Approximating extent measures of points
Journal of the ACM (JACM)
A space-optimal data-stream algorithm for coresets in the plane
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Adaptive sampling for geometric problems over data streams
Computational Geometry: Theory and Applications
Proceedings of the twenty-fourth annual symposium on Computational geometry
An Almost Space-Optimal Streaming Algorithm for Coresets in Fixed Dimensions
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Faster core-set constructions and data-stream algorithms in fixed dimensions
Computational Geometry: Theory and Applications
PODS '12 Proceedings of the 31st symposium on Principles of Database Systems
Processing a large number of continuous preference top-k queries
SIGMOD '12 Proceedings of the 2012 ACM SIGMOD International Conference on Management of Data
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Given a set P of n points in Rd, an ε-kernel K ⊆ P approximates the directional width of P in every direction within a relative (1-ε) factor. In this paper we study the stability of ε-kernels under dynamic insertion and deletion of points to P and by changing the approximation factor ε. In the first case, we say an algorithm for dynamically maintaining a ε-kernel is stable if at most O(1) points change in K as one point is inserted or deleted from P. We describe an algorithm to maintain an ε-kernel of size O(1/ε(d-1)/2) in O(1/ε(d-1)/2 + log n) time per update. Not only does our algorithm maintain a stable ε-kernel, its update time is faster than any known algorithm that maintains an ε-kernel of size O(1/ε(d-1)/2). Next, we show that if there is an ε-kernel of P of size κ, which may be dramatically less than O(1/ε(d-1)/2), then there is an (ε/2)-kernel of P of size O(min{1/ε(d-1)/2, κ⌊d/2⌋ logd-2(1/ε)}). Moreover, there exists a point set P in Rd and a parameter ε 0 such that if every ε-kernel of P has size at least κ, then any (ε/2)-kernel of P has size Ω(κ⌊d/2⌋).