Introduction to algorithms
On Vertex Ranking for Permutations and Other Graphs
STACS '94 Proceedings of the 11th Annual Symposium on Theoretical Aspects of Computer Science
On conflict-free coloring of points and simple regions in the plane
Proceedings of the nineteenth annual symposium on Computational geometry
On the chromatic number of some geometric hypergraphs
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
ACM SIGACT News
Conflict-free colorings of shallow discs
Proceedings of the twenty-second annual symposium on Computational geometry
How to play a coloring game against a color-blind adversary
Proceedings of the twenty-second annual symposium on Computational geometry
Conflict-free coloring for intervals: from offline to online
Proceedings of the eighteenth annual ACM symposium on Parallelism in algorithms and architectures
Conflict-free coloring for rectangle ranges using O(n.382) colors
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Deterministic conflict-free coloring for intervals: From offline to online
ACM Transactions on Algorithms (TALG)
Online conflict-free colouring for hypergraphs
Combinatorics, Probability and Computing
Conflict-Free coloring made stronger
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
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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(log2 n) colors. Next, we present two randomized solutions. The first algorithm requires an expected number of at most O(log2 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.