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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.
Cite as

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-dev.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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