Abstract
The recently introduced Broadcasting Automata model draws inspiration from a variety of sources such as Ad-Hoc radio networks, cellular automata, neighbourhood sequences and nature, employing many of the same pattern forming methods that can be seen in the superposition of waves and resonance. Algorithms for the broadcasting automata model are in the same vain as those encountered in distributed algorithms using a simple notion of waves, messages passed from automata to automata throughout the topology, to construct computations. The waves generated by activating processes in a digital environment can be used for designing a variety of wave algorithms. In this chapter we aim to study the geometrical shapes of informational waves on integer grid generated in broadcasting automata model as well as their potential use for metric approximation in a discrete space. An exploration of the ability to vary the broadcasting radius of each node leads to results of categorisations of digital discs, their form, composition, encodings and generation. Results pertaining to the nodal patterns generated by arbitrary transmission radii on the plane are exploredwith a connection to broadcasting sequences and approximation of discrete metrics of which results are given for the approximation of astroids, a previously unachievable concave metric, through a novel application of the aggregation of waves via a number of explored functions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bhowmick, P., Bhattacharya, B.B.: Number-theoretic interpretation and construction of a digital circle. Discrete Applied Mathematics 156(2), 2381–2399 (2008), http://www.sciencedirect.com/science/article/pii/S0166218X07004817 , doi:10.1016/j.dam.2007.10.022, ISSN 0166-218X
Bresenham, J.: A linear algorithm for incremental digital display of circular arcs. Commun. ACM 20(2), 100–106 (1977), http://doi.acm.org/10.1145/359423.359432 , doi:10.1145/359423.359432, ISSN 0001-0782
Chatterji, B.N., Das, P.P., Chakrabarti, P.P.: Generalized distances in digital geometry. Information Sciences 42, 51–67 (1987)
Conway, J.H.: Regular Algebra and Finite Machines. Dover Books on Mathematics Series. Dover Publications (2012), http://books.google.co.uk/books?id=1KAXc5TpEV8C , ISBN 9780486485836
Duncan, R.: A survey of parallel computer architectures. Computer 23(2), 5–16 (1990), doi:10.1109/2.44900, ISSN 0018-9162
Farkas, S., Bajk, J., Nagy, B.: Approximating the Euclidean circle in the square grid using neighbourhood sequences. Pure Math. Appl (PU.M.A.) 17, 309–322 (2006), ISSN 1218-4586
Farkas, S., Bajak, J., Nagy, B.: Approximating the Euclidean circle in the square grid using neighbourhood sequences. ArXiv e-prints (June 2010)
Feng, T.: A survey of interconnection networks. Computer 14(12), 12–27 (1981), doi:10.1109/C-M.1981.220290, ISSN 0018-9162
Feynman, R.P., Leighton, R.B., Sands, M.L.: The Feynman lectures on physics. Addison-Wesley World Student Series, vol. 1. Addison-Wesley Pub. Co. (1963), http://books.google.co.uk/books?id=_ZUfAQAAMAAJ
Freeman, H.: On the encoding of arbitrary geometric configurations. IRE Transactions on Electronic Computers, EC-10, 260–268 (1961), doi:10.1109/TEC.1961.5219197, ISSN 0367-9950
Gerhardt, M., Schuster, H., Tyson, J.J.: A cellular automaton model of excitable media: Ii. curvature, dispersion, rotating waves and meandering waves. Physica D: Nonlinear Phenomena 46(3), 392–415 (1990), http://www.sciencedirect.com/science/article/pii/016727899090101T , doi:10.1016/0167-2789(90)90101-T, ISSN 0167-2789
Hella, L., Järvisalo, M., Kuusisto, A., Laurinharju, J., Lempiäinen, T., Luosto, K., Suomela, J., Virtema, J.: Weak models of distributed computing, with connections to modal logic. CoRR, abs/1205.2051 (2012)
Hajdu, A.: Geometry of neighbourhood sequences. Pattern Recognition Letters 24(15), 2597–2606 (2003)
