Abstract
The notion of unbordered picture generalizes to two dimensions the notion of unbordered (or bifix-free) string. We extend to two dimensions Nielsen’s construction of unbordered strings ([23]) and describe an algorithm to construct the set U(m, n) of unbordered pictures of fixed size (m, n). The algorithm recursively computes the set of quasi-unbordered pictures Q(m, n), i.e. pictures that can possibly have some “large” borders.
Partially supported by MIUR Projects “Formal Languages and Automata: Mathematical Structures and Applicative Directions” and “PRISMA PON04a2 A/F”, and by FARB Projects of University of Catania, Roma “Tor Vergata”, Salerno.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Aigrain, P., Beauquier, D.: Polyomino tilings, cellular automata and codicity. Theoretical Computer Science 147, 165–180 (1995)
Anselmo, M., Giammarresi, D., Madonia, M.: Deterministic and unambiguous families within recognizable two-dimensional languages. Fund. Inform. 98(2–3), 143–166 (2010)
Anselmo, M., Giammarresi, D., Madonia, M.: Strong prefix codes of pictures. In: Muntean, T., Poulakis, D., Rolland, R. (eds.) CAI 2013. LNCS, vol. 8080, pp. 47–59. Springer, Heidelberg (2013)
Anselmo, M., Giammarresi, D., Madonia, M., Restivo, A.: Unambiguous recognizable two-dimensional languages. RAIRO -ITA 40(2), 227–294 (2006)
Anselmo, M., Giammarresi, D., Madonia, M.: A computational model for tiling recognizable two-dimensional languages. Theor. Comput. Sci. 410(37), 3520–3529 (2009)
Anselmo, M., Giammarresi, D., Madonia, M.: Prefix picture codes: a decidable class of two-dimensional codes. Int. J. Found. Comput. Sci. 25(8), 1017–1032 (2014)
Bajic, D., Loncar-Turukalo, T.: A simple suboptimal construction of cross-bifix-free codes. Cryptography and Communications 6(1), 27–37 (2014)
Beauquier, D., Nivat, M.: A codicity undecidable problem in the plane. Theoret. Comp. Sci 303, 417–430 (2003)
Berstel, J., Perrin, D., Reutenauer, C.: Codes and Automata. Cambridge University Press (2009)
Bilotta, S., Pergola, E., Pinzani, R.: A new approach to cross-bifix-free sets. IEEE Transactions on Information Theory 58(6), 4058–4063 (2012)
Blum, M., Hewitt, C.: Automata on a 2-dimensional tape. In: SWAT (FOCS), pp. 155–160 (1967)
Bozapalidis, S., Grammatikopoulou, A.: Picture codes. RAIRO - ITA 40(4), 537–550 (2006)
Chee, Y.M., Kiah, H.M., Purkayastha, P., Wang, C.: Cross-bifix-free codes within a constant factor of optimality. IEEE Transactions on Information Theory 59(7), 4668–4674 (2013)
Crochemore, M., Iliopoulos, C.S., Korda, M.: Two-dimensional prefix string matching and covering on square matrices. Algorithmica 20(4), 353–373 (1998)
Crochemore, M., Rytter, W.: Jewels of stringology. World Scientific (2002). http://www-igm.univ-mlv.fr/ mac/JOS/JOS.html
Gamard, G., Richomme, G.: Coverability in two dimensions. In: Dediu, A.-H., Formenti, E., Martín-Vide, C., Truthe, B. (eds.) LATA 2015. LNCS, vol. 8977, pp. 402–413. Springer, Heidelberg (2015)
Giammarresi, D., Restivo, A.: Recognizable picture languages. Int. Journal. Pattern Recognition and Artificial Intelligence 6(2–3), 241–256 (1992)
Giammarresi, D., Restivo, A.: Two-dimensional languages. In: Rozenberg, G.(ed.) Handbook of Formal Languages, vol. III, pp. 215–268. Springer Verlag (1997)
Gusfield, D.: Algorithms on Strings, Trees, and Sequences - Computer Science and Computational Biology. Cambridge University Press (1997)
Kari, J., Salo, V.: A survey on picture-walking automata. In: Kuich, W., Rahonis, G. (eds.) Algebraic Foundations in Computer Science. LNCS, vol. 7020, pp. 183–213. Springer, Heidelberg (2011)
Kolarz, M., Moczurad, W.: Multiset, set and numerically decipherable codes over directed figures. In: Smyth, B. (ed.) IWOCA 2012. LNCS, vol. 7643, pp. 224–235. Springer, Heidelberg (2012)
Na, J.C., Ferragina, P., Giancarlo, R., Park, K.: Indexed two-dimensional string matching. In: Encyclopedia of Algorithms (2015)
Nielsen, P.T.: A note on bifix-free sequences (corresp.). IEEE Transactions on Information Theory 19(5), 704–706 (1973)
Otto, F., Mráz, F.: Extended two-way ordered restarting automata for picture languages. In: Dediu, A.-H., Martín-Vide, C., Sierra-Rodríguez, J.-L., Truthe, B. (eds.) LATA 2014. LNCS, vol. 8370, pp. 541–552. Springer, Heidelberg (2014)
Pradella, M., Cherubini, A., Crespi-Reghizzi, S.: A unifying approach to picture grammars. Inf. Comput. 209(9), 1246–1267 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Anselmo, M., Giammarresi, D., Madonia, M. (2015). Unbordered Pictures: Properties and Construction. In: Maletti, A. (eds) Algebraic Informatics. CAI 2015. Lecture Notes in Computer Science(), vol 9270. Springer, Cham. https://doi.org/10.1007/978-3-319-23021-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-23021-4_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-23020-7
Online ISBN: 978-3-319-23021-4
eBook Packages: Computer ScienceComputer Science (R0)