## How to play a coloring game against a color-blind adversary (2006)

Venue: | In Proc. 22nd Annual ACM Symposium on Computational Geometry (SoCG 2006 |

Citations: | 11 - 2 self |

### BibTeX

@INPROCEEDINGS{Chen06howto,

author = {Ke Chen},

title = {How to play a coloring game against a color-blind adversary},

booktitle = {In Proc. 22nd Annual ACM Symposium on Computational Geometry (SoCG 2006},

year = {2006},

pages = {44--51}

}

### OpenURL

### Abstract

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(log n) 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. 1

### Citations

693 |
Online Computation and Competitive Analysis
- Borodin, El-Yaniv
- 1998
(Show Context)
Citation Context ...ous adversary must construct the input sequence in advance, while the adaptive adversary chooses the input sequence based on the online algorithm’s actions thus far. We refer the interested reader to =-=[BE98]-=-. As mentioned above, our randomized algorithms are considerably simpler than the previous algorithms of [FLM + 05, KS04]. The skeptical reader might assume that this holds because the adversary in ou... |

45 | Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks
- Even, Lotker, et al.
(Show Context)
Citation Context ...n for a client is a disk, and R is a set of disks in the plane. To use the network, the client needs a base station with a unique frequency within its reach. The problem was introduced by Even et al. =-=[ELRS03]-=-. They showed that one can find an assignment of O(log n) frequencies to the base stations which is conflict-free with respect to disks in the plane, and this is tight in the worst case. Har-Peled and... |

39 | Conflict-free coloring of points and simple regions in the plane
- Har-Peled, Smorodinsky
(Show Context)
Citation Context ... that one can find an assignment of O(log n) frequencies to the base stations which is conflict-free with respect to disks in the plane, and this is tight in the worst case. Har-Peled and Smorodinsky =-=[HS05]-=- extended those results by considering other cases, and provided some sufficient conditions for the CF chromatic ∗ Department of Computer Science, University of Illinois at Urbana-Champaign; 201 N. Go... |

27 |
On the chromatic number of some geometric hypergraphs
- Smorodinsky
(Show Context)
Citation Context ...the CF coloring problem was studied in [ELRS03, HS05], where the colors are assigned to the given ranges, so that for any point, the set of ranges that contain the point is conflict-free. Smorodinsky =-=[Smo06]-=- improved several results studied in [HS05] by providing deterministic coloring algorithms for those problems. Elbassioni and Mustafa [EM06] asked the following question: Given a set of points in the ... |

14 | Conflict-free colorings of shallow discs
- Alon, Smorodinsky
- 2006
(Show Context)
Citation Context ... points into the plane, can the CF chromatic number (with respect to axis-parallel rectangles) be decreased? They showed that this is possible under certain conditions. Recently, Alon and Smorodinsky =-=[AS06]-=- showed that the CF chromatic number (with respect to disks) is O � log 3 k � when each disk intersects at most k others. There is also interest in CF coloring in dynamic settings [FLM + 05, KS04], in... |

14 | Conflict-free colorings of rectangles ranges
- Elbassioni, Mustafa
- 2006
(Show Context)
Citation Context ...anges that contain the point is conflict-free. Smorodinsky [Smo06] improved several results studied in [HS05] by providing deterministic coloring algorithms for those problems. Elbassioni and Mustafa =-=[EM06]-=- asked the following question: Given a set of points in the plane, if one is allowed to insert extra points into the plane, can the CF chromatic number (with respect to axis-parallel rectangles) be de... |

10 |
Online CF coloring for halfplanes, congruent disks, and axis-parallel rectangles
- Kaplan, Sharir
- 2004
(Show Context)
Citation Context ...east 1 − 1/n c for an arbitrary constant c.) They also showed that online CF coloring of n points in the plane, with respect to arbitrary disks, requires n colors in the worst case. Kaplan and Sharir =-=[KS04]-=- considered online CF coloring of n points in the plane with respect to halfplanes, congruent disks, and nearly-equal axis-parallel rectangles. They presented a fairly involved randomized algorithm th... |

9 | Online conflict-free coloring for intervals - Fiat, Levy, et al. - 2005 |

6 | Conflict-free colorings, in Discrete and Computational Geometry — The Goodman-Pollack Festschrift - Pach, Tóth - 2003 |