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An Efficient Algorithm for the Generation of Z-Convex Polyominoes

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Combinatorial Image Analysis (IWCIA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 8466))

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Abstract

We present a characterization of Z-convex polyominoes in terms of pairs of suitable integer vectors. This lets us design an algorithm which generates all Z-convex polyominoes of size n in constant amortized time.

Partially supported by Project M.I.U.R. PRIN 2010-2011: Automi e linguaggi formali: aspetti matematici e applicativi.

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References

  1. Barcucci, E., Frosini, A., Rinaldi, S.: On directed-convex polyominoes in a rectangle. Discrete Mathematics 298(1-3), 62–78 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Barcucci, E., Lungo, A.D., Nivat, M., Pinzani, R.: Reconstructing Convex Polyominoes from Horizontal and Vertical Projections. Theor. Comput. Sci. 155(2), 321–347 (1996)

    Article  MATH  Google Scholar 

  3. Barcucci, E., Lungo, A.D., Pergola, E., Pinzani, R.: ECO: a methodology for the Enumeration of Combinatorial Objects. J. of Diff. Eq. and App. 5, 435–490 (1999)

    Article  MATH  Google Scholar 

  4. Battaglino, D., Fedou, J.M., Frosini, A., Rinaldi, S.: Encoding Centered Polyominoes by Means of a Regular Language. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 464–465. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  5. Beauquier, D., Nivat, M.: On Translating One Polyomino to Tile the Plane. Discrete & Computational Geometry 6, 575–592 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bender, E.A.: Convex n-ominoes. Discrete Math. 8, 219–226 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bousquet-Mélou, M.: A method for the enumeration of various classes of column-convex polygons. Discrete Math. 154(1-3), 1–25 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Brlek, S., Provençal, X., Fedou, J.-M.: On the tiling by translation problem. Discrete Applied Mathematics 157(3), 464–475 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brocchi, S., Frosini, A., Pinzani, R., Rinaldi, S.: A tiling system for the class of L-convex polyominoes. Theor. Comput. Sci. 475, 73–81 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Carli, F.D., Frosini, A., Rinaldi, S., Vuillon, L.: On the Tiling System Recognizability of Various Classes of Convex Polyominoes. Ann. Comb. 13, 169–191 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Castiglione, G., Frosini, A., Munarini, E., Restivo, A., Rinaldi, S.: Combinatorial aspects of L-convex polyominoes. Eur. J. Comb. 28(6), 1724–1741 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Castiglione, G., Frosini, A., Restivo, A., Rinaldi, S.: A Tomographical Characterization of L-Convex Polyominoes. In: Andrès, É., Damiand, G., Lienhardt, P. (eds.) DGCI 2005. LNCS, vol. 3429, pp. 115–125. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  13. Castiglione, G., Frosini, A., Restivo, A., Rinaldi, S.: Enumeration of L-convex polyominoes by rows and columns. Theor. Comput. Sci. 347(1-2), 336–352 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Castiglione, G., Restivo, A.: Reconstruction of L-convex Polyominoes. Electronic Notes in Discrete Mathematics 12, 290–301 (2003)

    Article  MathSciNet  Google Scholar 

  15. Delest, M.P., Viennot, G.: Algebraic Languages and Polyominoes Enumeration. Theor. Comput. Sci. 34, 169–206 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  16. Duchi, E., Rinaldi, S., Schaeffer, G.: The number of Z-convex polyominoes. Advances in Applied Mathematics 40(1), 54–72 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Golomb, W.S.: Checker Boards and Polyominoes. The American Mathematical Monthly 61, 675–682 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  18. Klarner, D.A., Rivest, R.R.: Asymptotic bounds for the number of convex n-ominoes. Discrete Math. 8, 31–40 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kuba, A., Balogh, E.: Reconstruction of convex 2D discrete sets in polynomial time. Theor. Comput. Sci. 283(1), 223–242 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Mantaci, R., Massazza, P.: From Linear Partitions to Parallelogram Polyominoes. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 350–361. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  21. Massazza, P.: On the generation of L-convex polyominoes. In: Proc. of GASCom 2012, Bordeaux, June 25-27 (2012)

    Google Scholar 

  22. Massazza, P.: On the Generation of Convex Polyominoes. Discrete Applied Mathematics (to appear)

    Google Scholar 

  23. Massé, A.B., Garon, A., Labbé, S.: Combinatorial properties of double square tiles. Theor. Comput. Sci. 502, 98–117 (2013)

    Article  MATH  Google Scholar 

  24. Micheli, A., Rossin, D.: Counting k-Convex Polyominoes. Electr. J. Comb. 20(2) (2013)

    Google Scholar 

  25. Ollinger, N.: Tiling the Plane with a Fixed Number of Polyominoes. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds.) LATA 2009. LNCS, vol. 5457, pp. 638–647. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  26. Tawbe, K., Vuillon, L.: 2L-convex polyominoes: Geometrical aspects. Contributions to Discrete Mathematics 6(1) (2011)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica ed Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123, Palermo, Italy

    Giusi Castiglione

  2. Dipartimento di Scienze, Teoriche e Applicate - Sezione Informatica, Università degli Studi dell’Insubria, Via Mazzini 5, 21100, Varese, Italy

    Paolo Massazza

Authors
  1. Giusi Castiglione
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  2. Paolo Massazza
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Editor information

Editors and Affiliations

  1. Department of Computer and Information Sciences, State University of New York at Fredonia, 280 Central Ave, 14063, Fredonia, NY, USA

    Reneta P. Barneva

  2. Mathematics Department, SUNY Buffalo State College, 14222, Buffalo, NY, USA

    Valentin E. Brimkov

  3. Department of Mathematics, Brno University of Technology, 616 69, Brno, Czech Republic

    Josef Šlapal

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Castiglione, G., Massazza, P. (2014). An Efficient Algorithm for the Generation of Z-Convex Polyominoes. In: Barneva, R.P., Brimkov, V.E., Šlapal, J. (eds) Combinatorial Image Analysis. IWCIA 2014. Lecture Notes in Computer Science, vol 8466. Springer, Cham. https://doi.org/10.1007/978-3-319-07148-0_6

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  • DOI: https://doi.org/10.1007/978-3-319-07148-0_6

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-07147-3

  • Online ISBN: 978-3-319-07148-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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