How hard is half-space range searching?

We investigate the complexity of half-space range searching: given n points in d-space, build a data structure that allows us to determine efficiently how many points lie in a query half-space. We establish a tradeoff between the storage m and the worst-case query time t in the Fredman/Yao arithmetic model of computation. We show that t must be at least on the order of {Mathematical expression} Although the bound is unlikely to be optimal, it falls reasonably close to the recent upper bound of O(n/m ^1/d ) established by Matoušek. We also show that it is possible to devise a sequence of n inserts and half-space range queries that require a total time of n ^2-O(1/d) . Our results imply the first nontrivial lower bounds for spherical range searching in any fixed dimension. For example, they show that, with linear storage, circular range queries in the plane require Ω(n ^1/3) time (modulo a logarithmic factor).