Lower bounds for orthogonal range searching: part II. The arithmetic model

Lower bounds on the complexity of orthogonal range searching in thestatic case are established. Specifically, we consider the followingdominance search problem: Given a collection of<?Pub Fmt italic>n<?Pub Fmt /italic> weighted points in<?Pub Fmt italic>d<?Pub Fmt /italic>-space and a query point<?Pub Fmt italic>q<?Pub Fmt /italic>, compute the cumulative weight ofthe points dominated (in all coordinates) by<?Pub Fmt italic>q<?Pub Fmt /italic>. It is assumed that the weights arechosen in a commutative semigroup and that the query time measures onlythe number of arithmetic operations needed to compute the answer. It isproved that if <?Pub Fmt italic>m<?Pub Fmt /italic> units of storage areavailable, then the query time is at least proportional to (log<?Pub Fmt italic>n<?Pub Fmt /italic>/log(2<?Pub Fmt italic>m<?Pub Fmt /italic>/<?Pub Fmt italic>n<?Pub Fmt /italic>))<supscrpt><?Pub Fmt italic>d<?Pub Fmt /italic>–<?Pub Caret1>1</supscrpt>in both the worst and average cases. This lower bound is provably tightfor <?Pub Fmt italic>m<?Pub Fmt /italic> =&OHgr;(<?Pub Fmt italic>n<?Pub Fmt /italic>(log<?Pub Fmt italic>n<?Pub Fmt /italic>)<supscrpt><?Pub Fmt italic>d<?Pub Fmt /italic>–1+ε)</supscrpt> and any fixed ε> 0. A lower bound of &OHgr;(<?Pub Fmt italic>n<?Pub Fmt /italic>/loglog<?Pub Fmt italic>n<?Pub Fmt /italic>)<supscrpt><?Pub Fmt italic>d<?Pub Fmt /italic></supscrpt>)on the time required for executing <?Pub Fmt italic>n<?Pub Fmt /italic>inserts and queries is also established. <abstrbyl><italic>—Author's Abstract</italic>