We consider an online version of the conflict-free coloring of a set of points on the line, where each newly inserted point nmst be assigned a color upon insertion, and at all times the coloring has to be conflict-free, in the sense that in every interval I there is a color that appears exactly once in I. We present several deterministic and randomized algorithms for achieving this goal, and analyze their performance, that is, the maximum number of colors that they need to use, as a function of the number n of inserted points. We first show that a natural and simple (deterministic) approach may perform rather poorly, requiring fl(x/~ ) colors in the worst case. We then modify this approach, to obtain an efficient deterministic algorithm that uses a maxinmm of O(log 2 n) colors. Next, we present two randomized solutions. The first algorithm requires an expected number of at most O(log 2 n) colors, and produces a coloring which is valid with high probability, and the second one, which is a variant of our efficient deterministic Mgorithm, requires an expected number of at most O(lognloglogn) colors but always produces a valid coloring. We also analyze the performance of the simplest proposed algorithm when the points are inserted in a randon1 order, and present an incomplete analysis that indicates that, with high probability, it uses only O(log n) colors. Finally, we show that in the extension of this problem to two dimensions, where the relevant ranges are disks, n colors -"
Chen et al. REF consider the online version of the conflict-free coloring of a set of points on the line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has to be conflict-free.
Online conflict-free coloring for intervals
SODA '05
Computer Science Mathematics