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.
@cite_18: This paper introduces a class of probabilistic counting algorithms with which one can estimate the number of distinct elements in a large collection of data (typically a large file stored on disk) in a single pass using only a small additional storage (typically less than a hundred binary words) and only a few operations per element scanned. The algorithms are based on statistical observations made on bits of hashed values of records. They are by construction totally insensitive to the replicative structure of elements in the file; they can be used in the context of distributed systems without any degradation of performances and prove especially useful in the context of data bases query optimisation.
@cite_14: A new class of algorithms to estimate the cardinality of very large multisets using constant memory and doing only one pass on the data is introduced here. It is based on order statistics rather than on bit patterns in binary representations of numbers. Three families of estimators are analyzed. They attain a standard error of 1M using M units of storage, which places them in the same class as the best known algorithms so far. The algorithms have a very simple internal loop, which gives them an advantage in terms of processing speed. For instance, a memory of only 12 kB and only few seconds are sufficient to process a multiset with several million elements and to build an estimate with accuracy of order 2 percent. The algorithms are validated both by mathematical analysis and by experimentations on real internet traffic.
@cite_4: Massive data sets often arise as physically distributed, parallel data streams. We present algorithms for estimating simple functions on the union of such data streams, while using only logarithmic space per stream. Each processor observes only its own stream, and communicates with the other processors only after observing its entire stream. This models the set-up in current network monitoring products. Our algorithms employ a novel coordinated sampling technique to extract a sample of the union; this sample can be used to estimate aggregate functions on the union. The technique can also be used to estimate aggregate functions over the distinct “labels” in one or more data streams, e.g., to determine the zeroth frequency moment (i.e., the number of distinct labels) in one or more data streams. Our space and time bounds are the best known for these problems, and our logarithmic space bounds for coordinated sampling contrast with polynomial lower bounds for independent sampling. We relate our distributed streams model to previously studied non-distributed (i.e., merged) streams models, presenting tight bounds on the gap between the distributed and merged models for deterministic algorithms.
@cite_22: The task of estimating the number of distinct values (DVs) in a large dataset arises in a wide variety of settings in computer science and elsewhere. We provide DV estimation techniques for the case in which the dataset of interest is split into partitions. We create for each partition a synopsis that can be used to estimate the number of DVs in the partition. By combining and extending a number of results in the literature, we obtain both suitable synopses and DV estimators. The synopses can be created in parallel, and can be easily combined to yield synopses and DV estimates for "compound" partitions that are created from the base partitions via arbitrary multiset union, intersection, or difference operations. Our synopses can also handle deletions of individual partition elements. We prove that our DV estimators are unbiased, provide error bounds, and show how to select synopsis sizes in order to achieve a desired estimation accuracy. Experiments and theory indicate that our synopses and estimators lead to lower computational costs and more accurate DV estimates than previous approaches.
@cite_21: 
@cite_23: Many datasets, including market basket data, text or hypertext documents, and events recorded in different locations or time periods, can be modeled as a collection of sets over a ground set of keys. Common queries over such data, including similarity or association rules are represented as the weight or selectivity of keys that satisfy some selection predicate defined over keys' attributes and memberships in particular sets. On massive data sets, exact computation of such aggregates can be inefficient or infeasible, and therefore, approximate queries are processed over sketches of the sets. Sketches based on coordinated random samples are scalable and flexible and well suited for many applications. Queries are resolved by producing a sketch of the union of sets used in the predicate from the sketches of these sets and then applying an estimator to this union-sketch. We derive novel tighter (unbiased) estimators that leverage sampled keys that are present in the union of applicable sketches but excluded from the union sketch. We establish analytically that our estimators dominate estimators applied to the union-sketch for all queries and data sets. Empirical evaluation on synthetic and real data reveals that on typical applications we can expect a 25 4 fold reduction in estimation error.
@cite_10: 
K'th Minimum Value sketches were introduced by Bar- @cite_10 , and developed into an unbiased scheme that handles set expressions by @cite_22 . Our own scheme is closely related to the schemes proposed and analyzed in Cohen and Kaplan @cite_23 , and in Gibbons and Tirthapura @cite_4 . Chen, Cao and Bu @cite_21 propose a somewhat different scheme for estimating unique counts with set expressions that is based on a data-structure related to the probabilistic counting'' sketches of @cite_18 , and also to the multi-bucket KMV sketches of @cite_14 (with @math ). However, the guarantees proved by @cite_21 are asymptotic in nature, and their system's union sketches are the same size as base sketches, and therefore do not provide the increased accuracy that is possible with a growing'' union rule as in @cite_23 , in @cite_4 , and in this paper's scheme.