Online conflict-free colorings for hypergraphs

Abstract

We provide a framework for online conflict-free coloring (CFcoloring) of any hypergraph. We use this framework to obtain an efficient randomized online algorithm for CF-coloring any k-degenerate hypergraph. Our algorithm uses O(k log n) colors with high probability and this bound is asymptotically optimal for any constant k. Moreover, our algorithm uses O(k log k log n) random bits with high probability. As a corollary, we obtain asymptotically optimal randomized algorithms for online CF-coloring some hypergraphs that arise in geometry. Our algorithm uses exponentially fewer random bits compared to previous results. We introduce deterministic online CF-coloring algorithms for points on the line with respect to intervals and for points on the plane with respect to halfplanes (or unit discs) that use ⊖(logn) colors and recolor O(n) points in total. © Springer-Verlag Berlin Heidelberg 2007.