 
Summary: The Effect of Corners on the Complexity of Approximate Range
Searching
Sunil Arya
Theocharis Malamatos
David M. Mount
January 31, 2007
Abstract
Given an nelement point set in Rd
, the range searching problem involves preprocessing these
points so that the total weight, or for our purposes the semigroup sum, of the points lying within
a given query range can be determined quickly. In approximate range searching we assume
that is bounded, and the sum is required to include all the points that lie within and may
additionally include any of the points lying within distance · diam() of 's boundary.
In this paper we contrast the complexity of approximate range searching based on properties
of the semigroup and range space. A semigroup (S, +) is idempotent if x + x = x for all
x S, and it is integral if for all k 2, the kfold sum x + · · · + x is not equal to x. Recent
research has shown that the computational complexity of approximate spherical range searching
is significantly lower for idempotent semigroups than it is for integral semigroups in terms of
the dependencies on . In this paper we consider whether these results can be generalized to
other sorts of ranges. We show that, as with integrality, allowing sharp corners on ranges has
