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Sudoku, also spelled Su Doku, is a logic-based placement puzzle. It consists in placing numbers from 1 to 9 in a 9-by-9 grid made up of nine 3-by-3 subgrids, called *regions* or *boxes*or *blocks*, starting with various numerals given in some cells, the *givens* or *clues*. It can be described with a single rule:

*Each row, column and region must contain all numbers from 1 to 9.*

We can immediately deduce that for each row, column, and region the values in the cells have to be different. Moreover, this condition is sufficient; thus, the unique rule could be reformulated as:

*Each row, column and region must contain numbers from 1 to 9 that are all different.*

Figure 1 shows a easy Sudoku puzzle containing 26 givens and Figure 2 shows its solution:

*Figure 1. An easy Sudoku puzzle*

*Figure 2. Solution to the Sudoku puzzle*

The Wikipedia entry on Sudoku has more information about the puzzle. Although first published in 1979, Sudoku initially became popular in Japan in 1986 and attained international popularity in 2005, both in newspapers (many newspapers publish a daily Sudoku puzzle) and on the Web. For example, see Koalog's Free Sudoku of the Dayto play Sudoku online.

While Sudoku lovers invent very complex techniques with weird names (*X-wing*, *Swordfish*, *Nishio*), to solve Sudokus without guessing, computer programs usually use brute-force methods. In this article, we will learn how to solve Sudokus using Constraint Programming in Java. This approach has many advantages:

- It can solve any Sudoku.
- It is fast.
- It is as powerful as the most complex human techniques: most of the Sudokus can be solved without
*trial and error*. - It is much more robust than the brute force approach: variants of the problem, with additional constraints, could be easily solved the same way.

## Constraint Programming

Constraint Programming (CP) is a technique of choice for solving combinatorial problems such as the Sudoku problem.

Firstly, it consists of modelling the problem using mathematical relations, or constraints. Precisely, the term *constraint*denotes the implementation of the mathematical relation. Hence, a Constraint Satisfaction Problem (CSP) is defined by:

- A set of variables.
- A set of domains (possible values) for the variables.
- a set of constraints/relations on the variables.

Secondly, a strategy for enumerating the different solutions has to be defined. CP differs from naive enumeration methods by the fact that constraints contain powerful filtering algorithms (usually inspired by Operational Research techniques) that will drastically reduce the search space by dynamically reducing the domains of the variables during the enumeration phase (also known as the "search phase").

We will use Koalog Constraint Solver, a Java library for Constraint Programming, to write the code. None of the techniques presented in this article depend on the choice of the solver, and could be implemented using any commercial or open source solver.

## Modelling the Sudoku Problem

Let us first create a problem object that will be the placeholder for the problem modelization. Using Koalog Constraint Solver, this can be done by subclassing a`BaseProblem`

:

public class SudokuProblem extends BaseProblem { /** * Sole constructor. * @param givens the givens * (from 0 to 9, * 0 being used for unknown values) */ public SudokuProblem(int[][] givens) { super(); // the forthcoming model } }

An important step in Constraint Programming is defining the problem variables. In our case, we want to find the value for each cell. Note that other models are possible: for example, we could define, for each value, its position in each row or column or block. It is also possible to mix several models. In our case, the following simple model will be sufficient.

IntegerVariable[][] num = new IntegerVariable[9][9];

where the local variable `num`

will be indexed by rows, from 0 to 8, and then by columns, from 0 to 8.

for (int i=0; i<9; i++) { for (int j=0; j<9; j++) { if (givens[i][j] != 0) { num[i][j] = new IntegerVariable("num"+i+"_"+j, new SparseDomain (givens[i][j])); } else { num[i][j] = new IntegerVariable("num"+i+"_"+j, new SparseDomain (1, 9)); } } }

We have defined the initial domain of each variable, taking into account the givens. `SparseDomain`

means that the domain of each variable will be represented by a set of possible values. Note that it is also possible to represent a domain by an interval of the form `[min, max]`

using the`MinMaxDomain`

class.

