Weakening the online adversary just enough to get optimal conflict-free colorings for intervals
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
Deterministic conflict-free coloring for intervals: From offline to online
ACM Transactions on Algorithms (TALG)
Online conflict-free coloring for halfplanes, congruent disks, and axis-parallel rectangles
ACM Transactions on Algorithms (TALG)
Dynamic Offline Conflict-Free Coloring for Unit Disks
Approximation and Online Algorithms
Conflict-free colourings of graphs and hypergraphs
Combinatorics, Probability and Computing
Online conflict-free colouring for hypergraphs
Combinatorics, Probability and Computing
Graph unique-maximum and conflict-free colorings
Journal of Discrete Algorithms
The potential to improve the choice: list conflict-free coloring for geometric hypergraphs
Proceedings of the twenty-seventh annual symposium on Computational geometry
Ordered coloring grids and related graphs
SIROCCO'09 Proceedings of the 16th international conference on Structural Information and Communication Complexity
Graph unique-maximum and conflict-free colorings
CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
Unique-maximum and conflict-free coloring for hypergraphs and tree graphs
SOFSEM'12 Proceedings of the 38th international conference on Current Trends in Theory and Practice of Computer Science
Conflict-Free Coloring of points on a line with respect to a set of intervals
Computational Geometry: Theory and Applications
Online conflict-free colorings for hypergraphs
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Hi-index | 0.01 |
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 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 $\Omega(\sqrt{n})$ colors in the worst case. We then derive two efficient variants of this simple algorithm. The first is deterministic and uses $O(\log^2 n)$ colors, and the second is randomized and uses $O(\log n)$ colors with high probability. We also show that the $O(\log^2 n)$ bound on the number of colors used by our deterministic algorithm is tight on the worst case. 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.