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

Abstract

Lower bounds on the complexity of orthogonal range searching in the static case are established. Specifically, we consider the following dominance search problem: Given a collection of n weighted points in d -space and a query point q , compute the cumulative weight of the points dominated (in all coordinates) by q . It is assumed that the weights are chosen in a commutative semigroup and that the query time measures only the number of arithmetic operations needed to compute the answer. It is proved that if m units of storage are available, then the query time is at least proportional to (log n /log(2 m / n )) d –*1 in both the worst and average cases. This lower bound is provably tight for m = Ω( n (log n ) d –1+ϵ) and any fixed ϵ > 0. A lower bound of Ω( n /log log n ) d ) on the time required for executing n inserts and queries is also established. —Author's Abstract