sudoku--Knuth's Algorithm X

http://en.wikipedia.org/wiki/Knuth%27s_Algorithm_X

http://www.cs.mcgill.ca/~aassaf9/python/algorithm_x.html

#!/usr/bin/env python3# Author: Ali Assaf <[e-mail]># Copyright: (C) 2010 Ali Assaf# License: GNU General Public License <http://www.gnu.org/licenses/>from itertools import productdef solve_sudoku(size, grid):    """ An efficient Sudoku solver using Algorithm X.    >>> grid = [    ...     [5, 3, 0, 0, 7, 0, 0, 0, 0],    ...     [6, 0, 0, 1, 9, 5, 0, 0, 0],    ...     [0, 9, 8, 0, 0, 0, 0, 6, 0],    ...     [8, 0, 0, 0, 6, 0, 0, 0, 3],    ...     [4, 0, 0, 8, 0, 3, 0, 0, 1],    ...     [7, 0, 0, 0, 2, 0, 0, 0, 6],    ...     [0, 6, 0, 0, 0, 0, 2, 8, 0],    ...     [0, 0, 0, 4, 1, 9, 0, 0, 5],    ...     [0, 0, 0, 0, 8, 0, 0, 7, 9]]    >>> for solution in solve_sudoku((3, 3), grid):    ...     print(*solution, sep='\\n')    [5, 3, 4, 6, 7, 8, 9, 1, 2]    [6, 7, 2, 1, 9, 5, 3, 4, 8]    [1, 9, 8, 3, 4, 2, 5, 6, 7]    [8, 5, 9, 7, 6, 1, 4, 2, 3]    [4, 2, 6, 8, 5, 3, 7, 9, 1]    [7, 1, 3, 9, 2, 4, 8, 5, 6]    [9, 6, 1, 5, 3, 7, 2, 8, 4]    [2, 8, 7, 4, 1, 9, 6, 3, 5]    [3, 4, 5, 2, 8, 6, 1, 7, 9]    """    R, C = size    N = R * C    X = ([("rc", rc) for rc in product(range(N), range(N))] +         [("rn", rn) for rn in product(range(N), range(1, N + 1))] +         [("cn", cn) for cn in product(range(N), range(1, N + 1))] +         [("bn", bn) for bn in product(range(N), range(1, N + 1))])    Y = dict()    for r, c, n in product(range(N), range(N), range(1, N + 1)):        b = (r // R) * R + (c // C) # Box number        Y[(r, c, n)] = [            ("rc", (r, c)),            ("rn", (r, n)),            ("cn", (c, n)),            ("bn", (b, n))]    X, Y = exact_cover(X, Y)    for i, row in enumerate(grid):        for j, n in enumerate(row):            if n:                select(X, Y, (i, j, n))    for solution in solve(X, Y, []):        for (r, c, n) in solution:            grid[r][c] = n        yield griddef exact_cover(X, Y):    X = {j: set() for j in X}    for i, row in Y.items():        for j in row:            X[j].add(i)    return X, Ydef solve(X, Y, solution):    if not X:        yield list(solution)    else:        c = min(X, key=lambda c: len(X[c]))        for r in list(X[c]):            solution.append(r)            cols = select(X, Y, r)            for s in solve(X, Y, solution):                yield s            deselect(X, Y, r, cols)            solution.pop()def select(X, Y, r):    cols = []    for j in Y[r]:        for i in X[j]:            for k in Y[i]:                if k != j:                    X[k].remove(i)        cols.append(X.pop(j))    return colsdef deselect(X, Y, r, cols):    for j in reversed(Y[r]):        X[j] = cols.pop()        for i in X[j]:            for k in Y[i]:                if k != j:                    X[k].add(i)if __name__ == "__main__":    import doctest    doctest.testmod()