Package: exact_cover
Finding Exact Covers in NumPyThis is a Python 3 package to solve exact cover problems using Numpy. It is based onhttps://github.com/moygit/exact_cover_npby Moy Easwaran. Jack Grahl ported it to Python 3, fixed some bugs and made lots of small improvements to the packaging.The original package by Moy was designed to solve sudoku. Now this package is only designed to solve exact cover problems given as boolean arrays. It can be used to solve sudoku and a variety of combinatorial problems. However the code to reduce a sudoku to an exact cover problem is no longer part of this project. It can be found at:https://pypi.org/project/xudoku/https://github.com/jwg4/xudokuAnother project, 'polyomino' by Jack Grahl uses this algorithm to solve polyomino tiling problems. It can be found at:https://pypi.org/project/polyomino/https://github.com/jwg4/polyominoSummaryThe exact cover problem is as follows: given a set X and a
collection S of subsets of X, we want to find a subcollection S*
of S that is an exact cover or partition of X.  In other words,
S* is a bunch of subsets of X whose union is X, and which have
empty intersection with each other.  (Example below; more detailson
wikipedia.)This NumPy module uses Donald Knuth's Algorithm X
(also known as Dancing Links) to find
exact covers of sets.
For details on Algorithm X please see eitherthe Wikipedia pageorKnuth's paper.
Specifically, we use the Knuth/Hitotsumatsu/Noshita method of
Dancing Links for efficient backtracking.  Please seeKnuth's paperfor details.How to Use ItSuppose X = {0,1,2,3,4}, and suppose S = {A,B,C,D}, whereA = {0, 3}
B = {0, 1, 2}
C = {1, 2}
D = {4}.Here we can just eyeball these sets and conclude that S* = {A,C,D} forms an
exact cover: each element of X is in one of these sets (i.e. is
"covered" by one of these sets), and no element of X is in more than
one.We'd useexact_coverto solve the problem as follows:
using 1 to denote that a particular member of X is in a subset and 0 to
denote that it's not, we can represent the sets asA = 1,0,0,1,0    # The 0th and 3rd entries are 1 since 0 and 3 are in A; the rest are 0.
B = 1,1,1,0,0    # The 0th, 1st, and 2nd entries are 1, and the rest are 0,
C = 0,1,1,0,0    # etc.
D = 0,0,0,0,1Now we can callexact_cover:>>> import numpy as np
>>> import exact_cover as ec
>>> S = np.array([[1,0,0,1,0],[1,1,1,0,0],[0,1,1,0,0],[0,0,0,0,1]], dtype=bool)
>>> print(ec.get_exact_cover(S))
[0 2 3]This is telling us that the 0th row (i.e. A), the 2nd row (i.e. C),
and the 3rd row (i.e. D) together form an exact cover.See the file examples.md for more detailed examples of use.Implementation OverviewThe NumPy module (exact_cover) is implemented in four pieces:The lowest level isquad_linked_list, which implements a circular
linked-list with left-, right-, up-, and down-links.This is used insparse_matrixto implement the type of sparse
representation of matrices that Knuth describes in his paper (in
brief, each column contains all its non-zero entries, and each
non-zero cell also points to the (horizontally) next non-zero cell
in either direction).Sparse matrices are used indlxto implement Knuth's Dancing
Links version of his Algorithm X, which calculates exact covers.exact_coverprovides the glue code letting us invokedlxon NumPy arrays.The package now has some pure Python modules for helper functions, with the main algorithm in the C-only packageexact_cover_impl.How to developThe package uses poetry and most of the setup for development uses that tool.To install locally (as an editable package):poetry installTo build:poetry buildTo run tests:poetry run testorpoetry run doctestTo open a Python shell with the package available:poetry run pythonThe exception is running the C unit tests:make c_testsRepositorybuild/ The location where files are built.dist/ The location for fully prepared files.exact_cover/ The Python code.obj/ Where the compiled C code is going to be output.src/ The C sources.tests/ Tests for both the Python package and the C code.tools/ Code used in analysing and working with the library. This is not distributed with the package.AcknowledgementsThanks very much to Moy Easwaran (https://github.com/moygit) for his inspiring work!Munit aka µnit (https://nemequ.github.io/munit/) is a wonderful unit testing framework for C code.