# Solving Sudoku using Knuth's Algorithm X

For my Algorithms 1 course in Fall 2013, the final project was a choice between building a web search engine and an NxN Sudoku solver. Our prof told us that for the search engine he would give us a bunch of guidelines and classes to start us off, but wouldn’t give us anything for the Sudoku problem – it was billed as the more ‘open ended’ and ‘challenging’ of the two. Being a nerd, I decided that I had to do the Sudoku Solver, and started reading online about Sudoku solving. I read that no less than Dr Donald Knuth had thought and written about this, and he had described an algorithm to solve a Sudoku puzzle – Algorithm X+Dancing Links. I decided I would take a shot at using his technique to write a Sudoku Solver in Java. And thus an assigment that should have taken a week to do ended up swallowing the better part of a month as I struggled with understanding Knuth’s paper and then implementing a Solver in Java. I believe my Algorithm X implementation is one of the few ones in Java (at least, I couldn’t find much online).

Anyway, here goes.

### Algorithm X, Exact Cover Problem And Dancing Links Implementation

My class AlgorithmXSolver takes an unsolved Sudoku puzzle as an int[][] (the Grid) and outputs the solved Sudoku puzzle. I convert the Sudoku puzzle into an exact cover problem, solve that using the Dancing Links algorithm as described by Dr Donald Knuth, and then get the solution and map it onto the Grid.

### Exact Cover And Dancing Links

An exact cover problem can be represented as a sparse matrix where the rows represent possibilities, and the columns represent constraints. Every row will have a 1 in every column (constraint) that it satisfies, and a 0 otherwise. A set of rows that together have exactly one 1 for each column can be said to be the solution set of the exact cover problem.

Now, Dancing Links is an efficient way of solving such a problem. The idea is to take the exact cover matrix and put it into a toroidal circular doubly-linked list. Thus, every node in such a list will be connected to 4 other nodes and the list will be circular i.e. the last element will point to the first one. In the case of Dancing Links, for every column of the linked list, there is a special ColumnNode (which extends the normal Node) that contains identifying information about that particular column as well as the size of the column i.e. the number of nodes in it. Each Node points to four other nodes as mentioned, as well as its ColumnNode.

## Solving

To solve the Exact Cover problem i.e. come up with a set of rows that contain exactly one 1 for every column/constraint, we search recursively using backtracking. It chooses a column, ‘covers’ it i.e. removes that column from the linked list completely, stores it in a solution list, and then try to recursively solve the rest of the table. If it’s not possible, backtrack, restore the column (uncover it), and try a different column. For this project I assumed that the Sudoku problem being provided has a solution.

### Sudoku Application

For Sudoku, there are 4 constraints.

1) Only 1 instance of a number can be in a row

2) Only 1 instance of a number can be in a column

3) Only 1 instance of a number can be in a block

4) There can be only one number in a cell

The rows represent every single possible position for every number. Every row would have 4 1s, representing one possible place for the number (satisfying all 4 constraints). To implement my solution, I created a class AlgorithmXSolver that contained all the methods and the data structures required to solve the problem. I instantiated an instance of this class in the solve() method, and then ran it. I had to convert the given Grid into a sparse matrix, accounting for the given clues (filled in values). Then, this matrix is converted into a linked list as talked about above and solved using the Dancing Links approach. We store possible solutions in an ArrayList ‘solution’. Once we get a set of Nodes that solves the problem, we take the solution list and iterate over every single Node and map the solution over the original Grid.

### Code

https://github.com/ShivanKaul/Sudoku-DLX

**Caveat Emptor:** written when I was in my first year of a CS degree.
**Bonus:** written when I was in my first year of a CS degree, so features endearingly exhaustive comments.

### Testing

~~I tested my solver using the puzzles provided by Prof Blanchette http://www.cs.mcgill.ca/~blanchem/250/hw5/sudoku.html~~ (Link doesn’t work)

### References

(1) Dr Donald Knuth’s original paper http://www.ocf.berkeley.edu/~jchu/publicportal/sudoku/0011047.pdf on Dancing Links.
(2) Jonathan Chu’s paper for the pseudocode for the Dancing Links implementation http://www.ocf.berkeley.edu/~jchu/publicportal/sudoku/sudoku.paper.html

(3) The Wikipedia pages on Dancing Links, Exact Cover problem, Algorithm X for helping to understand Knuth’s paper

(4) This StackOverflow discussion to intuitively understand the Dancing Links implementation: http://stackoverflow.com/questions/1518335/the-dancing-links-algorithm-an-explanation-that-is-less-explanatory-but-more-o

(5) Xi Chen’s implementation in C to get an understanding of the data structures http://uaa.wtf.im/?page_id=27
(6) Alex Rudnick’s implementation in Python for getting ideas on how to implement some of the methods https://code.google.com/p/narorumo/wiki/SudokuDLX