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Gabriel Graphs in Arbitrary Metric Space and their Cellular Automaton for Many Grids

Published:01 June 2011Publication History
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Abstract

Gabriel graphs are subgraphs of Delaunay graphs that are used in many domains such as sensor networks and computer graphics. Although very useful in their original form, their definition is bounded to applications involving Euclidean spaces only, but their principles seem to be applicable to a wider range of applications. In this article, we generalize this construct and define metric Gabriel graphs that transport the principles of Gabriel graphs on arbitrary metric space, allowing their use in domains like cellular automata and amorphous computing, or any other domains where a non-Euclidean metric is used. We study global/local properties of metric Gabriel graphs and use them to design a cellular automaton that draws the metric Gabriel graph of its input. This cellular automaton only uses seven states to achieve this goal and has been tested on hexagonal grids, 4-connected, and 8-connected square grids.

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              • Published in

                cover image ACM Transactions on Autonomous and Adaptive Systems
                ACM Transactions on Autonomous and Adaptive Systems  Volume 6, Issue 2
                June 2011
                106 pages
                ISSN:1556-4665
                EISSN:1556-4703
                DOI:10.1145/1968513
                Issue’s Table of Contents

                Copyright © 2011 ACM

                Publisher

                Association for Computing Machinery

                New York, NY, United States

                Publication History

                • Published: 1 June 2011
                • Accepted: 1 June 2010
                • Revised: 1 April 2010
                • Received: 1 August 2009
                Published in taas Volume 6, Issue 2

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