Approximating the a;-level in three-dimensional plane arrangements

Sariel Har-Peled, Haim Kaplan, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contribution


Let H be a set of n non-vertical planes in three dimensions, and let r < n be a parameter. We give a simple alternative proof of the existence of a 0(1/r)-cutting of the first n/r levels of A(H), which consists of 0(r) semi-unbounded vertical triangular prisms. The same construction yields an approximation of the (n/r)-level by a terrain consisting of 0(r/ϵ3) triangular faces, which lies entirely between the levels (1 ± ϵ)n/r. The proof does not use sampling, and exploits techniques based on planar separators and various structural properties of levels in three-dimensional arrangements and of planar maps. The proof is constructive, and leads to a simple randomized algorithm, that computes the terrain in 0(n + r2ϵ-6 log3 r) expected time. An application of this technique allows us to mimic Matousek's construction of cuttings in the plane [36], to obtain a similar construction of "layered" (l/r)-cutting of the entire arrangement A(H), of optimal size 0(r3). Another application is a simplified optimal approximate range counting algorithm in three dimensions, competing with that of Afshani and Chan [1].

Original languageEnglish (US)
Title of host publication27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016
EditorsRobert Krauthgamer
PublisherAssociation for Computing Machinery
Number of pages20
ISBN (Electronic)9781510819672
StatePublished - 2016
Event27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 - Arlington, United States
Duration: Jan 10 2016Jan 12 2016

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Other27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016
Country/TerritoryUnited States

ASJC Scopus subject areas

  • Software
  • General Mathematics


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