This is an attempt to solving the two bars tiling problem on a region with holes, s.t each bar is of size 3x1 or 1x3

To use: run 'python3 main.py'. Modify values for grid, holes as required. Rendering output is optional using flag. In addiiton, to run tests: 'python3 -m unittest'

Previous Work

It has been shown in (DBRR 1995) that the problem of tiling with two bars of length 3 on a region with holes is NP-Complete. This necessitates the use of a search based approach e.g backtracking. While considering regions with holes, we can additionally optimize using early exits in the case of early detection of untileable components. Finally, recent work has shown that we can solve the 2BAR-tiling problem in the case of a region without holes, in particular with 3x1, 1x3 bars, in linear time in terms of the area. This allows us to solve several subproblems with early exit conditions.

Chosen Approach

In our approach we implement a simple backtracking based approach since it allows us to keep code clean, concise and easy to adapt to modification, in particular, we should be able to reuse the codebase accessible should we want to optimize the algorithm by doing a linear time check on a subproblem instance without holes using algorithm in (C.Kenyon, R. Kenyon 1992)

With this in mind, we will attempt to solve the tiling problem using:

1. A backtracking based search algorithm. (Brute Force)
2. Backtracking with 'early exit' conditions:
   • Unreachable tiles i.e either the whole grid or individual strongly connect component which are not tileable (implemented)
   • Linear time calculations on regions without holes (future work)

Alternative approaches

There is an additonal popular approach for this problem. It considers framing 2BAR tiling as an instance of polynomino tiling. In addition, polynomino tiling can be considered an instance of Exact Cover for which (D. Knuth 2000) has proposed the Dancing Links approach. Finally, this problem is additonally representable as a Boolean Satifiability problem for which many solvers exist.

Software Hacks

Some other software optimizations considered:

1. Represent grid Boolean Numpy Array. Allows swift computation of certain logic operation in backtracking
2. Further optimize storage by using BitField instead of NumPy array
3. Early exit if any strongly connected component within the grid is no longer satisifiable, i.e if size of component mod 3 != 0
4. Using a fast permutation algorithm to optimize storage and accessibility of remaining grid points
5. Store remaining coordinates in a set

Acknowledgements

Finally, the author deeply thanks the team at Gradient.tech for motivating such an interesting problem. In particular, this problem has been a great source of entertainment and interest in a strange quarantine experience amid the COVID-19 Pandemic. With links to topology, group theory and other areas of mathematics, this project has been, above all, a fun endeavor in learning.

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References

1. Danièle Beauquier, Maurice Nivat, Eric Remila, Mike Robson, Tiling figures of the plane with two bars, Computational Geometry, Volume 5, Issue 1, 1995, Pages 1-25, ISSN 0925-7721, https://doi.org/10.1016/0925-7721(94)00015-N. (http://www.sciencedirect.com/science/article/pii/092577219400015N)
2. Don Sheehy, The Complexity of Domino Tiling Problems, 2005
3. Horiyama, T., Ito, T., Nakatsuka, K. et al. Complexity of Tiling a Polygon with Trominoes or Bars. Discrete Comput Geom 58, 686–704 (2017). https://doi.org/10.1007/s00454-017-9884-9
4. Donald E. Knuth: “Dancing links”, 2000, Millenial Perspectives in Computer Science, 2000, 187--214; [http://arxiv.org/abs/cs/0011047 arXiv:cs/0011047].
5. C. Kenyon and R. Kenyon, "Tiling a polygon with rectangles," Proceedings., 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, PA, USA, 1992, pp. 610-619, doi: 10.1109/SFCS.1992.267790.