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Online Conflict-Free Colouring for Hypergraphs

Published online by Cambridge University Press:  11 December 2009

A. BAR-NOY ,
P. CHEILARIS ,
S. OLONETSKY  and
S. SMORODINSKY
A. BAR-NOY
Affiliation:
Computer and Information Science Department, Brooklyn College, USA (e-mail: amotz@sci.brooklyn.cuny.edu)
P. CHEILARIS
Affiliation:
Center for Advanced Studies in Mathematics, Ben-Gurion University, Israel (e-mail: panagiot@math.bgu.ac.il)
S. OLONETSKY
Affiliation:
School of Computer Science, Tel-Aviv University, Israel (e-mail: olonetsk@post.tau.ac.il)
S. SMORODINSKY
Affiliation:
Department of Mathematics, Ben-Gurion University, Israel (e-mail:shakhar@math.bgu.ac.il)
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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.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 4 , July 2010 , pp. 493 - 516
Copyright
Copyright © Cambridge University Press 2009

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