In-Range Farthest Point Queries and Related Problem in High Dimensions
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Range-aggregate query is an important type of queries with numerous applications. It aims to obtain some structural information (defined by an aggregate function F(·)) of the points (from a point set P) inside a given query range B. In this paper, we study the range-aggregate query problem in high dimensional space for two aggregate functions: (1) F(P ∩ B) is the farthest point in P ∩ B to a query point q in ℝ^d and (2) F(P ∩ B) is the minimum enclosing ball (MEB) of P ∩ B. For problem (1), called In-Range Farthest Point (IFP) Query, we develop a bi-criteria approximation scheme: For any ϵ>0 that specifies the approximation ratio of the farthest distance and any γ>0 that measures the "fuzziness" of the query range, we show that it is possible to pre-process P into a data structure of size Õ_ϵ,γ(dn^1+ρ) in Õ_ϵ,γ(dn^1+ρ) time such that given any ℝ^d query ball B and query point q, it outputs in Õ_ϵ,γ(dn^ρ) time a point p that is a (1-ϵ)-approximation of the farthest point to q among all points lying in a (1+γ)-expansion B(1+γ) of B, where 0<ρ<1 is a constant depending on ϵ and γ and the hidden constants in big-O notations depend only on ϵ, γ and Polylog(nd). For problem (2), we show that the IFP result can be applied to develop query scheme with similar time and space complexities to achieve a (1+ϵ)-approximation for MEB.
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