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.
Babcock and Olston @cite_24 designed some deterministic heuristics called as to compute top- @math frequent items. Fuller and Kantardzid modified their technique and proposed @cite_33 , a heuristic method, to track the heavy hitters while reducing communication cost and improving overall quality of results. Manjhi al @cite_44 studied @math -heavy hitter tracking in a hierarchical communication model.
@cite_24 @cite_44 @cite_33
1997642935 2949236774 1995073517 1989921461
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 show that randomization can lead to significant improvements for a few fundamental problems in distributed tracking. Our basis is the count-tracking problem, where there are @math players, each holding a counter @math that gets incremented over time, and the goal is to track an @math -approximation of their sum @math continuously at all times, using minimum communication. While the deterministic communication complexity of the problem is @math , where @math is the final value of @math when the tracking finishes, we show that with randomization, the communication cost can be reduced to @math . Our algorithm is simple and uses only O(1) space at each player, while the lower bound holds even assuming each player has infinite computing power. Then, we extend our techniques to two related distributed tracking problems: frequency-tracking and rank-tracking , and obtain similar improvements over previous deterministic algorithms. Both problems are of central importance in large data monitoring and analysis, and have been extensively studied in the literature. In 1966, Graham showed that a simple procedure called list scheduling yields a 2-approximation algorithm for the central problem of scheduling precedence constrained jobs on identical machines to minimize makespan. To date this has remained the best algorithm, and whether it can or cannot be improved has become a major open problem in scheduling theory. We address this problem by establishing a quite surprising relation between the approximability of the considered problem and that of scheduling precedence constrained jobs on a single machine to minimize weighted completion time. More specifically, we prove that if the single machine problem is hard to approximate within a factor of @math , then the considered parallel machine problem, even in the case of unit processing times, is hard to approximate within a factor of @math , where @math tends to 0 as @math tends to 0. Combining this with Bansal and Khot's recent hardness result for the single machine problem gives that it is NP-hard to improve upon the approximation ratio obtained by Graham, assuming a new variant of the unique games conjecture. Given @math elements with non-negative integer weights @math and an integer capacity @math , we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most @math . We give the first deterministic, fully polynomial-time approximation scheme (FPTAS) for estimating the number of solutions to any knapsack constraint (our estimate has relative error @math ). Our algorithm is based on dynamic programming. Previously, randomized polynomial-time approximation schemes (FPRAS) were known first by Morris and Sinclair via Markov chain Monte Carlo techniques, and subsequently by Dyer via dynamic programming and rejection sampling. In addition, we present a new method for deterministic approximate counting using read-once branching programs. Our approach yields an FPTAS for several other counting problems, including counting solutions for the multidimensional knapsack problem with a constant number of constraints, the general integer knapsack problem, and the contingency tables problem with a constant number of rows.