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2006
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On the importance of idempotence

10 years 3 months ago
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...
Sunil Arya, Theocharis Malamatos, David M. Mount
Added 03 Dec 2009
Updated 03 Dec 2009
Type Conference
Year 2006
Where STOC
Authors Sunil Arya, Theocharis Malamatos, David M. Mount
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