- COPYRIGHT AND LICENSE
sudoku.c Solve a sudoku by brute force search Read input data from file or string l_int32 *sudokuReadFile() l_int32 *sudokuReadString() Create/destroy L_SUDOKU *sudokuCreate() void sudokuDestroy() Solve the puzzle l_int32 sudokuSolve() static l_int32 sudokuValidState() static l_int32 sudokuNewGuess() static l_int32 sudokuTestState() Test for uniqueness l_int32 sudokuTestUniqueness() static l_int32 sudokuCompareState() static l_int32 *sudokuRotateArray() Generation L_SUDOKU *sudokuGenerate() Output l_int32 sudokuOutput() Solving sudokus is a somewhat addictive pastime. The rules are simple but it takes just enough concentration to make it rewarding when you find a number. And you get 50 to 60 such rewards each time you complete one. The downside is that you could have been doing something more creative, like keying out a new plant, staining the deck, or even writing a computer program to discourage your wife from doing sudokus. My original plan for the sudoku solver was somewhat grandiose. The program would model the way a person solves the problem. It would examine each empty position and determine how many possible numbers could fit. The empty positions would be entered in a priority queue keyed on the number of possible numbers that could fit. If there existed a position where only a single number would work, it would greedily take it. Otherwise it would consider a positions that could accept two and make a guess, with backtracking if an impossible state were reached. And so on. Then one of my colleagues announced she had solved the problem by brute force and it was fast. At that point the original plan was dead in the water, because the two top requirements for a leptonica algorithm are (1) as simple as possible and (2) fast. The brute force approach starts at the UL corner, and in succession at each blank position it finds the first valid number (testing in sequence from 1 to 9). When no number will fit a blank position it backtracks, choosing the next valid number in the previous blank position. This is an inefficient method for pruning the space of solutions (imagine backtracking from the LR corner back to the UL corner and starting over with a new guess), but it nevertheless gets the job done quickly. I have made no effort to optimize it, because it is fast: a 5-star (highest difficulty) sudoku might require a million guesses and take 0.05 sec. (This BF implementation does about 20M guesses/sec at 3 GHz.) Proving uniqueness of a sudoku solution is tricker than finding a solution (or showing that no solution exists). A good indication that a solution is unique is if we get the same result solving by brute force when the puzzle is also rotated by 90, 180 and 270 degrees. If there are multiple solutions, it seems unlikely that you would get the same solution four times in a row, using a brute force method that increments guesses and scans LR/TB. The function sudokuTestUniqueness() does this. And given a function that can determine uniqueness, it is easy to generate valid sudokus. We provide sudokuGenerate(), which starts with some valid initial solution, and randomly removes numbers, stopping either when a minimum number of non-zero elements are left, or when it becomes difficult to remove another element without destroying the uniqueness of the solution. For further reading, see the Wikipedia articles: (1) http://en.wikipedia.org/wiki/Algorithmics_of_sudoku (2) http://en.wikipedia.org/wiki/Sudoku How many 9x9 sudokus are there? Here are the numbers. - From ref(1), there are about 6 x 10^27 "latin squares", where each row and column has all 9 digits. - There are 7.2 x 10^21 actual solutions, having the added constraint in each of the 9 3x3 squares. (The constraint reduced the number by the fraction 1.2 x 10^(-6).) - There are a mere 5.5 billion essentially different solutions (EDS), when symmetries (rotation, reflection, permutation and relabelling) are removed. - Thus there are 1.3 x 10^12 solutions that can be derived by symmetry from each EDS. Can we account for these? - Sort-of. From an EDS, you can derive (3!)^8 = 1.7 million solutions by simply permuting rows and columns. (Do you see why it is not (3!)^6 ?) - Also from an EDS, you can derive 9! solutions by relabelling, and 4 solutions by rotation, for a total of 1.45 million solutions by relabelling and rotation. Then taking the product, by symmetry we can derive 1.7M x 1.45M = 2.45 trillion solutions from each EDS. (Something is off by about a factor of 2 -- close enough.) Another interesting fact is that there are apparently 48K EDS sudokus (with unique solutions) that have only 17 givens. No sudokus are known with less than 17, but there exists no proof that this is the minimum.
L_SUDOKU * sudokuCreate ( l_int32 *array )
sudokuCreate() Input: array (of 81 numbers, 9 rows of 9 numbers each) Return: l_sudoku, or null on error Notes: (1) The input array has 0 for the unknown values, and 1-9 for the known initial values. It is generated from a file using sudokuReadInput(), which checks that the file data has 81 numbers in 9 rows.
void sudokuDestroy ( L_SUDOKU **psud )
sudokuDestroy() Input: &l_sudoku (<to be nulled>) Return: void
L_SUDOKU * sudokuGenerate ( l_int32 *array, l_int32 seed, l_int32 minelems, l_int32 maxtries )
sudokuGenerate() Input: array (of 81 numbers, 9 rows of 9 numbers each) seed (random number) minelems (min non-zero elements allowed; <= 80) maxtries (max tries to remove a number and get a valid sudoku) Return: l_sudoku, or null on error Notes: (1) This is a brute force generator. It starts with a completed sudoku solution and, by removing elements (setting them to 0), generates a valid (unique) sudoku initial condition. (2) The process stops when either @minelems, the minimum number of non-zero elements, is reached, or when the number of attempts to remove the next element exceeds @maxtries. (3) No sudoku is known with less than 17 nonzero elements.
l_int32 sudokuOutput ( L_SUDOKU *sud, l_int32 arraytype )
sudokuOutput() Input: l_sudoku (at any stage) arraytype (L_SUDOKU_INIT, L_SUDOKU_STATE) Return: void Notes: (1) Prints either the initial array or the current state of the solution.
l_int32 * sudokuReadFile ( const char *filename )
sudokuReadFile() Input: filename (of formatted sudoku file) Return: array (of 81 numbers), or null on error Notes: (1) The file format has: * any number of comment lines beginning with '#' * a set of 9 lines, each having 9 digits (0-9) separated by a space
l_int32 * sudokuReadString ( const char *str )
sudokuReadString() Input: str (of input data) Return: array (of 81 numbers), or null on error Notes: (1) The string is formatted as 81 single digits, each separated by 81 spaces.
l_int32 sudokuSolve ( L_SUDOKU *sud )
sudokuSolve() Input: l_sudoku (starting in initial state) Return: 1 on success, 0 on failure to solve (note reversal of typical unix returns)
l_int32 sudokuTestUniqueness ( l_int32 *array, l_int32 *punique )
sudokuTestUniqueness() Input: array (of 81 numbers, 9 lines of 9 numbers each) &punique (<return> 1 if unique, 0 if not) Return: 0 if OK, 1 on error Notes: (1) This applies the brute force method to all four 90 degree rotations. If there is more than one solution, it is highly unlikely that all four results will be the same; consequently, if they are the same, the solution is most likely to be unique.
Zakariyya Mughal <firstname.lastname@example.org>
COPYRIGHT AND LICENSE
This software is copyright (c) 2014 by Zakariyya Mughal.
This is free software; you can redistribute it and/or modify it under the same terms as the Perl 5 programming language system itself.