Dynamic Conflict-Free Colorings in the Plane
Mark de Berg, Aleksandar Markovic

TL;DR
This paper introduces new dynamic conflict-free coloring algorithms for various geometric objects in the plane, achieving efficient recoloring and coloring bounds for insertions and deletions, including the first such results for certain object classes.
Contribution
It presents novel dynamic CF-coloring methods for unit squares, rectangles, disks, and arbitrary objects, with improved bounds and the first fully-dynamic solutions in the plane.
Findings
Maintains $O( ext{log} n)$ colors for unit squares with $O( ext{log} n)$ recolorings per update.
Extends to rectangles with side lengths in $[1,c]$, using $O( ext{log}^2 n)$ colors.
Provides the first fully-dynamic CF-colorings for points and rectangles in the plane.
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 of unit squares with respect to points. Our method maintains a CF-coloring that uses colors at any time, where is the current number of squares in , at the cost of only recolorings per insertion or deletion of a square. We generalize the method to rectangles whose sides have lengths in the range , where is a fixed constant. Here the number of used colors becomes . The method also extends to arbitrary rectangles whose coordinates come from a fixed universe of size , yielding colors. The number of recolorings for both methods stays in . - We then present a general framework to maintain…
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Taxonomy
TopicsAlgorithms and Data Compression · Genome Rearrangement Algorithms · Computational Geometry and Mesh Generation
