Dynamic Offline Conflict-Free Coloring for Unit Disks

Abstract

Abstract.A conflict-free coloring for a given set of disks is a coloring of the disks such that for any point p on the plane there is a disk among the disks covering p having a color different from that of the rest of the disks that covers p. In the dynamic offline setting, a sequence of disks is given, we have to color the disks one-by-one according to the order of the sequence and maintain the conflict-free property at any time for the disks that are colored. This paper focuses on unit disks, i.e., disks with radius one. We give an algorithm that colors a sequence of n unit disks in the dynamic offline setting using O(log n) colors. The algorithm is asymptotically optimal because Ω(log n) colors is necessary to color some set of n unit disks for any value of n [9].

Online Conflict-Free Colouring for Hypergraphs

“…In the end the vertices look like Observe that entries 3 and 5, respectively, do not have a vertex coloured with f(3) and f (5), respectively. As a consequence colours 3, 5 do not appear in the conflict-free colouring although colours 1, 2, 4, 6 do.…”

“…The main result of [14] is a general randomized algorithm which conflict-free colours any set of n 'simple' regions (not necessarily convex) whose union has 'low' complexity, using a 'small' number of colours. Many more variants and extensions were studied in several recent works: see, e.g., [1,5,6,8,9,10,11,14,16,18,20,21].…”