Online conflict-free colouring for hypergraphs

  • Authors:

  • A. Bar-noy;P. Cheilaris;S. Olonetsky;S. Smorodinsky

  • Affiliations:

  • Computer and information science department, brooklyn college, usa (e-mail: amotz@sci.brooklyn.cuny.edu);Center for advanced studies in mathematics, ben-gurion university, israel (e-mail: panagiot@math.bgu.ac.il);School of computer science, tel-aviv university, israel (e-mail: olonetsk@post.tau.ac.il);Department of mathematics, ben-gurion university, israel (e-mail:shakhar@math.bgu.ac.il)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2010

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Abstract

We provide a framework for online conflict-free colouring of any hypergraph. We introduce the notion of a degenerate hypergraph, which characterizes hypergraphs that arise in geometry. We use our framework to obtain an efficient randomized online algorithm for conflict-free colouring of any k-degenerate hypergraph with n vertices. Our algorithm uses O(k log n) colours with high probability and this bound is asymptotically optimal. Moreover, our algorithm uses O(k log k log n) random bits with high probability. We introduce algorithms that are allowed to perform a few recolourings of already coloured points. We provide deterministic online conflict-free colouring algorithms for points on the line with respect to intervals and for points on the plane with respect to half-planes (or unit disks) that use O(log n) colours and perform a total of at most O(n) recolourings.