Abstract
In the cellular automata domain, the discrete convex hull computation rules proposed until now only deal with a connected set of seeds in infinite space, or with distant set of seeds in finite space. We present a cellular automata rule that constructs the discrete convex hull of arbitrary set of seeds in infinite spaces. The rule is expressed using intrinsic and general properties of the cellular spaces, considering them as metric spaces. In particular, this rule is a direct application of metric Gabriel graphs. This allows the rule and its components to be used on all common 2D and 3D grids used in cellular automata.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Adamatzky, A.: Computing in nonlinear media and automata collectives. IOP Publishing Ltd., Bristol (2001)
Aurenhammer, F.: Voronoi diagrams - a survey of a fundamental geometric data structure. ACM Comput. Surv. 23(3), 345–405 (1991)
Chan, T.M.: Optimal output-sensitive convex hull algorithms in two and three dimensions. Discrete & Computational Geometry 16, 361–368 (1996)
Clarridge, A.G., Salomaa, K.: An improved cellular automata based algorithm for the 45-convex hull problem. Journal of Cellular Automata (2008)
Gabriel, R.K., Sokal, R.R.: A new statistical approach to geographic variation analysis. Systematic Zoology 18(3), 259–278 (1969)
Graham, R.L.: An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters 1(4), 132 – 133 (1972), http://dx.doi.org/10.1016/0020-0190 (72)90045-2
Ilachinski, A.: Cellular Automata: A Discrete Universe. World Scientific Publishing Co., Inc., River Edge (2001)
Jarvis, R.A.: On the identification of the convex hull of a finite set of points in the plane. Information Processing Letters 2(1), 18 (1973), http://dx.doi.org/10.1016/0020-0190 (73)90020-3
Kirkpatrick, D.G., Seidel, R.: The ultimate planar convex hull algorithm? SIAM Journal on Computing 15(1), 287–299 (1986), http://link.aip.org/link/?SMJ/15/287/1
Levi, F.: On helly’s theorem and the axioms of convexity. J. of Indian Math. Soc., 65–76 (1951)
Maignan, L., Gruau, F.: Integer gradient for cellular automata: Principle and examples. In: IEEE International Conference on Self-Adaptive and Self-Organizing Systems Workshops, pp. 321–325 (2008)
Maignan, L., Gruau, F.: Gabriel graphs in arbitrary metric space and their cellular automaton for many grids. ACM Trans. Auton. Adapt. Syst. (2010) (in press)
Singer, I.: Abstract convex analysis. John Wiley & Sons Inc., Chichester (1997)
Torbey, S., Akl, S.G.: An exact and optimal local solution to the two-dimensional convex hull of arbitrary points problem. Journal of Cellular Automata (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Maignan, L., Gruau, F. (2010). Convex Hulls on Cellular Automata. In: Bandini, S., Manzoni, S., Umeo, H., Vizzari, G. (eds) Cellular Automata. ACRI 2010. Lecture Notes in Computer Science, vol 6350. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15979-4_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-15979-4_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15978-7
Online ISBN: 978-3-642-15979-4
eBook Packages: Computer ScienceComputer Science (R0)