Online conflict-free coloring for geometric hypergraphs

(i) We provide a framework for online conflict-free coloring (CF-coloring) 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. We obtain asymptotically optimal randomized algorithms for online CF-coloring some hypergraphs that arise in geometry and model an important version of the frequency assignment task for cellular networks. Our algorithms use exponentially fewer random bits compared to previous results for these special cases (O(log n) bits instead of Θ(n log n) bits). (ii) We initiate the study of deterministic online CFcoloring with recoloring. The goal is to use few colors, but also resort to recoloring as little as possible. We provide an algorithm for CF-coloring with respect to halfplanes using O(log n) colors and O(n) recolorings.