Dynamic conflict-free colorings in the plane

M.T. de Berg, A. Markovic

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

1 Citation (Scopus)

Abstract

We study dynamic conflict-free colorings in the plane, where the goal is to maintain a conflict-free coloring (CF-coloring for short) under insertions and deletions. - First we consider CF-colorings of a set S of unit squares with respect to points. Our method maintains a CF-coloring that uses O(log n) colors at any time, where n is the current number of squares in S, at the cost of only O(log n) recolorings per insertion or deletion We generalize the method to rectangles whose sides have lengths in the range [1, c], where c is a fixed constant. Here the number of used colors becomes O(log^2 n). The method also extends to arbitrary rectangles whose coordinates come from a fixed universe of size N, yielding O(log^2 N log^2 n) colors. The number of recolorings for both methods stays in O(log n). - We then present a general framework to maintain a CF-coloring under insertions for sets of objects that admit a unimax coloring with a small number of colors in the static case. As an application we show how to maintain a CF-coloring with O(log^3 n) colors for disks (or other objects with linear union complexity) with respect to points at the cost of O(log n) recolorings per insertion. We extend the framework to the fully-dynamic case when the static unimax coloring admits weak deletions. As an application we show how to maintain a CF-coloring with O(sqrt(n) log^2 n) colors for points with respect to rectangles, at the cost of O(log n) recolorings per insertion and O(1) recolorings per deletion. These are the first results on fully-dynamic CF-colorings in the plane, and the first results for semi-dynamic CF-colorings for non-congruent objects.
Original languageEnglish
Title of host publicationISAAC 2017 : 28th International Symposium on Algorithms and Computation
EditorsTakeshi Tokuyama, Yoshio Okamoto
Place of PublicationDagstuhl
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
Pages27:1-27:13
Volume92
ISBN (Electronic)9783959770545
ISBN (Print)978-3-95977-054-5
DOIs
Publication statusPublished - 2017
Event28th International Symposium on Algorithms and Computation (ISAAC 2017) - Phuket, Thailand
Duration: 9 Dec 201712 Dec 2017
Conference number: 28
http://aiat.in.th/isaac2017/

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume92
ISSN (Print)1868-8969

Conference

Conference28th International Symposium on Algorithms and Computation (ISAAC 2017)
Abbreviated titleISAAC 2017
CountryThailand
CityPhuket
Period9/12/1712/12/17
Internet address

Keywords

  • Conflict-free colorings
  • Dynamic data structures

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