Analysing local algorithms in location-aware quasi-unit-disk graphs

  • Authors:

  • Marja Hassinen;Joel Kaasinen;Evangelos Kranakis;Valentin Polishchuk;Jukka Suomela;Andreas Wiese

  • Affiliations:

  • Helsinki Institute for Information Technology HIIT, University of Helsinki, P.O. Box 68, 00014, Finland;Helsinki Institute for Information Technology HIIT, University of Helsinki, P.O. Box 68, 00014, Finland;School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada;Helsinki Institute for Information Technology HIIT, University of Helsinki, P.O. Box 68, 00014, Finland;Helsinki Institute for Information Technology HIIT, University of Helsinki, P.O. Box 68, 00014, Finland;Institut für Mathematik, Technische Universität Berlin, Straíe des 17.Juni 136, 10623 Berlin, Germany

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2011

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Abstract

A local algorithm with local horizon r is a distributed algorithm that runs in r synchronous communication rounds; here r is a constant that does not depend on the size of the network. As a consequence, the output of a node in a local algorithm only depends on the input within r hops from the node. We give tight bounds on the local horizon for a class of local algorithms for combinatorial problems on unit-disk graphs (UDGs). Most of our bounds are due to a refined analysis of existing approaches, while others are obtained by suggesting new algorithms. The algorithms we consider are based on network decompositions guided by a rectangular tiling of the plane. The algorithms are applied to matching, independent set, graph colouring, vertex cover, and dominating set. We also study local algorithms on quasi-UDGs, which are a popular generalisation of UDGs, aimed at more realistic modelling of communication between the network nodes. Analysing the local algorithms on quasi-UDGs allows one to assume that the nodes know their coordinates only approximately, up to an additive error. Despite the localisation error, the quality of the solution to problems on quasi-UDGs remains the same as for the case of UDGs with perfect location awareness. We analyse the increase in the local horizon that comes along with moving from UDGs to quasi-UDGs.