Local Construction and Coloring of Spanners of Location Aware Unit Disk Graphs

(Extended Abstract)
  • Andreas Wiese
  • Evangelos Kranakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5344)

Abstract

We investigate the problem of locally coloring and constructing special spanners of location aware Unit Disk Graphs (UDGs). First we present a local approximation algorithm for the vertex coloring problem in UDGs which uses at most four times as many colors as required by an optimal solution. Then we look at the colorability of spanners of UDGs. In particular we present a local algorithm for constructing a 4-colorable spanner of a unit disk graph. The output consists of the spanner and the 4-coloring. The computed spanner also has the properties that it is planar, the degree of a vertex in the spanner is at most 5 and the angles between two edges are at least π/3. By enlarging the locality distance (i.e. the size of the neighborhood which a vertex has to explore in order to compute its color) we can ensure the total weight of the spanner to be arbitrarily close to the weight of a minimum spanning tree.

We prove that a local algorithm cannot compute a bipartite spanner of a unit disk graph and therefore our algorithm needs at most one color more than any local algorithm for the task requires. Moreover, we prove that there is no local algorithm for 3-coloring UDGs or spanners of UDGs, even if the 3-colorability of the graph (or the spanner respectively) is guaranteed in advance.

Keywords

Approximation Ratio Minimum Span Tree Competitive Ratio Local Algorithm Locality Distance 
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.
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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Andreas Wiese
    • 1
  • Evangelos Kranakis
    • 2
  1. 1.Institut für MathematikTechnische Universität BerlinGermany
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada

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