, Volume 1, Issue 1, pp 83-93

Halfspace range search: An algorithmic application ofk-sets

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Given a fixed setS ofn points inE 3 and a query planeπ, the halfspace range search problem asks for the retrieval of all points ofS on a chosen side ofπ. We prove that withO(n(logn)8 (loglogn)4) storage it is possible to solve this problem inO(k+logn) time, wherek is the number of points to be reported. This result rests crucially on a new combinatorial derivation. We show that the total number ofj-sets (j=1, ...,k) realized by a set ofn points inE 3 isO(nk 5); ak-set is any subset ofS of sizek which can be separated from the rest ofS by a plane.

Supported in part by NSF grants MCS 83-03925 and the Office of Naval Research and the Defense Advanced Research Projects Agency under contract N00014-83-K-0146 and ARPA Order No. 4786.
Supported in part by Joint Services Electronics Program under Contract N00014-79-C-0424.