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.
Later, Huang al @cite_25 proposed a randomized algorithm that uses @math space at each site and @math total communication and tracks heavy hitters in a distributed stream. For each item @math in the stream a site chooses to send a message with a probability @math where @math is a @math -approximation of the total count. It then sends @math the total count of messages at site @math where @math , to the coordinator. Again an approximation heavy-hitter count @math can be used at each site to reduce space.
@cite_25
1997642935 2290712804 2950318219 2517585112
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. Let @math be a uniformly distributed random @math -SAT formula with @math variables and @math clauses. For clauses variables ratio @math the formula @math is satisfiable with high probability. However, no efficient algorithm is known to provably find a satisfying assignment beyond @math with a non-vanishing probability. Non-rigorous statistical mechanics work on @math -CNF led to the development of a new efficient "message passing algorithm" called [M ', Science 2002]. Experiments conducted for @math suggest that the algorithm finds satisfying assignments close to @math . However, in the present paper we prove that the basic version of Survey Propagation Guided Decimation fails to solve random @math -SAT formulas efficiently already for @math with @math almost a factor @math below @math . We resolve several fundamental questions in the area of distributed functional monitoring, initiated by Cormode, Muthukrishnan, and Yi (SODA, 2008). In this model there are @math sites each tracking their input and communicating with a central coordinator that continuously maintain an approximate output to a function @math computed over the union of the inputs. The goal is to minimize the communication. We show the randomized communication complexity of estimating the number of distinct elements up to a @math factor is @math , improving the previous @math bound and matching known upper bounds up to a logarithmic factor. For the @math -th frequency moment @math , @math , we improve the previous @math communication bound to @math . We obtain similar improvements for heavy hitters, empirical entropy, and other problems. We also show that we can estimate @math , for any @math , using @math communication. This greatly improves upon the previous @math bound of Cormode, Muthukrishnan, and Yi for general @math , and their @math bound for @math . For @math , our bound resolves their main open question. Our lower bounds are based on new direct sum theorems for approximate majority, and yield significant improvements to problems in the data stream model, improving the bound for estimating @math in @math passes from @math to @math , giving the first bound for estimating @math in @math passes of @math bits of space that does not use the gap-hamming problem. We study a noiseless broadcast link serving @math users whose requests arise from a library of @math files. Every user is equipped with a cache of size @math files each. It has been shown that by splitting all the files into packets and placing individual packets in a random independent manner across all the caches prior to any transmission, at most @math file transmissions are required for any set of demands from the library. The achievable delivery scheme involves linearly combining packets of different files following a greedy clique cover solution to the underlying index coding problem. This remarkable multiplicative gain of random placement and coded delivery has been established in the asymptotic regime when the number of packets per file @math scales to infinity. The asymptotic coding gain obtained is roughly @math . In this paper, we initiate the finite-length analysis of random caching schemes when the number of packets @math is a function of the system parameters @math , and @math . Specifically, we show that the existing random placement and clique cover delivery schemes that achieve optimality in the asymptotic regime can have at most a multiplicative gain of 2 even if the number of packets is exponential in the asymptotic gain @math . Furthermore, for any clique cover-based coded delivery and a large class of random placement schemes that include the existing ones, we show that the number of packets required to get a multiplicative gain of @math is at least @math . We design a new random placement and an efficient clique cover-based delivery scheme that achieves this lower bound approximately. We also provide tight concentration results that show that the average (over the random placement involved) number of transmissions concentrates very well requiring only a polynomial number of packets in the rest of the system parameters.