An algorithm for deciding if a polyomino tiles the plane

Ian Gambini; Laurent Vuillon

RAIRO - Theoretical Informatics and Applications (2007)

  • Volume: 41, Issue: 2, page 147-155
  • ISSN: 0988-3754

Abstract

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For polyominoes coded by their boundary word, we describe a quadratic O(n2) algorithm in the boundary length n which improves the naive O(n4) algorithm. Techniques used emanate from algorithmics, discrete geometry and combinatorics on words.

How to cite

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Gambini, Ian, and Vuillon, Laurent. "An algorithm for deciding if a polyomino tiles the plane." RAIRO - Theoretical Informatics and Applications 41.2 (2007): 147-155. <http://eudml.org/doc/250073>.

@article{Gambini2007,
abstract = { For polyominoes coded by their boundary word, we describe a quadratic O(n2) algorithm in the boundary length n which improves the naive O(n4) algorithm. Techniques used emanate from algorithmics, discrete geometry and combinatorics on words. },
author = {Gambini, Ian, Vuillon, Laurent},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Polyominoes; tiling the plane by translation; theorem of Beauquier-Nivat; pseudo-square; pseudo-hexagon; enumeration of special classes of polyominoes},
language = {eng},
month = {7},
number = {2},
pages = {147-155},
publisher = {EDP Sciences},
title = {An algorithm for deciding if a polyomino tiles the plane},
url = {http://eudml.org/doc/250073},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Gambini, Ian
AU - Vuillon, Laurent
TI - An algorithm for deciding if a polyomino tiles the plane
JO - RAIRO - Theoretical Informatics and Applications
DA - 2007/7//
PB - EDP Sciences
VL - 41
IS - 2
SP - 147
EP - 155
AB - For polyominoes coded by their boundary word, we describe a quadratic O(n2) algorithm in the boundary length n which improves the naive O(n4) algorithm. Techniques used emanate from algorithmics, discrete geometry and combinatorics on words.
LA - eng
KW - Polyominoes; tiling the plane by translation; theorem of Beauquier-Nivat; pseudo-square; pseudo-hexagon; enumeration of special classes of polyominoes
UR - http://eudml.org/doc/250073
ER -

References

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  1. E. Barcucci, A. Del Lungo, E. Pergola and R. Pinzani, ECO: a methodology for the Enumeration of Combinatorial Objects. J. Difference Equ. Appl.5 (1999) 435–490.  
  2. D. Beauquier and M. Nivat, On translating one polyomino to tile the plane. Discrete Comput. Geom.6 (1991) 575–592.  
  3. M. Bousquet-Mélou, A method for the enumeration of various classes of column-convex polygons. Discrete Math.154 (1996) 1–25.  
  4. M. Bousquet-Mélou. Habilitation. LABRI Université de Bordeaux 1 (1996).  
  5. S.J. Chang and K.Y. Lin. Rigorous results for the number of convex polygons on the square and honeycomb lattices. J. Phys. A21 (1988) 2635–2642.  
  6. T.H. Cormen, C.E. Leiserson and R.L. Rivest, Introduction to algorithms. Chapt. 34, MIT Press (1990) 853–885.  
  7. A. Daurat and M. Nivat. Salient and Reentrant Points of Discrete Sets, in Proc. of the nineth International Workshop on Combinatorial Image Analysis (IWCIA 2003), volume 12 of Electronic Notes in Discrete Mathematics. Elsevier (2003).  
  8. A. Del Lungo, E. Duchi, A. Frosini and S. Rinaldi, Enumeration of convex polyominoes using the ECO method, in Discrete Models for Complex Systems, DMCS'03, edited by M. Morvan and É. Rémila, Discrete Mathematics and Theoretical Computer Science Proceedings AB, 103–116.  
  9. M. Delest and X. Viennot, Algebraic languages and polyominoes enumeration. Theoret. Comput. Sci.34 (1984) 169–206.  
  10. I. Gambini, A Method for Cutting Squares Into Distinct Squares. Discrete Appl. Math.98 (1999) 65–80.  
  11. S.W. Golomb. Polyominoes, Princeton science library (1994).  
  12. P. Hubert and L. Vuillon. Complexity of cutting words on regular tilings. Eur. J. Combin.28 (2007) 429–438.  
  13. D.E. Knuth, J.H. Morris and V.R. Pratt. Fast pattern matching in strings. SIAM J. Comput.6 (1997) 323–350.  
  14. P. Leroux, E. Rassart and A. Robitaille, Enumeration of symmetry classes of convex polyminoes in the square lattice. Adv. Appl. Math.21 (1998) 343–380.  
  15. P. Leroux and E. Rassart, Enumeration of symmetry classes of parallelogram polyminoes. Ann. Sci. Math. Québec25 (2001) 53–72.  

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