# Dispersion in unit disks

* Corresponding author
Abstract : We present two new approximation algorithms with (improved) constant ratios for selecting $n$ points in $n$ unit disks such that the minimum pairwise distance among the points is maximized. (I) A very simple $O(n \log{n})$-time algorithm with ratio 0.5110 for disjoint unit disks. In combination with an algorithm of Cabello \cite{Ca07}, it yields a $O(n^2)$-time algorithm with ratio of 0.4487 for dispersion in $n$ not necessarily disjoint unit disks. (II) A more sophisticated LP-based algorithm with ratio 0.6495 for disjoint unit disks that uses a linear number of variables and constraints, and runs in polynomial time. The algorithm introduces a novel technique which combines linear programming and projections for approximating distances. The previous best approximation ratio for disjoint unit disks was 1/2. Our results give a partial answer to an open question raised by Cabello \cite{Ca07}, who asked whether 1/2 could be improved.
Keywords :
Document type :
Conference papers
Domain :

Cited literature [8 references]

https://hal.inria.fr/inria-00455298
Contributor : Publications Loria <>
Submitted on : Wednesday, February 10, 2010 - 9:38:36 AM
Last modification on : Thursday, April 4, 2019 - 11:30:03 AM
Long-term archiving on: : Friday, June 18, 2010 - 7:50:13 PM

### File

dumitrescu.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : inria-00455298, version 1

### Citation

Adrian Dumitrescu, Minghui Jiang. Dispersion in unit disks. 27th International Symposium on Theoretical Aspects of Computer Science - STACS 2010, Inria Nancy Grand Est & Loria, Mar 2010, Nancy, France. pp.299-310. ⟨inria-00455298⟩

Record views