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# Source code for networkx.generators.sudoku

    """Generator for Sudoku graphs

    This module gives a generator for n-Sudoku graphs. It can be used to develop
    algorithms for solving or generating Sudoku puzzles.

    A completed Sudoku grid is a 9x9 array of integers between 1 and 9, with no
    number appearing twice in the same row, column, or 3x3 box.

    +---------+---------+---------+
    | | 8 6 4 | | 3 7 1 | | 2 5 9 |
    | | 3 2 5 | | 8 4 9 | | 7 6 1 |
    | | 9 7 1 | | 2 6 5 | | 8 4 3 |
    +---------+---------+---------+
    | | 4 3 6 | | 1 9 2 | | 5 8 7 |
    | | 1 9 8 | | 6 5 7 | | 4 3 2 |
    | | 2 5 7 | | 4 8 3 | | 9 1 6 |
    +---------+---------+---------+
    | | 6 8 9 | | 7 3 4 | | 1 2 5 |
    | | 7 1 3 | | 5 2 8 | | 6 9 4 |
    | | 5 4 2 | | 9 1 6 | | 3 7 8 |
    +---------+---------+---------+

    The Sudoku graph is an undirected graph with 81 vertices, corresponding to
    the cells of a Sudoku grid. It is a regular graph of degree 20. Two distinct
    vertices are adjacent if and only if the corresponding cells belong to the
    same row, column, or box. A completed Sudoku grid corresponds to a vertex
    coloring of the Sudoku graph with nine colors.

    More generally, the n-Sudoku graph is a graph with n^4 vertices, corresponding
    to the cells of an n^2 by n^2 grid. Two distinct vertices are adjacent if and
    only if they belong to the same row, column, or n by n box.

    References
    ----------
    .. [1] Herzberg, A. M., & Murty, M. R. (2007). Sudoku squares and chromatic
        polynomials. Notices of the AMS, 54(6), 708-717.
    .. [2] Sander, Torsten (2009), "Sudoku graphs are integral",
        Electronic Journal of Combinatorics, 16 (1): Note 25, 7pp, MR 2529816
    .. [3] Wikipedia contributors. "Glossary of Sudoku." Wikipedia, The Free
        Encyclopedia, 3 Dec. 2019. Web. 22 Dec. 2019.
    """

    import networkx as nx
    from networkx.exception import NetworkXError

    __all__ = ["sudoku_graph"]

    @nx._dispatchable(graphs=None, returns_graph=True)
    def sudoku_graph(n=3):
        """Returns the n-Sudoku graph. The default value of n is 3.

        The n-Sudoku graph is a graph with n^4 vertices, corresponding to the
        cells of an n^2 by n^2 grid. Two distinct vertices are adjacent if and
        only if they belong to the same row, column, or n-by-n box.

        Parameters
        ----------
        n: integer
           The order of the Sudoku graph, equal to the square root of the
           number of rows. The default is 3.

        Returns
        -------
        NetworkX graph
            The n-Sudoku graph Sud(n).

        Examples
        --------
        >>> G = nx.sudoku_graph()
        >>> G.number_of_nodes()
        81
        >>> G.number_of_edges()
        810
        >>> sorted(G.neighbors(42))
        [6, 15, 24, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 51, 52, 53, 60, 69, 78]
        >>> G = nx.sudoku_graph(2)
        >>> G.number_of_nodes()
        16
        >>> G.number_of_edges()
        56

        References
        ----------
        .. [1] Herzberg, A. M., & Murty, M. R. (2007). Sudoku squares and chromatic
           polynomials. Notices of the AMS, 54(6), 708-717.
        .. [2] Sander, Torsten (2009), "Sudoku graphs are integral",
           Electronic Journal of Combinatorics, 16 (1): Note 25, 7pp, MR 2529816
        .. [3] Wikipedia contributors. "Glossary of Sudoku." Wikipedia, The Free
           Encyclopedia, 3 Dec. 2019. Web. 22 Dec. 2019.
        """

        if n < 0:
            raise NetworkXError("The order must be greater than or equal to zero.")

        n2 = n * n
        n3 = n2 * n
        n4 = n3 * n

        # Construct an empty graph with n^4 nodes
        G = nx.empty_graph(n4)

        # A Sudoku graph of order 0 or 1 has no edges
        if n < 2:
            return G

        # Add edges for cells in the same row
        for row_no in range(n2):
            row_start = row_no * n2
            for j in range(1, n2):
                for i in range(j):
                    G.add_edge(row_start + i, row_start + j)

        # Add edges for cells in the same column
        for col_no in range(n2):
            for j in range(col_no, n4, n2):
                for i in range(col_no, j, n2):
                    G.add_edge(i, j)

        # Add edges for cells in the same box
        for band_no in range(n):
            for stack_no in range(n):
                box_start = n3 * band_no + n * stack_no
                for j in range(1, n2):
                    for i in range(j):
                        u = box_start + (i % n) + n2 * (i // n)
                        v = box_start + (j % n) + n2 * (j // n)
                        G.add_edge(u, v)

        return G