Online conflict-free coloring for intervals Conference Paper


Author(s): Fiat, Amos; Levy, Meital B; Matoušek, Jiří; Pach, Elchanan M; Sharir, Micha; Smorodinsky, Shakhar; Wagner, Uli; Welzl, Emo
Title: Online conflict-free coloring for intervals
Affiliation
Abstract: We consider an online version of the conflict-free coloring of a set of points on the line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has to be conflict-free, in the sense that in every interval I there is a color that appears exactly once in I. We present several deterministic and randomized algorithms for achieving this goal, and analyze their performance, that is, the maximum number of colors that they need to use, as a function of the number n of inserted points. We first show that a natural and simple (deterministic) approach may perform rather poorly, requiring Ω(√n) colors in the worst case. We then modify this approach, to obtain an efficient deterministic algorithm that uses a maximum of Θ(log 2 n) colors. Next, we present two randomized solutions. The first algorithm requires an expected number of at most O(log 2 n) colors, and produces a coloring which is valid with high probability, and the second one, which is a variant of our efficient deterministic algorithm, requires an expected number of at most O(log n log log n) colors but always produces a valid coloring. We also analyze the performance of the simplest proposed algorithm when the points are inserted in a random order, and present an incomplete analysis that indicates that, with high probability, it uses only O(log n) colors. Finally, we show that in the extension of this problem to two dimensions, where the relevant ranges are disks, n colors may be required in the worst case. The average-case behavior for disks, and cases involving other planar ranges, are still open.
Keywords: Deterministic algorithms; Online conflict-free coloring; Planar range; Randomized solutions
Conference Title: SODA: Symposium on Discrete Algorithms
Conference Dates: January 23-25, 2005
Conference Location: Vancouver, BC, Canada
ISBN: 1557-9468
Publisher: SIAM  
Date Published: 2005-01-01
Start Page: 545
End Page: 554
DOI: 10.1137/S0097539704446682
Open access: no
IST Austria Authors
  1. Uli Wagner
    50 Wagner
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