Abstract
A sequence of approximating geometric patterns that define in the limit, the Sierpiński space filling curve is considered and the problem of generation of the infinite set of these patterns is investigated. A P system model in the bio-inspired area of membrane computing is constructed to generate the language of chain code kind of words that describe the approximating geometric patterns of Sierpiński curve.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Aftosmis, M.J., Berger, M.J., Murman, S.M.: Applications of space-filling curves to Cartesian methods for CFD. In: 42nd Aerospace Sciences Meeting and Exhibit, Reno, Nevada (2004)
Ali, M.A., Ladhake, S.A.: Overview of space-filling curves and their applications in scheduling. Int. J. Adv. Eng. Technol. 1(4), 148–154 (2011)
Bader, M.: Space-filling Curves - An Introduction with Applications in Scientific Computing. Texts in Computational Science and Engineering. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-31046-1
Bader, M., Schraufstetter, S., Vigh, C.A., Behrens, J.: Memory efficient adaptive mesh generation and implementation of multigrid algorithms using Sierpiński curves. Int. J. Comput. Sci. Eng. 4(1), 12–21 (2008)
Ceterchi, R., Subramanian, K.G., Venkat, I.: P systems with parallel rewriting for chain code picture languages. In: Beckmann, A., Mitrana, V., Soskova, M. (eds.) CiE 2015. LNCS, vol. 9136, pp. 145–155. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-20028-6_15
Dassow, J., Habel, A., Taubenberger, S.: Chain-code pictures and collages generated by hyperedge replacement. In: Cuny, J., Ehrig, H., Engels, G., Rozenberg, G. (eds.) Graph Grammars 1994. LNCS, vol. 1073, pp. 412–427. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61228-9_102
Drewes, F.: Some remarks on the generative power of collage grammars and chain-code grammars. In: Ehrig, H., Engels, G., Kreowski, H.-J., Rozenberg, G. (eds.) TAGT 1998. LNCS, vol. 1764, pp. 1–14. Springer, Heidelberg (2000). https://doi.org/10.1007/978-3-540-46464-8_1
Giammarresi, D., Restivo, A.: Two-dimensional languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 3, pp. 215–267. Springer, Heidelberg (1997). https://doi.org/10.1007/978-3-642-59126-6_4
He, Y., Li, L., Zhai, H., Dang, X., Liang, C.-H., Liu, Q.H.: Sierpinski space-filling curves and their application in high-speed circuits for ultrawideband SSN suppression. IEEE Antennas Wirel. Propag. Lett. 9, 568–571 (2010)
Hilbert, D.: Über die stetige Abbildung einer Linie auf ein Flächenstück. Math. Annln. 38, 459–460 (1891)
Kitaev, S., Mansour, T., Seebold, P.: The Peano curve and counting occurrences of some patterns. J. Autom. Lang. Combin. 9(4), 439–455 (2004)
Lawder, J.: The application of space-filling curves to the storage and retrieval of multi-dimensional data. Ph.D thesis. University of London (1999)
Maurer, H.A., Rozenberg, G., Welzl, E.: Using string languages to describe picture languages. Inform. Control 54, 155–185 (1982)
Pǎun, Gh.: Computing with membranes. J. Comput. Syst. Sci. 61, 108–143 (2000)
Păun, Gh.: Membrane Computing: An Introduction. Springer, Heidelbrg (2000). https://doi.org/10.1007/978-3-642-56196-2
Păun, Gh., Rozenberg, G., Salomaa, A.: The Oxford Handbook of Membrane Computing. Oxford University Press, New York (2010)
Peano, G.: Sur une courbe qui remplit toute une aire plane. Math. Annln. 36, 157–160 (1890)
Rozenberg, G., Salomaa, A. (eds.): Handbook of Formal Languages (3 Volumes). Springer, Berlin (1997). https://doi.org/10.1007/978-3-642-59126-6
Sagan, H.: Space-Filling Curves. Springer, New York (1994). https://doi.org/10.1007/978-1-4612-0871-6
Salomaa, A.: Formal Languages. Academic Press, London (1973)
Seebold, P.: Tag system for the Hilbert curve. Discrete Math. Theoret. Comput. Sci. 9, 213–226 (2007)
Sierpiński, W.: Sur une nouvelle courbe continnue qui remplit toute une aire plane. Bull. Acad. Sci. de Cracovie (Sci. math et nat.) Série A, 462–478 (1912)
Siromoney, R., Subramanian, K.G.: Space-filling curves and infinite graphs. In: Ehrig, H., Nagl, M., Rozenberg, G. (eds.) Graph Grammars 1982. LNCS, vol. 153, pp. 380–391. Springer, Heidelberg (1983). https://doi.org/10.1007/BFb0000120
Subramanian, K.G.: P systems and picture languages. In: Durand-Lose, J., Margenstern, M. (eds.) MCU 2007. LNCS, vol. 4664, pp. 99–109. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74593-8_9
Subramanian, K.G., Venkat, I., Pan, L.: P systems generating chain code picture languages. In: Proceedings of Asian Conference on Membrane Computing, pp. 115–123 (2012)
Zhang, G., Pérez-Jiménez, M.J., Gheorghe, M.: Real-life Applications with Membrane Computing. ECC, vol. 25. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-55989-6
Acknowledgement
The authors thank the reviewers for their useful comments which helped for a better presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Dharani, A., Stella Maragatham, R., Nagar, A.K., Subramanian, K.G. (2018). Chain Code P System for Generation of Approximation Patterns of Sierpiński Curve. In: Barneva, R., Brimkov, V., Tavares, J. (eds) Combinatorial Image Analysis. IWCIA 2018. Lecture Notes in Computer Science(), vol 11255. Springer, Cham. https://doi.org/10.1007/978-3-030-05288-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-05288-1_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-05287-4
Online ISBN: 978-3-030-05288-1
eBook Packages: Computer ScienceComputer Science (R0)