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Addressing, distances and routing in triangular systems with applications in cellular networks

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Abstract

Triangular systems are the subgraphs of the regular triangular grid which are formed by a simple circuit of the grid and the region bounded by this circuit. They are used to model cellular networks where nodes are base stations. In this paper, we propose an addressing scheme for triangular systems by employing their isometric embeddings into the Cartesian product of three trees. This embedding provides a simple representation of any triangular system with only three small integers per vertex, and allows to employ the compact labeling schemes for trees for distance queries and routing. We show that each such system with n vertices admits a labeling that assigns O(log 2 n) bit labels to vertices of the system such that the distance between any two vertices u and v can be determined in constant time by merely inspecting the labels of u and v, without using any other information about the system. Furthermore, there is a labeling, assigning labels of size O(log n) bits to vertices, which allows, given the label of a source vertex and the label of a destination, to compute in constant time the port number of the edge from the source that heads in the direction of the destination. These results are used in solving some problems in cellular networks. Our addressing and distance labeling schemes allow efficient implementation of distance and movement based tracking protocols in cellular networks, by providing information, generally not available to the user, and means for accurate cell distance determination. Our routing and distance labeling schemes provide elegant and efficient routing and connection rerouting protocols for cellular networks.

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Authors and Affiliations

  1. Laboratoire d’Informatique Fondamentale de Marseille, Université de la Méditerranée, Faculté des Sciences de Luminy, F-13288, Marseille Cedex 9, France

    Victor Chepoi & Yann Vaxès

  2. Department of Computer Science, Kent State University, Kent, Ohio, 44242, USA

    Feodor F. Dragan

Authors
  1. Victor Chepoi
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  2. Feodor F. Dragan
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  3. Yann Vaxès
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Correspondence to Feodor F. Dragan.

Additional information

Victor Chepoi received the M.S. degree in Applied Mathematics and Computer Science from Moldova State University, in 1983, and the PhD degree in Theoretical Computer Science from the Belorussian Academy of Sciences, in 1987. He was an Assistant and then an Associate Professor at the Mathematics and Computer Science Department of Moldova State University from 1987 to 1994. He was awarded the Alexander von Humboldt Shtiftung Fellowship from 1994 to 1995 at the University of Hamburg, Germany. During 1995 to 1997, he was a Visiting Professor at the Laboratoire de Biomathematiques, Universite de la Mediterranee, France. During 1998, he was a Fellow at SFB343 “Diskrete Strukturen in der Mathematik”, University of Bielefeld, Germany. Since September 1998 he has been a Professor of Computer Science at Faculte des Sciences de Luminy, Universite de la Maditerranee, France. His research interests include graph theory and combinatorics, design and analysis of network and graph algorithms, geometry and algorithmics of metric spaces, computational geometry, and approximation algorithms.

Feodor F. Dragan received the M.S. degree in Applied Mathematics and Computer Science from Moldova State University, in 1985, and the PhD degree in Theoretical Computer Science from the Belorussian Academy of Sciences, in 1990. He was an Assistant and then an Associate Professor at the Mathematics and Computer Science Department of Moldova State University from 1988 to 1999. From 1994 to 1999, he was on leave of absence and worked in Germany as a Research Associate on a Volkswagen Foundation (VW) project and on a German Research Community (DFG) project. He was also awarded a DAAD Research Fellowship (Germany) from 1994 to 1995. During 1999 to 2000, he was a Research Associate at the Computer Science Department of University of California, Los Angeles. Since August 2000 he has been with Kent State University and he is currently an Associate Professor of Computer Science. He has authored more than 70 refereed scientific publications. His research interests include design and analysis of network algorithms, algorithmic graph and hypergraph theory, computational geometry, VLSI CAD, and combinatorial optimization.

Yann Vaxes received the PhD degree in Computer Science from the Universite de la Mediterranee, in 1998. Then, he joined the Computer Science Department of this university as an Assistant Professor. His research interests include design and analysis of network algorithms, algorithmic graph theory and combinatorial optimization.

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Chepoi, V., Dragan, F.F. & Vaxès, Y. Addressing, distances and routing in triangular systems with applications in cellular networks. Wireless Netw 12, 671–679 (2006). https://doi.org/10.1007/s11276-006-6527-0

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  • DOI: https://doi.org/10.1007/s11276-006-6527-0

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