Hajdu, A., Hajdu, L.: Approximating the euclidean distance using non-periodic neighbourhood sequences. Discrete Mathematics 283(1-3), 101–111 (2004), http://www.sciencedirect.com/science/article/pii/S0012365X04001116 , doi:10.1016/j.disc.2003.12.016, ISSN 0012-365X
Kari, J.: Theory of cellular automata: a survey. Theor. Comput. Sci. 334, 3–33 (2005), http://dl.acm.org/citation.cfm?id=1083031.1083033 , doi:10.1016/j.tcs.2004.11.021, ISSN 0304-3975
Lee, G., Chong, N.Y.: A geometric approach to deploying robot swarms. Annals of Mathematics and Artificial Intelligence 52 257–280 (2008), http://dl.acm.org/citation.cfm?id=1527581.1527606 , doi: 10.1007/s10472-009-9125-x, ISSN 1012-2443
Lee, G., Yoon, S.: A mobile sensor network forming concentric circles through local interaction and consensus building. Journal of Robotics and Mechatronics 21, 469–477 (2009), ISSN 1883-8049
Linz, P.: An Introduction to Formal Languages and Automata. Theory of Computation Series. Jones and Bartlett (2001), http://books.google.co.uk/books?id=Cgooanwdo9AC , ISBN 9780763714222
Martin, R., Nickson, T., Potapov, I.: Geometric computations by broadcasting automata on the integer grid. In: Calude, C.S., Kari, J., Petre, I., Rozenberg, G. (eds.) UC 2011. LNCS, vol. 6714, pp. 138–151. Springer, Heidelberg (2011)
Martin, R., Nickson, T., Potapov, I.: Geometric computations by broadcasting automata. Natural Computing, 1–13 (2012), http://dx.doi.org/10.1007/s11047-012-9330-0 , doi:10.1007/s11047-012-9330-0, ISSN 1567-7818
Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics. Springer (2002), http://books.google.co.uk/books?id=MzFzCZAAk8MC ISBN 9780387953731
Mazoyer, J.: An overview of the firing squad synchronization problem. In: Choffrut, C. (ed.) Automata Networks. LNCS, vol. 316, pp. 82–94. Springer, Heidelberg (1988), http://dx.doi.org/10.1007/3-540-19444-4_16 , doi:10.1007/3-540-19444-4_16, ISBN 978-3-540-19444-6
Nagy, B.: Metric and non-metric distances on zn by generalized neighbourhood sequences. In: Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, ISPA 2005, pp. 215–220 (September 2005), doi:10.1109/ISPA.2005.195412
Nagy, B., Strand, R.: Approximating euclidean distance using distances based on neighbourhood sequences in non-standard three-dimensional grids. In: Reulke, R., Eckardt, U., Flach, B., Knauer, U., Polthier, K. (eds.) IWCIA 2006. LNCS, vol. 4040, pp. 89–100. Springer, Heidelberg (2006)
Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. In: Encyclopedia of Mathematics and Its Applications. Cambridge University Press (1993), http://books.google.co.uk/books?id=2QhT8UCKx2kC
Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences (1995), http://oeis.org/A001481A001481 ; Numbers that are the sum of 2 nonnegative squares
Suzuki, I., Yamashita, M.: Distributed anonymous mobile robots: Formation of geometric patterns. SIAM Journal on Computing 28, 1347–1363 (1999)
Thiaville, A.: Extensions of the geometric solution of the two dimensional coherent magnetization rotation model. Journal of Magnetism and Magnetic Materials 182(1-2), 5–18 (1998), http://www.sciencedirect.com/science/article/pii/S0304885397010147 , doi:10.1016/S0304-8853(97)01014-7, ISSN 0304-8853
Wolfram, S.: Universality and complexity in cellular automata. Physica D: Nonlinear Phenomena 10(1-2), 1–35 (1984), http://www.sciencedirect.com/science/article/pii/0167278984902458 , doi:10.1016/0167-2789(84)90245-8, ISSN 0167-2789
Yates, R.C.: Curves and their properties. Classics in mathematics education. National Council of Teachers of Mathematics (1974), http://books.google.co.uk/books?id=UPs-AAAAIAAJ
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Nickson, T., Potapov, I. (2015). Broadcasting Automata and Patterns on ℤ2 . In: Adamatzky, A. (eds) Automata, Universality, Computation. Emergence, Complexity and Computation, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-09039-9_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-09039-9_14
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09038-2
Online ISBN: 978-3-319-09039-9
eBook Packages: EngineeringEngineering (R0)