We consider the problem of private computation of approximate Heavy Hitters. Alice and Bob each hold a vector and, in the vector sum, they want to find the B largest values along with their indices. While the exact problem requires linear communication, protocols in the literature solve this problem approximately using polynomial computation time, polylogarithmic communication, and constantly many rounds. We show how to solve the problem privately with comparable cost, in the sense that nothing is learned by Alice and Bob beyond what is implied by their input, the ideal top-B output, and goodness of approximation (equivalently, the Euclidean norm of the vector sum). We give lower bounds showing that the Euclidean norm must leak by any efficient algorithm.
(MATH) A vector A of length N is defined implicitly, via a stream of updates of the form "add 5 to A3." We give a sketching algorithm, that constructs a small sketch from the stream of updates, and a reconstruction algorithm, that produces a B-bucket piecewise-constant representation (histogram) H for A from the sketch, such that ||A—H||x(1+e)||A—Hopt|&#124, where the error ||A—H|| is either @math (absolute) or @math (root-mean-square) error. The time to process a single update, time to reconstruct the histogram, and size of the sketch are each bounded by poly(B,log(N),log||A,1 e. Our result is obtained in two steps. First we obtain what we call a robust histogram approximation for A, a histogram such that adding a small number of buckets does not help improve the representation quality significantly. From the robust histogram, we cull a histogram of desired accruacy and B buckets in the second step. This technique also provides similar results for Haar wavelet representations, under @math error. Our results have applications in summarizing data distributions fast and succinctly even in distributed settings.  The frequency moments of a sequence containingmielements of typei, 1?i?n, are the numbersFk=?ni=1mki. We consider the space complexity of randomized algorithms that approximate the numbersFk, when the elements of the sequence are given one by one and cannot be stored. Surprisingly, it turns out that the numbersF0,F1, andF2can be approximated in logarithmic space, whereas the approximation ofFkfork?6 requiresn?(1)space. Applications to data bases are mentioned as well. Most database management systems maintain statistics on the underlying relation. One of the important statistics is that of the "hot items" in the relation: those that appear many times (most frequently, or more than some threshold). For example, end-biased histograms keep the hot items as part of the histogram and are used in selectivity estimation. Hot items are used as simple outliers in data mining, and in anomaly detection in networking applications.We present a new algorithm for dynamically determining the hot items at any time in the relation that is undergoing deletion operations as well as inserts. Our algorithm maintains a small space data structure that monitors the transactions on the relation, and when required, quickly outputs all hot items, without rescanning the relation in the database. With user-specified probability, it is able to report all hot items. Our algorithm relies on the idea of "group testing", is simple to implement, and has provable quality, space and time guarantees. Previously known algorithms for this problem that make similar quality and performance guarantees can not handle deletions, and those that handle deletions can not make similar guarantees without rescanning the database. Our experiments with real and synthetic data shows that our algorithm is remarkably accurate in dynamically tracking the hot items independent of the rate of insertions and deletions. This paper presents algorithms for tracking (approximate) join and self-join sizes in limited storage, in the presence of insertions and deletions to the data set(s). Such algorithms detect changes in join and self-join sizes without an expensive recomputation from the base data, and without the large space overhead required to maintain such sizes exactly. Query optimizers rely on fast, high-quality estimates of join sizes in order to select between various join plans, and estimates of self-join sizes are used to indicate the degree of skew in the data. For self-joins, we consider two approaches proposed in N. (The space complexity of approximating the frequency moments, J. Comput. System Sci.58 (1999), 137?147), which we denote tug-of-war and sample-count . We present fast algorithms for implementing these approaches, and extensions to handle deletions as well as insertions. We also report on the first experimental study of the two approaches, on a range of synthetic and real-world data sets. Our study shows that tug-of-war provides more accurate estimates for a given storage limit than sample-count, which in turn is far more accurate than a standard sampling-based approach. For example, tug-of-war needed only 4-256 memory words, depending on the data set, in order to estimate the self-join size to within a 15 relative error; on average, this is over 4 times (50 times) fewer memory words than needed by sample-count (standard sampling, resp.) to obtain a similar accuracy. For joins, we propose schemes based on maintaining a small signature of each relation independently, such that join sizes can be quickly and accurately estimated between any pair of relations using only these signatures. We show that taking random samples for join signatures can lead to an inaccurate estimation unless the sample size is quite large; moreover, we show that no other signature scheme can significantly improve upon sampling without further assumptions. These negative results are shown to hold even in the presence of sanity bounds. On the other hand, we present a fast join signature scheme based on tug-of-war signatures that provides guarantees on join size estimation as a function of the self-join sizes of the joining relations; this scheme can significantly improve upon the sampling scheme.