Tracking and approximating data matrices in streaming fashion is a fundamental challenge. The problem requires more care and attention when data comes from multiple distributed sites, each receiving a stream of data. This paper considers the problem of "tracking approximations to a matrix" in the distributed streaming model. In this model, there are m distributed sites each observing a distinct stream of data (where each element is a row of a distributed matrix) and has a communication channel with a coordinator, and the goal is to track an eps-approximation to the norm of the matrix along any direction. To that end, we present novel algorithms to address the matrix approximation problem. Our algorithms maintain a smaller matrix B, as an approximation to a distributed streaming matrix A, such that for any unit vector x: | ||A x||^2 - ||B x||^2 | <= eps ||A||_F^2. Our algorithms work in streaming fashion and incur small communication, which is critical for distributed computation. Our best method is deterministic and uses only O((m eps) log(beta N)) communication, where N is the size of stream (at the time of the query) and beta is an upper-bound on the squared norm of any row of the matrix. In addition to proving all algorithmic properties theoretically, extensive experiments with real large datasets demonstrate the efficiency of these protocols.
Yi and Zhang @cite_47 provided a deterministic algorithm with communication cost @math and @math space at each site to continuously track @math -heavy hitters and the @math -quantiles. In their method, every site and the coordinator have as many counters as the type of items plus one more counter for the total items. Every site keeps track of number of items it receives in each round, once this number reaches roughly @math times of the total counter at the coordinator, the site sends the counter to the coordinator. After the coordinator receives @math such messages, it updates its counters and broadcasts them to all sites. Sites reset their counter values and continue to next round. To lower space usage at sites, they suggested using space-saving sketch @cite_10 . The authors also gave matching lower bounds on the communication costs for both problems, showing their algorithms are optimal in the deterministic setting.
@cite_47 @cite_10
1997642935 1970779762
We consider the problem of tracking heavy hitters and quantiles in the distributed streaming model. The heavy hitters and quantiles are two important statistics for characterizing a data distribution. Let A be a multiset of elements, drawn from the universe U= 1,ź,u . For a given 0≤ź≤1, the ź-heavy hitters are those elements of A whose frequency in A is at least ź|A|; the ź-quantile of A is an element x of U such that at most ź|A| elements of A are smaller than A and at most (1źź)|A| elements of A are greater than x. Suppose the elements of A are received at k remote sites over time, and each of the sites has a two-way communication channel to a designated coordinator, whose goal is to track the set of ź-heavy hitters and the ź-quantile of A approximately at all times with minimum communication. We give tracking algorithms with worst-case communication cost O(k ∈źlogn) for both problems, where n is the total number of items in A, and ∈ is the approximation error. This substantially improves upon the previous known algorithms. We also give matching lower bounds on the communication costs for both problems, showing that our algorithms are optimal. We also consider a more general version of the problem where we simultaneously track the ź-quantiles for all 0≤ź≤1. We propose an approximate integrated approach for solving both problems of finding the most popular k elements, and finding frequent elements in a data stream coming from a large domain. Our solution is space efficient and reports both frequent and top-k elements with tight guarantees on errors. For general data distributions, our top-k algorithm returns k elements that have roughly the highest frequencies; and it uses limited space for calculating frequent elements. For realistic Zipfian data, the space requirement of the proposed algorithm for solving the exact frequent elements problem decreases dramatically with the parameter of the distribution; and for top-k queries, the analysis ensures that only the top-k elements, in the correct order, are reported. The experiments, using real and synthetic data sets, show space reductions with hardly any loss in accuracy. Having proved the effectiveness of the proposed approach through both analysis and experiments, we extend it to be able to answer continuous queries about frequent and top-k elements. Although the problems of incremental reporting of frequent and top-k elements are useful in many applications, to the best of our knowledge, no solution has been proposed.