### Samenvatting

We introduce the dynamic conflict-free coloring problem for a set S of intervals in R 1 with respect to points, where the goal is to maintain a conflict-free coloring for S under insertions and deletions. We investigate trade-offs between the number of colors used and the number of intervals that are recolored upon insertion or deletion of an interval. Our results include:

- a lower bound on the number of recolorings as a function of the number of colors, which implies that with O(1) recolorings per update the worst-case number of colors is Ω(logn/loglogn) , and that any strategy using O(1/ε) colors needs Ω(εn ε ) recolorings;

- a coloring strategy that uses O(logn) colors at the cost of O(logn) recolorings, and another strategy that uses O(1/ε) colors at the cost of O(n ε /ε) recolorings;

- stronger upper and lower bounds for special cases.

We also consider the kinetic setting where the intervals move continuously (but there are no insertions or deletions); here we show how to maintain a coloring with only four colors at the cost of three recolorings per event and show this is tight.

- a lower bound on the number of recolorings as a function of the number of colors, which implies that with O(1) recolorings per update the worst-case number of colors is Ω(logn/loglogn) , and that any strategy using O(1/ε) colors needs Ω(εn ε ) recolorings;

- a coloring strategy that uses O(logn) colors at the cost of O(logn) recolorings, and another strategy that uses O(1/ε) colors at the cost of O(n ε /ε) recolorings;

- stronger upper and lower bounds for special cases.

We also consider the kinetic setting where the intervals move continuously (but there are no insertions or deletions); here we show how to maintain a coloring with only four colors at the cost of three recolorings per event and show this is tight.

Originele taal-2 | Engels |
---|---|

Artikelnummer | 1701.03388 |

Aantal pagina's | 17 |

Tijdschrift | arXiv |

Nummer van het tijdschrift | 1701.03388 |

Status | Gepubliceerd - 2017 |

## Vingerafdruk Duik in de onderzoeksthema's van 'Dynamic and kinetic conflict-free coloring of intervals with respect to points'. Samen vormen ze een unieke vingerafdruk.

## Citeer dit

de Berg, M. T., Leijsen, T., van Renssen, A. M., Roeloffzen, M. J. M., Markovic, A., & Woeginger, G. (2017). Dynamic and kinetic conflict-free coloring of intervals with respect to points.

*arXiv*, (1701.03388), [1701.03388]. https://arxiv.org/abs/1701.03388