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132 lines
4.2 KiB
132 lines
4.2 KiB
"""Generator for Sudoku graphs
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This module gives a generator for n-Sudoku graphs. It can be used to develop
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algorithms for solving or generating Sudoku puzzles.
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A completed Sudoku grid is a 9x9 array of integers between 1 and 9, with no
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number appearing twice in the same row, column, or 3x3 box.
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+---------+---------+---------+
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| | 8 6 4 | | 3 7 1 | | 2 5 9 |
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| | 3 2 5 | | 8 4 9 | | 7 6 1 |
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| | 9 7 1 | | 2 6 5 | | 8 4 3 |
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+---------+---------+---------+
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| | 4 3 6 | | 1 9 2 | | 5 8 7 |
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| | 1 9 8 | | 6 5 7 | | 4 3 2 |
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| | 2 5 7 | | 4 8 3 | | 9 1 6 |
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+---------+---------+---------+
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| | 6 8 9 | | 7 3 4 | | 1 2 5 |
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| | 7 1 3 | | 5 2 8 | | 6 9 4 |
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| | 5 4 2 | | 9 1 6 | | 3 7 8 |
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+---------+---------+---------+
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The Sudoku graph is an undirected graph with 81 vertices, corresponding to
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the cells of a Sudoku grid. It is a regular graph of degree 20. Two distinct
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vertices are adjacent if and only if the corresponding cells belong to the
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same row, column, or box. A completed Sudoku grid corresponds to a vertex
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coloring of the Sudoku graph with nine colors.
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More generally, the n-Sudoku graph is a graph with n^4 vertices, corresponding
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to the cells of an n^2 by n^2 grid. Two distinct vertices are adjacent if and
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only if they belong to the same row, column, or n by n box.
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References
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----------
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.. [1] Herzberg, A. M., & Murty, M. R. (2007). Sudoku squares and chromatic
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polynomials. Notices of the AMS, 54(6), 708-717.
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.. [2] Sander, Torsten (2009), "Sudoku graphs are integral",
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Electronic Journal of Combinatorics, 16 (1): Note 25, 7pp, MR 2529816
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.. [3] Wikipedia contributors. "Glossary of Sudoku." Wikipedia, The Free
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Encyclopedia, 3 Dec. 2019. Web. 22 Dec. 2019.
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"""
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import networkx as nx
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from networkx.exception import NetworkXError
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__all__ = ["sudoku_graph"]
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@nx._dispatchable(graphs=None, returns_graph=True)
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def sudoku_graph(n=3):
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"""Returns the n-Sudoku graph. The default value of n is 3.
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The n-Sudoku graph is a graph with n^4 vertices, corresponding to the
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cells of an n^2 by n^2 grid. Two distinct vertices are adjacent if and
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only if they belong to the same row, column, or n-by-n box.
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Parameters
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----------
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n: integer
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The order of the Sudoku graph, equal to the square root of the
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number of rows. The default is 3.
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Returns
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-------
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NetworkX graph
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The n-Sudoku graph Sud(n).
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Examples
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--------
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>>> G = nx.sudoku_graph()
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>>> G.number_of_nodes()
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81
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>>> G.number_of_edges()
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810
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>>> sorted(G.neighbors(42))
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[6, 15, 24, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 51, 52, 53, 60, 69, 78]
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>>> G = nx.sudoku_graph(2)
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>>> G.number_of_nodes()
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16
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>>> G.number_of_edges()
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56
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References
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----------
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.. [1] Herzberg, A. M., & Murty, M. R. (2007). Sudoku squares and chromatic
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polynomials. Notices of the AMS, 54(6), 708-717.
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.. [2] Sander, Torsten (2009), "Sudoku graphs are integral",
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Electronic Journal of Combinatorics, 16 (1): Note 25, 7pp, MR 2529816
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.. [3] Wikipedia contributors. "Glossary of Sudoku." Wikipedia, The Free
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Encyclopedia, 3 Dec. 2019. Web. 22 Dec. 2019.
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"""
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if n < 0:
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raise NetworkXError("The order must be greater than or equal to zero.")
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n2 = n * n
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n3 = n2 * n
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n4 = n3 * n
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# Construct an empty graph with n^4 nodes
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G = nx.empty_graph(n4)
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# A Sudoku graph of order 0 or 1 has no edges
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if n < 2:
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return G
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# Add edges for cells in the same row
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for row_no in range(n2):
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row_start = row_no * n2
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for j in range(1, n2):
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for i in range(j):
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G.add_edge(row_start + i, row_start + j)
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# Add edges for cells in the same column
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for col_no in range(n2):
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for j in range(col_no, n4, n2):
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for i in range(col_no, j, n2):
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G.add_edge(i, j)
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# Add edges for cells in the same box
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for band_no in range(n):
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for stack_no in range(n):
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box_start = n3 * band_no + n * stack_no
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for j in range(1, n2):
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for i in range(j):
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u = box_start + (i % n) + n2 * (i // n)
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v = box_start + (j % n) + n2 * (j // n)
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G.add_edge(u, v)
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return G
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