How to play a coloring game against a color-blind adversary

摘要

We study the problem of conflict-free (CF) coloring of a set of points in the plane, in an online fashion, with respect to halfplanes, nearly-equal axis-parallel rectangles, and congruent disks. As a warm-up exercise, the online CF coloring of points on the line with respect to intervals is also considered. We present randomized algorithms in the oblivious adversary model, where the adversary does not see the colors used. For the problems considered, the algorithms always produce valid CF colorings, and use O(logn) colors with high probability (these bounds are optimal in the worst case). Our randomized online algorithms are considerably simpler than previous algorithms for this problem and use fewer colors.We also present a deterministic algorithm for the CF coloring of points in the plane with respect to nearly-equal axis-parallel rectangles, using O(polylog(n)) colors. This is the first efficient deterministic online CF coloring algorithm for this problem.