Fiat, Amos ; Levy, Meital B ; Matoušek, Jiří ; Pach, Elchanan M ; Sharir, Micha ; Smorodinsky, Shakhar ; Wagner, UliIST Austria ; Welzl, Emo
We consider an 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, 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 Ω(√n) colors in the worst case. We then modify this approach, to obtain an efficient deterministic algorithm that uses a maximum of Θ(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 algorithm, requires an expected number of at most O(log n log log n) colors but always produces a valid coloring. We also analyze the performance of the simplest proposed algorithm when the points are inserted in a random 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 may be required in the worst case. The average-case behavior for disks, and cases involving other planar ranges, are still open.
545 - 554
SODA: Symposium on Discrete Algorithms
Fiat A, Levy M, Matoušek J, et al. Online conflict-free coloring for intervals. In: SIAM; 2005:545-554. doi:10.1137/S0097539704446682
Fiat, A., Levy, M., Matoušek, J., Pach, E., Sharir, M., Smorodinsky, S., … Welzl, E. (2005). Online conflict-free coloring for intervals (pp. 545–554). Presented at the SODA: Symposium on Discrete Algorithms, SIAM. https://doi.org/10.1137/S0097539704446682
Fiat, Amos, Meital Levy, Jiří Matoušek, Elchanan Pach, Micha Sharir, Shakhar Smorodinsky, Uli Wagner, and Emo Welzl. “Online Conflict-Free Coloring for Intervals,” 545–54. SIAM, 2005. https://doi.org/10.1137/S0097539704446682.
A. Fiat et al., “Online conflict-free coloring for intervals,” presented at the SODA: Symposium on Discrete Algorithms, 2005, pp. 545–554.
Fiat A, Levy M, Matoušek J, Pach E, Sharir M, Smorodinsky S, Wagner U, Welzl E. 2005. Online conflict-free coloring for intervals. SODA: Symposium on Discrete Algorithms 545–554.
Fiat, Amos, et al. Online Conflict-Free Coloring for Intervals. SIAM, 2005, pp. 545–54, doi:10.1137/S0097539704446682.