right is a sudoku grid with all the small 3 3 squares denoted by Bi,j with 1 i, j 3
Summary
We use the established terminology (Delahaye, 2006) - a Sudoku grid is a square 9 9 table with 81 cells. To uniquely identify any Sudoku grid we need only 73 bits This encoding is optimal (we cannot represent one of n possible values with less than log n bits), both the sending and the receiving device must contain a database of all possible Sudoku grids which put heavy memory and computational burden on the two devices. At the sending device, given a grid, we compute one puzzle (this is the compression algorithm), we send an encoding of the puzzle through the Internet. In (McGuire et al, 2014), it was proved that any solvable puzzle needs at least 17 clues This approach raises two questions: (1) Does a 17-clue puzzle exist for any sudoku grid and (2) given a sudoku grid, how to efficiently compute at least one of its puzzles with minimal number of clues, they are not considered in our paper. Since we use a fixed pattern for the empty cells, the linearization contains only content of the clue cells
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