Small-size ε-nets for axis-parallel rectangles and boxes

Boris Aronov, Esther Ezra, Micha Sharir

Research output: Contribution to journalArticlepeer-review

85 Scopus citations


We show the existence of ε-nets of size O (1/ε log log 1/ε) for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane and "fat" triangular ranges and for point sets in R3 and axis-parallel boxes; these are the first known nontrivial bounds for these range spaces. Our technique also yields improved bounds on the size of ε-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of ε-nets of size O (1/ε log log log 1/ε) for the dual range space of "fat" regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Brönnimann and Goodrich or of Even, Rawitz, and Shahar, we obtain improved approximation factors (computable in expected polynomial time by a randomized algorithm) for the HITTING SET or the SET COVER problems associated with the corresponding range spaces.

Original languageEnglish
Pages (from-to)3248-3282
Number of pages35
JournalSIAM Journal on Computing
Issue number7
StatePublished - 2010
Externally publishedYes


  • Exponential decay lemma
  • Geometric range spaces
  • ε-nets


Dive into the research topics of 'Small-size ε-nets for axis-parallel rectangles and boxes'. Together they form a unique fingerprint.

Cite this