Dynamic Conflict-Free Colorings in the Plane

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 $S$ 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 of a square. 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.