ACM Computing Surveys (CSUR)
Algorithmic geometry
Space-Time Tradeoffs for Emptiness Queries
SIAM Journal on Computing
Computational Geometry: Theory and Applications
Space-efficient approximate Voronoi diagrams
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Linear-size approximate voronoi diagrams
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
A Replacement for Voronoi Diagrams of Near Linear Size
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Space-time tradeoffs for approximate spherical range counting
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
The effect of corners on the complexity of approximate range searching
Proceedings of the twenty-second annual symposium on Computational geometry
Space-time tradeoffs for approximate nearest neighbor searching
Journal of the ACM (JACM)
Enclosing weighted points with an almost-unit ball
Information Processing Letters
Approximate range searching: The absolute model
Computational Geometry: Theory and Applications
Tight lower bounds for halfspace range searching
Proceedings of the twenty-sixth annual symposium on Computational geometry
A unified approach to approximate proximity searching
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
approximate range searching: the absolute model
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
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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 η can be determined quickly. In the approximate version of the problem we assume that η is bounded, and we are given an approximation parameter ε 0. We are to determine the semigroup sum of all the points contained within η and may additionally include any of the points lying within distance ε • diam(η) of η's boundar.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, +) is integral. To date, all upper and lower bounds hold irrespective of the semigroup. We show that semigroup properties do indeed make a difference for both exact and approximate range searching, and in the case of approximate range searching the differences are dramatic.First, we consider exact halfspace range searching. The assumption that the semigroup is integral allows us to improve the best lower bounds in the semigroup arithmetic model. For example, assuming O(n) storage in the plane and ignoring polylog factors, we provide an Ω*(n2/5) lower bound for integral semigroups, improving upon the best lower bound of Ω*(n1/3), thus closing the gap with the O(n1/2) upper bound.We also consider approximate range searching for Euclidean ball ranges. We present lower bounds and nearly matching upper bounds for idempotent semigroups. We also present lower bounds for range searching for integral semigroups, which nearly match existing upper bounds. These bounds show that the advantages afforded by idempotency can result in major improvements. In particular, assuming roughly linear space, the exponent in the ε-dependencies is smaller by a factor of nearly 1/2. All our results are presented in terms of space-time tradeoffs, and our lower and upper bounds match closely throughout the entire spectrum.To our knowledge, our results provide the first proof that semigroup properties affect the computational complexity of range searching in the semigroup arithmetic model. These are the first lower bound results for any approximate geometric retrieval problems. The existence of nearly matching upper bounds, throughout the range of space-time tradeoffs, suggests that we are close to resolving the computational complexity of both idempotent and integral approximate spherical range searching in the semigroup arithmetic model.