Space-time tradeoffs for approximate nearest neighbor searching
Journal of the ACM (JACM)
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
Approximate Halfspace Range Counting
SIAM Journal on Computing
Approximate static and continuous range search in mobile navigation
Proceedings of the 5th International Conference on Ubiquitous Information Management and Communication
Optimal area-sensitive bounds for polytope approximation
Proceedings of the twenty-eighth annual symposium on Computational geometry
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Given an n-element 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 η can be determined quickly. In ε-approximate range searching we assume that η is bounded, and the sum is required to include all the points that lie within η and may additionally include any of the points lying within distance ε⋅diam(η) of η’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 k-fold 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 ε. 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 an adverse effect on the complexity of the problem. In particular, we establish lower bounds on the worst-case complexity of approximate range searching in the semigroup arithmetic model for ranges consisting of d-dimensional unit hypercubes under rigid motions. We show that for arbitrary (including idempotent) semigroups and linear space, the query time is at least $\varOmega(1/{\varepsilon }^{d-2\sqrt{d}})$. In the case of integral semigroups we prove a tighter lower bound of Ω(1/ε d−2). These lower bounds nearly match existing upper bounds for arbitrary semigroups. In contrast, we show that the improvements offered by idempotence do apply to smooth convex ranges. We say that a range is smooth if at every boundary point there is an incident Euclidean sphere that lies entirely within the range whose radius is proportional to the range’s diameter. We show that for smooth ranges and idempotent semigroups, ε-approximate range queries can be answered in O(log n+(1/ε)(d−1)/2log (1/ε)) time using O(n/ε) space. We show that this is nearly tight by presenting a lower bound of Ω(log n+(1/ε)(d−1)/2). This bound is in the decision-tree model and holds irrespective of space.