Halfspace range search: an algorithmic application of K-sets

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