On the importance of idempotence

Range searching is among the most fundamental problems in computational geometry. An n-element point set in Rd is given along with an assignment of weights to these points from some commutative semigroup. Subject to a fixed space of possible range shapes, the problem is to preprocess the points so that the total semigroup sum of the points lying within a given query range can be determined quickly. In the approximate version of the problem we assume that is bounded, and we are given an approximation parameter > 0. We are to determine the semigroup sum of all the points contained within and may additionally include any of the points lying within distance ? diam() of 's boundary. In this paper we contrast the complexity of range searching based on semigroup properties. A semigroup (S, +) is idempotent if x + x = x for all x S, and it is integral if for all k 2, the k-fold sum x + ? ? ? + x is not equal to x. For example, (R, min) and ({0, 1}, ) are both idempotent, and (N...

• Type: Conference
• Year: 2006
• Where: STOC
• Authors: Sunil Arya, Theocharis Malamatos, David M. Mount
• Keywords: Algorithms, Semigroup Arithmetic Model, Semigroup Properties, STOC 2006, Total Semigroup Sum