Online conflict-free coloring for intervals

Amos Fiat, Meital Levy, Jiří Matoušek, Elchanan Mossel János Pach, Micha Sharir, Shakhar Smorodinsky, Uli Wagner, Emo Welzl

Research output: Contribution to conferencePaperpeer-review

16 Scopus citations

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.

Original languageEnglish
Pages545-554
Number of pages10
StatePublished - 1 Jul 2005
Externally publishedYes
EventSixteenth Annual ACM-SIAM Symposium on Discrete Algorithms - Vancouver, BC, United States
Duration: 23 Jan 200525 Jan 2005

Conference

ConferenceSixteenth Annual ACM-SIAM Symposium on Discrete Algorithms
Country/TerritoryUnited States
CityVancouver, BC
Period23/01/0525/01/05

ASJC Scopus subject areas

  • Software
  • Mathematics (all)

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