Given @math distributed data streams @math , we consider the problem of estimating the number of unique identifiers in streams defined by set expressions over @math . We identify a broad class of algorithms for solving this problem, and show that the estimators output by any algorithm in this class are perfectly unbiased and satisfy strong variance bounds. Our analysis unifies and generalizes a variety of earlier results in the literature. To demonstrate its generality, we describe several novel sampling algorithms in our class, and show that they achieve a novel tradeoff between accuracy, space usage, update speed, and applicability.
Tirthapura and Woodruff @cite_12 give sketching algorithms for estimating @math queries for a special class of properties @math . Specifically, they consider streams that contain tuples of the form @math , where @math is a numerical parameter, and the subpopulation @math is specified via a lower or upper bound on @math .
@cite_12
2963274201 1995437102 2078764670 2950897836
This paper resolves one of the longest standing basic problems in the streaming computational model. Namely, optimal construction of quantile sketches. An e approximate quantile sketch receives a stream of items x1,·,xn and allows one to approximate the rank of any query item up to additive error e n with probability at least 1-δ.The rank of a query x is the number of stream items such that xi ≤ x. The minimal sketch size required for this task is trivially at least 1 e.Felber and Ostrovsky obtain a O((1 e)log(1 e)) space sketch for a fixed δ.Without restrictions on the nature of the stream or the ratio between e and n, no better upper or lower bounds were known to date. This paper obtains an O((1 e)log log (1 δ)) space sketch and a matching lower bound. This resolves the open problem and proves a qualitative gap between randomized and deterministic quantile sketching for which an Ω((1 e)log(1 e)) lower bound is known. One of our contributions is a novel representation and modification of the widely used merge-and-reduce construction. This modification allows for an analysis which is both tight and extremely simple. The same technique was reported, in private communications, to be useful for improving other sketching objectives and geometric coreset constructions. There is growing interest in algorithms for processing and querying continuous data streams (i.e., data seen only once in a fixed order) with limited memory resources. In its most general form, a data stream is actually an update stream, i.e., comprising data-item deletions as well as insertions. Such massive update streams arise naturally in several application domains (e.g., monitoring of large IP network installations or processing of retail-chain transactions). Estimating the cardinality of set expressions defined over several (possibly distributed) update streams is perhaps one of the most fundamental query classes of interest; as an example, such a query may ask “what is the number of distinct IP source addresses seen in passing packets from both router R1 and R 2 but not router R3?”. Earlier work only addressed very restricted forms of this problem, focusing solely on the special case of insert-only streams and specific operators (e.g., union). In this paper, we propose the first space-efficient algorithmic solution for estimating the cardinality of full-fledged set expressions over general update streams. Our estimation algorithms are probabilistic in nature and rely on a novel, hash-based synopsis data structure, termed ”2-level hash sketch”. We demonstrate how our 2-level hash sketch synopses can be used to provide low-error, high-confidence estimates for the cardinality of set expressions (including operators such as set union, intersection, and difference) over continuous update streams, using only space that is significantly sublinear in the sizes of the streaming input (multi-)sets. Furthermore, our estimators never require rescanning or resampling of past stream items, regardless of the number of deletions in the stream. We also present lower bounds for the problem, demonstrating that the space usage of our estimation algorithms is within small factors of the optimal. Finally, we propose an optimized, time-efficient stream synopsis (based on 2-level hash sketches) that provides similar, strong accuracy-space guarantees while requiring only guaranteed logarithmic maintenance time per update, thus making our methods applicable for truly rapid-rate data streams. Our results from an empirical study of our synopsis and estimation techniques verify the effectiveness of our approach. In most algorithmic applications which compare two distributions, information theoretic distances are more natural than standard l p norms. In this paper we design streaming and sublinear time property testing algorithms for entropy and various information theoretic distances. posed the problem of property testing with respect to the Jensen-Shannon distance. We present optimal algorithms for estimating bounded, symmetric f-divergences (including the Jensen-Shannon divergence and the Hellinger distance) between distributions in various property testing frameworks. Along the way, we close a (log n) H gap between the upper and lower bounds for estimating entropy H, yielding an optimal algorithm over all values of the entropy. In a data stream setting (sublinear space), we give the first algorithm for estimating the entropy of a distribution. Our algorithm runs in polylogarithmic space and yields an asymptotic constant factor approximation scheme. An integral part of the algorithm is an interesting use of an F 0 (the number of distinct elements in a set) estimation algorithm; we also provide other results along the space time approximation tradeoff curve.Our results have interesting structural implications that connect sublinear time and space constrained algorithms. The mediating model is the random order streaming model, which assumes the input is a random permutation of a multiset and was first considered by Munro and Paterson in 1980. We show that any property testing algorithm in the combined oracle model for calculating a permutation invariant functions can be simulated in the random order model in a single pass. This addresses a question raised by regarding the relationship between property testing and stream algorithms. Further, we give a polylog-space PTAS for estimating the entropy of a one pass random order stream. This bound cannot be achieved in the combined oracle (generalized property testing) model. We undertake a systematic study of sketching a quadratic form: given an @math matrix @math , create a succinct sketch @math which can produce (without further access to @math ) a multiplicative @math -approximation to @math for any desired query @math . While a general matrix does not admit non-trivial sketches, positive semi-definite (PSD) matrices admit sketches of size @math , via the Johnson-Lindenstrauss lemma, achieving the "for each" guarantee, namely, for each query @math , with a constant probability the sketch succeeds. (For the stronger "for all" guarantee, where the sketch succeeds for all @math 's simultaneously, again there are no non-trivial sketches.) We design significantly better sketches for the important subclass of graph Laplacian matrices, which we also extend to symmetric diagonally dominant matrices. A sequence of work culminating in that of Batson, Spielman, and Srivastava (SIAM Review, 2014), shows that by choosing and reweighting @math edges in a graph, one achieves the "for all" guarantee. Our main results advance this front. @math For the "for all" guarantee, we prove that 's bound is optimal even when we restrict to "cut queries" @math . In contrast, previous lower bounds showed the bound only for spectral-sparsifiers . @math For the "for each" guarantee, we design a sketch of size @math bits for "cut queries" @math . We prove a nearly-matching lower bound of @math bits. For general queries @math , we construct sketches of size @math bits.