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

Andreas Wiese, Evangelos Kranakis

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


We look at the problem of coloring locally specially constructed spanners of unit disk graphs. First we present a local approximation algorithm for the vertex coloring problem in Unit Disk Graphs (UDGs) which uses at most four times as many colors as an optimal solution requires. Next 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.

Original languageEnglish
Pages (from-to)555-588
Number of pages34
JournalDiscrete Mathematics, Algorithms and Applications
Issue number4
StatePublished - 1 Dec 2009
Externally publishedYes


  • Local algorithms
  • approximation algorithms
  • coloring
  • spanner
  • unit disk graphs


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