In each of the nine blocks, we want the values to be different. This means that we should enforce 36 (the number of distinct pairs that can be selected from nine values: 9*8/2) inequations between the nine `num`

variables of each block. Instead, we simply use a *global* constraint called`AllDifferent_SPARSE`

; it is suffixed by`_SPARSE`

(a convention in Koalog Constraint Solver) to take into account the *sparse* nature of the domains. These constraints are added to the problem using its `add`

method:

for (int i=0; i<3; i++) { for (int j=0; j<3; j++) { add(new AllDifferent_SPARSE (new IntegerVariable[]{ num[3*i][3*j], num[3*i][1+3*j], num[3*i][2+3*j], num[1+3*i][3*j], num[1+3*i][1+3*j], num[1+3*i][2+3*j], num[2+3*i][3*j], num[2+3*i][1+3*j], num[2+3*i][2+3*j]})); } }

We also want the values to be different in each row and in each column. In order to do that, we can use the`SquareMatrix`

class, which provides utility methods to access the rows and columns of a square matrix:

SquareMatrix mnum = new SquareMatrix(num); for (int j=0; j<9; j++) { add(new AllDifferent_SPARSE((IntegerVariable[]) mnum.getColumn(j))); } for (int i=0; i<9; i++) { add(new AllDifferent_SPARSE((IntegerVariable[]) mnum.getLine(i))); }

The advantage of global constraints over a large number of small constraints is not only that they make the constraint model easier to read, but that they are also more powerful, since they usually incorporate complex filtering algorithms. For example, the`AllDifferent_SPARSE`

constraint uses a*n ^{2.5}* filtering algorithm (based on a maximal matching algorithm from the graph theory). An

*interval*version of the same constraint exists. It is called

`AllDifferent`

; it uses a *nxlog(n)*filtering algorithm. The

`AllDifferent`

constraint is thus faster but less powerful than the `AllDifferent_SPARSE`

constraint, which is why we used the latter in our model.Finally, we define the array of variables that will be used by the solver to display the solution:

setVariables((IntegerVariable[]) mnum.getElements());

We are done with the modelling, which only consists of 27`AllDifferent_SPARSE`

constraints: nine for the blocks, nine for the rows, and nine for the columns. The complete model can be found in the source file, in the Resources section.

## Solving the Sudoku Problem

This problem is simple enough to use one of the predefined solvers available in Koalog Constraint Solver:

Solver solver = new DefaultFFSolver(problem); solver.solve();where

`problem`

is a Sudoku problem instance and`DefaultFFSolver`

is a solver conforming to the first-fail heuristic. More precisely, this is syntactic sugar for: Solver solver = new SplitSolver (problem, new SmallestDomainVariableHeuristic(), new IncreasingOrderDomainHeuristic());

where `SmallestDomainVariableHeuristic`

selects the variable with the smallest domain (with the hope of cutting large branches from the search tree) and`IncreasingOrderDomainHeuristic`

tries to instantiate the selected variable by choosing the first value in its domain. Note that these types of solver and heuristics are very common and are available in any constraint solver.

Using this very simple model, most of the Sudokus are solved by deduction (i.e., with no search). For example, here is an extract of the output produced by Koalog Constraint Solver when solving the Sudoku given to illustrate the beginning of this article. The test was performed on a 1.6GHz PC running Linux SuSE 9.3:

solution found in 104 ms 0 backtrack(s), 160 filter(s), max depth 0 no more solution solved in 109 ms

This means that:

- A solution was found in 104ms by filtering 160 constraints (a constraint solver needs to filter each constraint more than once since the domains of the variables get reduced during the process).
- No trial and error was required.
- This solution is unique (i.e., the Sudoku is well-formed).
- The proof of this last result took only (109-104=) 5ms!

These results mean that the filtering algorithm, used by the 27`AllDifferent_SPARSE`

constraints, is powerful enough to deduce the unique solution of the problem.

A small percentage of Sudokus still require some*backtracking* (backtracking is an enumeration algorithm), but this remains very limited. In any case, solutions are found very quickly (in less than 200ms).

## Going Further

Helmut Simonis' article "Sudoku as a Constraint Problem" presents redundant constraints dedicated to the Sudoku problem, further reducing the need for backtracking. In particular, one could take advantage of the `LatinSquare`

constraint, which will be available in the next release of Koalog Constraint Solver, version 3.0. In any case, the solving of problems such as the Sudoku problem is made very easy with the use of Constraint Programming.

There are two other interesting, but harder, problems related to Sudokus:

**Generating well-formed Sudokus:**more precisely, generating givens that lead to a unique solution, which is the case of the Sudoku provided as an example at the beginning of this article.**Explaining Sudokus:**more precisely, generating a human-readable explanation of the resolution of a Sudoku.

Both problems can be addressed with Constraint Programming, but are out of the scope of this article.

## Resources

- Sample code for this article
- Wikipedia: Sudoku
- Wikipedia: Constraint Programming
- Wikipedia: Operational Research
- Koalog Constraint Solver
- A detailed article about the modelling of Sudokus using Constraint Programming
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- Koalog's free Sudoku of the day

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