Connected dominating sets on dynamic geometric graphs

  • Authors:
  • Leonidas Guibas;Nikola Milosavljević;Arik Motskin

  • Affiliations:
  • Computer Science Department, Gates Building, Stanford University, Stanford, CA 94305, USA;Institute of Formal Methods in Computer Science, University of Stuttgart, Universitätsstr. 38, D-70569 Stuttgart, Germany;Google Inc., 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2013

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Abstract

We propose algorithms for efficiently maintaining a constant-approximate minimum connected dominating set (MCDS) of a geometric graph under node insertions and deletions, and under node mobility. Assuming that two nodes are adjacent in the graph if and only if they are within a fixed geometric distance, we show that an O(1)-approximate MCDS of a graph in R^d with n nodes can be maintained in O(log^2^dn) time per node insertion or deletion. We also show that @W(n) time per operation is necessary to maintain exact MCDS. This lower bound holds even for d=1, even for randomized algorithms, and even when running time is amortized over a sequence of insertions/deletions, or over continuous motion. The crucial fact is that a single operation may affect the entire exact solution, while an approximate solution is affected only in a small neighborhood of the node that was inserted or deleted. In the approximate case, we show how to compute these local changes by a few range searching queries and a few bichromatic closest pair queries.