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.
This paper's Alpha Algorithm'' is inspired by the elegant Approximate Counting method of Morris @cite_13 , that has previously been applied to the estimation of the frequency moments @math , for @math . By contrast, our task is to estimate @math . The Alpha Algorithm is able to do this because its Approximate Counting process is tightly interleaved with another process that removes duplicates from the input stream while maintaining a small memory footprint by using feedback from the approximate counter.
@cite_13
2032665768 2952007101 2951542269 2439904216
In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals. Given data stream @math of size @math of numbers from @math , the frequency of @math is defined as @math . The @math -th of @math is defined as @math . We consider the problem of approximating frequency moments in insertion-only streams for @math . For any constant @math we show an @math upper bound on the space complexity of the problem. Here @math is the iterative @math function. To simplify the presentation, we make the following assumptions: @math and @math are polynomially far; approximation error @math and parameter @math are constants. We observe a natural bijection between streams and special matrices. Our main technical contribution is a non-uniform sampling method on matrices. We call our method a ; it samples a heavy element (i.e., element @math with frequency @math ) with probability @math and gives approximation @math . In addition, the estimations never exceed the real values, that is @math for all @math . As a result, we reduce the space complexity of finding a heavy element to @math bits. We apply our method of recursive sketches and resolve the problem with @math bits. We adapt a well known streaming algorithm for approximating item frequencies to the matrix sketching setting. The algorithm receives the rows of a large matrix @math one after the other in a streaming fashion. It maintains a sketch matrix @math such that for any unit vector @math [ |Ax |^2 |Bx |^2 |Ax |^2 - |A |_ f ^2 .] Sketch updates per row in @math require @math operations in the worst case. A slight modification of the algorithm allows for an amortized update time of @math operations per row. The presented algorithm stands out in that it is: deterministic, simple to implement, and elementary to prove. It also experimentally produces more accurate sketches than widely used approaches while still being computationally competitive. Approximated algorithms are often used to estimate the frequency of items on high volume, fast data streams. The most common ones are variations of Count-Min sketch, which use sub-linear space for the count, but can produce errors in the counts of the most frequent items and can misclassify low-frequency items. In this paper, we improve the accuracy of sketch-based algorithms by increasing the frequency estimation accuracy of the most frequent items and reducing the possible misclassification of low-frequency items, while also improving the overall throughput. Our solution, called Augmented Sketch (ASketch), is based on a pre-filtering stage that dynamically identifies and aggregates the most frequent items. Items overflowing the pre-filtering stage are processed using a conventional sketch algorithm, thereby making the solution general and applicable in a wide range of contexts. The pre-filtering stage can be efficiently implemented with SIMD instructions on multi-core machines and can be further parallelized through pipeline parallelism where the filtering stage runs in one core and the sketch algorithm runs in another core.