At SODA'10, Agarwal and Sharathkumar presented a streaming algorithm for approximating the minimum enclosing ball of a set of points in d-dimensional Euclidean space. Their algorithm requires one pass, uses O(d) space, and was shown to have approximation factor at most (1+sqrt{3})/2 + eps ~ 1.3661. We prove that the same algorithm has approximation factor less than 1.22, which brings us much closer to a (1+sqrt{2})/2 ~ 1.207 lower bound given by Agarwal and Sharathkumar.
We also apply this technique to the dynamic version of the minimum enclosing ball problem (in the non-streaming setting). We give an O(dn)-space data structure that can maintain a 1.22-approximate minimum enclosing ball in O(d log n) expected amortized time per insertion/deletion.
We present three results related to dynamic convex hulls:
Dynamic connectivity is a well-studied problem, but so far the most compelling progress has been confined to the edge-update model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two more challenging, yet equally fundamental problems:
Subgraph connectivity asks to maintain an understanding of connectivity under vertex updates: updates can turn vertices on and off, and queries refer to the subgraph induced by on vertices. (For instance, this is closer to applications in networks of routers, where node faults may occur.) We describe a data structure supporting vertex updates in O~(m^{2/3}) amortized time, where m denotes the number of edges in the graph. This greatly improves over the previous result [STOC'02], which required fast matrix multiplication and had an update time of O(m^{0.94}). The new data structure is also simpler.
Geometric connectivity asks to maintain a dynamic set of n geometric objects, and query connectivity in their intersection graph. (For instance, the intersection graph of balls describes connectivity in a network of sensors with bounded transmission radius.) Previously, nontrivial fully dynamic results were known only for special cases like axis-parallel line segments and rectangles. We provide similarly improved update times, O~(n^{2/3}), for these special cases. Moreover, we show how to obtain sublinear update bounds for virtually all families of geometric objects which allow sublinear-time range queries. In particular, we obtain the first sublinear update time for arbitrary 2D line segments: O~(n^{9/10}); for d-dimensional simplices: O~(n^{1-1/(d(2d+1))}); and for d-dimensional balls: O~(n^{1-1/((d+1)(2d+3))}).
We give a dynamic data structure that can maintain an epsilon-coreset of n points, with respect to the extent measure, in O(log n) time for any constant epsilon > 0 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 O(min{log n, log log U}) randomized amortized time 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.
In this paper we give a fully dynamic data structure to maintain the connectivity of the intersection graph of n axis-parallel rectangles. The amortized update time (insertion and deletion of rectangles) is O(n^{10/11} polylog n) and the query time (deciding whether two given rectangles are connected) is O(1). It slightly improves the update time (O(n^{0.94})) of the previous method while drastically reducing the query time (near O(n^{1/3})). Our method does not use fast matrix multiplication results and supports a wider range of queries.
We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O(log^3 n) expected amortized time, deletions take O(log^6 n) expected amortized time, and extreme-point queries take O(log^2 n) worst-case time. This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous O(n^epsilon)-time method by Agarwal and Matousek a decade ago. As a consequence, we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems (such as levels in three dimensions and dynamic Euclidean minimum spanning trees in the plane).
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