Halfspace range search: an algorithmic application of K-sets

Authors:
Bernard Chazelle;Franco P. Preparata
Affiliations:
Department of Computer Science, Brown University, Providence, RI;Coordinated Science Laboratory, University of Illinois, Urbana-Champaign, IL
Venue:
SCG '85 Proceedings of the first annual symposium on Computational geometry
Year:
1985

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Abstract

Given a fixed set S of n points in E3 and a query plane &pgr;, the halfspace range search problem asks for the retrieval of all points of S on a chosen side of &pgr;. We prove that with &Ogr;(n(log n)3(log log n)4) storage it is possible to solve this problem in &Ogr;(&kgr; + log n) time, where &kgr; is the number of points to be reported. This result rests crucially on a new combinatorial derivation. We show that the maximum number of &kgr;-sets realized by a set of n points in E3 is Ogr;(nkc) for a small positive constant c; a &kgr;-set is any subset of S of size &kgr; which can be separated from the rest of S by a plane. Incidentally, this result constitutes the only nontrivial upper bound, as a function of n and &kgr;, known to date.