Tight Lower Bounds for Halfspace Range Searching

We establish two new lower bounds for the halfspace range searching problem: Given a set of n points in R-d, where each point is associated with a weight from a commutative semigroup, compute the semigroup sum of the weights of the points lying within any query halfspace. Letting m denote the space requirements, we prove a lower bound for general semigroups of (Omega) over tilde (n(1-1/(d+1))/m(1/(d+1))) and for integral semigroups of (Omega) over tilde (n/m(1/d)). Our lower bounds are proved in the semigroup arithmetic model. Neglecting logarithmic factors, our result for integral semigroups matches the best known upper bound due to Matousek. Our result for general semigroups improves upon the best known lower bound due to Bronnimann, Chazelle, and Pach. Moreover, Fonseca and Mount have shown that, given uniformly distributed points, halfspace range queries over idempotent semigroups can be answered in O(n(1-1/(d+ 1))/m(1/(d+ 1))) time in the semigroup arithmetic model. As our lower bounds are established for uniformly distributed point sets, it follows that they also resolve the computational complexity of halfspace range searching over idempotent semigroups in this important special case.