Abstract
We introduce the fully-dynamic conflict-free coloring problem for a set [Formula: see text] of intervals in [Formula: see text] with respect to points, where the goal is to maintain a conflict-free coloring for [Formula: see text] under insertions and deletions. A coloring is conflict-free if for each point [Formula: see text] contained in some interval, [Formula: see text] is contained in an interval whose color is not shared with any other interval containing [Formula: see text]. 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 [Formula: see text] recolorings per update the worst-case number of colors is [Formula: see text], and that any strategy using [Formula: see text] colors needs [Formula: see text] recolorings; a coloring strategy that uses [Formula: see text] colors at the cost of [Formula: see text] recolorings, and another strategy that uses [Formula: see text] colors at the cost of [Formula: see text] 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.
Highlights
Consider a set S of fixed base stations that can be used for communication by mobile clients
Each base station has a transmission range, and a client can potentially communicate via that base station when it lies within the transmission range
Even et al [10] proved that any set of n disks in the plane admits a conflict-free coloring with O(log n) colors, and this bound is tight in the worst case
Summary
Consider a set S of fixed base stations that can be used for communication by mobile clients. Bar-Noy et al [2] considered the case where recolorings are allowed for each insertion They prove that for coloring points in the plane with respect to halfplanes, one can obtain a coloring with O(log n) colors in an online setting at the cost of O(n) recolorings in total. We are given a (dynamic) set S of intervals in R1, which we want to color such that for any point q ∈ R1 the set S(q) of intervals containing q contains an interval with a unique color This version of the problem can be used to model the case where the base stations are located along a highway, for instance, and 1-dimensional range and frequency assignment problems have already been studied in various settings [2, 7, 11]. Due to space constraints some proofs have been deferred to the full version [6]
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