Convex hull of imprecise points in o(n log n) time after preprocessing
Proceedings of the twenty-seventh annual symposium on Computational geometry
Streaming and dynamic algorithms for minimum enclosing balls in high dimensions
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Convex hull of points lying on lines in o(nlogn) time after preprocessing
Computational Geometry: Theory and Applications
Streaming and dynamic algorithms for minimum enclosing balls in high dimensions
Computational Geometry: Theory and Applications
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We give a dynamic data structure that can maintain an ε-coreset of n points, with respect to the extent measure, in O(log n) time per update for any constant ε0 and any constant dimension. The previous method by Agarwal, Har-Peled, and Varadarajan requires polylogarithmic update time. For points with integer coordinates bounded by U, we alternatively get O(log log U) time. Numerous applications follow, for example, on dynamically approximating the width, smallest enclosing cylinder, minimum bounding box, or minimum-width annulus. We can also use the same approach to maintain approximate k-centers in time O(log n) (or O(log log U) if the spread is bounded by U) for any constant k and any constant dimension. For the smallest enclosing cylinder problem, we also show that a constant-factor approximation can be maintained in O(1) randomized amortized time on the word RAM.