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DOI: 10.4230/LIPIcs.ISAAC.2017.26

Fully-Dynamic and Kinetic Conflict-Free Coloring of Intervals with Respect to Points

Authors: Mark de Berg, Tim Leijsen, Aleksandar Markovic, André van Renssen, Marcel Roeloffzen, and Gerhard Woeginger

Published in: LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

Abstract

We introduce the fully-dynamic conflict-free coloring problem for a set S of intervals in R^1 with respect to points, where the goal is to maintain a conflict-free coloring for S under insertions and deletions. A coloring is conflict-free if for each point p contained in some interval, p is contained in an interval whose color is not shared with any other interval containing p. We investigate trade-offs between the number of colors used and the number of intervals that are recolored upon insertion or deletion of an interval. Our results include: - a lower bound on the number of recolorings as a function of the number of colors, which implies that with O(1) recolorings per update the worst-case number of colors is Omega(log n/log log n), and that any strategy using O(1/epsilon) colors needs Omega(epsilon n^epsilon) recolorings; - a coloring strategy that uses O(log n) colors at the cost of O(log n) recolorings, and another strategy that uses O(1/epsilon) colors at the cost of O(n^epsilon/epsilon) recolorings; - stronger upper and lower bounds for special cases. We also consider the kinetic setting where the intervals move continuously (but there are no insertions or deletions); here we show how to maintain a coloring with only four colors at the cost of three recolorings per event and show this is tight.

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Mark de Berg, Tim Leijsen, Aleksandar Markovic, André van Renssen, Marcel Roeloffzen, and Gerhard Woeginger. Fully-Dynamic and Kinetic Conflict-Free Coloring of Intervals with Respect to Points. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 26:1-26:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{deberg_et_al:LIPIcs.ISAAC.2017.26,
  author =	{de Berg, Mark and Leijsen, Tim and Markovic, Aleksandar and van Renssen, Andr\'{e} and Roeloffzen, Marcel and Woeginger, Gerhard},
  title =	{{Fully-Dynamic and Kinetic Conflict-Free Coloring of Intervals with Respect to Points}},
  booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
  pages =	{26:1--26:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-054-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{92},
  editor =	{Okamoto, Yoshio and Tokuyama, Takeshi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.26},
  URN =		{urn:nbn:de:0030-drops-82683},
  doi =		{10.4230/LIPIcs.ISAAC.2017.26},
  annote =	{Keywords: Conflict-free colorings, Dynamic data structures, Kinetic data structures}
}

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Fully-Dynamic and Kinetic Conflict-Free Coloring of Intervals with Respect to Points

Abstract

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