 NSFPAR ID:
 10464817
 Date Published:
 Journal Name:
 Journal of the ACM
 ISSN:
 00045411
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Estimating ranks, quantiles, and distributions over streaming data is a central task in data analysis and monitoring. Given a stream of n items from a data universe equipped with a total order, the task is to compute a sketch (data structure) of size polylogarithmic in n. Given the sketch and a query item y, one should be able to approximate its rank in the stream, i.e., the number of stream elements smaller than or equal to y.more » « less

Estimating the εapproximate quantiles or ranks of a stream is a fundamental task in data monitoring. Given a stream x_1,..., x_n from a universe \mathcalU with total order, an additiveerror quantile sketch \mathcalM allows us to approximate the rank of any query y\in \mathcalU up to additive ε n error. In 2001, Greenwald and Khanna gave a deterministic algorithm (GK sketch) that solves the εapproximate quantiles estimation problem using O(ε^1 łog(ε n)) space \citegreenwald2001space ; recently, this algorithm was shown to be optimal by Cormode and Vesleý in 2020 \citecormode2020tight. However, due to the intricacy of the GK sketch and its analysis, oversimplified versions of the algorithm are implemented in practical applications, often without any known theoretical guarantees. In fact, it has remained an open question whether the GK sketch can be simplified while maintaining the optimal space bound. In this paper, we resolve this open question by giving a simplified deterministic algorithm that stores at most (2 + o(1))ε^1 łog (ε n) elements and solves the additiveerror quantile estimation problem; as a side benefit, our algorithm achieves a smaller constant factor than the \frac11 2 ε^1 łog(ε n) space bound in the original GK sketch~\citegreenwald2001space. Our algorithm features an easier analysis and still achieves the same optimal asymptotic space complexity as the original GK sketch. Lastly, our simplification enables an efficient data structure implementation, with a worstcase runtime of O(łog(1/ε) + łog łog (ε n)) perelement for the ordinary εapproximate quantile estimation problem. Also, for the related weighted'' quantile estimation problem, we give efficient data structures for our simplified algorithm which guarantee a worstcase perelement runtime of O(łog(1/ε) + łog łog (ε W_n/w_\textrmmin )), achieving an improvement over the previous upper bound of \citeassadi2023generalizing.

An εapproximate quantile sketch over a stream of n inputs approximates the rank of any query point q—that is, the number of input points less than q—up to an additive error of εn, generally with some probability of at least 1−1/ poly(n), while consuming o(n) space. While the celebrated KLL sketch of Karnin, Lang, and Liberty achieves a provably optimal quantile approximation algorithm over worstcase streams, the approximations it achieves in practice are often far from optimal. Indeed, the most commonly used technique in practice is Dunning’s tdigest, which often achieves much better approximations than KLL on realworld data but is known to have arbitrarily large errors in the worst case. We apply interpolation techniques to the streaming quantiles problem to attempt to achieve better approximations on realworld data sets than KLL while maintaining similar guarantees in the worst case.more » « less

Given a data set of size n in d'dimensional Euclidean space, the kmeans problem asks for a set of k points (called centers) such that the sum of the l_2^2distances between the data points and the set of centers is minimized. Previous work on this problem in the local differential privacy setting shows how to achieve multiplicative approximation factors arbitrarily close to optimal, but suffers high additive error. The additive error has also been seen to be an issue in implementations of differentially private kmeans clustering algorithms in both the central and local settings. In this work, we introduce a new locally private kmeans clustering algorithm that achieves nearoptimal additive error whilst retaining constant multiplicative approximation factors and round complexity. Concretely, given any c>sqrt(2), our algorithm achieves O(k^(1 + O(1/(2c^21))) * sqrt(d' n) * log d' * poly log n) additive error with an O(c^2) multiplicative approximation factor.more » « less

We present the first nearlineartime algorithm that computes a (1+ε)approximation of the diameter of a weighted unitdisk graph of n vertices. Our algorithm requires O(n log^2 n) time for any constant ε>0, so we considerably improve upon the nearO(n^{3/2})time algorithm of Gao and Zhang (2005). Using similar ideas we develop (1+ε)approximate \emph{distance oracles} of O(1) query time with a likewise improvement in the preprocessing time, specifically from near O(n^{3/2}) to O(n log^3 n). We also obtain similar new results for a number of related problems in the weighted unitdisk graph metric such as the radius and the bichromatic closest pair. As a further application we employ our distance oracle, along with additional ideas, to solve the (1+ε)approximate \emph{allpairs boundedleg shortest paths\/} (apBLSP) problem for a set of n planar points. Our data structure requires O(n^2 log n) space, O(loglog n) query time, and nearly O(n^{2.579}) preprocessing time for any constant ε>0, and is the first that breaks the nearcubic preprocessing time bound given by Roditty and Segal (2011).more » « less