A Tight Bound for Online Coloring of Disk Graphs

Abstract

Abstract. We present an improved upper bound on the competitiveness of the online coloring algorithm First-Fit in disk graphs which are graphs representing overlaps of disks on the plane. We also show that this bound is best possible for deterministic online coloring algorithms that do not use the disk representation of the input graph. We also present a related new lower bound for unit disk graphs.

OSERENA: a Coloring Algorithm Optimized for Dense Wireless Networks

“…In this section, we compare the behavior of OS-ERENA with the well-known centralized First Fit 3-hop node coloring [8]. More precisely, we compare the colors granted to nodes by both coloring algorithms.…”

“…Of crucial interest is the approximation ratio of coloring algorithms that is defined as the ratio of the number of colors obtained by the algorithm to the optimal number. A well-known coloring algorithm is First Fit [8] that sequentially assigns colors to nodes. Each node is colored with the first available color.…”

Local 7-coloring for planar subgraphs of unit disk graphs

“…For such graphs there have been several papers in the literature addressing the coloring problem. Among these it is worth mentioning the work by Marathe et al [13] (presenting an on-line coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs: the heuristic does not need a geometric representation of unit disk graphs which is used only in establishing the performance guarantees of the heuristics), Graf et al [9] (which improves on a result of Clark, Colbourn and Johnson (1990) and shows that the coloring problem for unit disk graphs remains NP-complete for any fixed number of colors k ≥ 3), Caragiannis et al [3] (which proves an improved upper bound on the competitiveness of the on-line coloring algorithm First-Fit in disk graphs which are graphs representing overlaps of disks on the plane) and Miyamoto et al [14] (which constructs multi-colorings of unit disk graphs represented on triangular lattice points).…”

Location Oblivious Distributed Unit Disk Graph Coloring

“…The currently greatest lower bound is 5/2, given by Caragiannis et al [2]. To prove a lower bound of b, we have to show that there exists a unit disk graph G for which there exists an ordering < of the nodes such that the number of colors used by the sequential coloring algorithm is at least b · χ(G).…”

“…The sequential coloring algorithm is the algorithm that colors the nodes of a graph in an arbitrary order, assigning to each node the lowest color that has not been assigned to one of its neighbors. In the literature, the greatest lower bound on the worst-case performance ratio of the sequential coloring algorithm over unit disk graphs is 5/2, by Caragiannis et al [2]. Therefore, it was unclear whether one slightly more complex algorithm with a performance ratio of three is better than the trivial sequential algorithm.…”