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
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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. Consider a complete communication network of @math nodes, where the nodes receive a common clock pulse. We study the synchronous @math -counting problem: given any starting state and up to @math faulty nodes with arbitrary behaviour, the task is to eventually have all correct nodes labeling the pulses with increasing values modulo @math in agreement. Thus, we are considering algorithms that are self-stabilising despite Byzantine failures. In this work, we give new algorithms for the synchronous counting problem that (1) are deterministic, (2) have optimal resilience, (3) have a linear stabilisation time in @math (asymptotically optimal), (4) use a small number of states, and consequently, (5) communicate a small number of bits per round. Prior algorithms either resort to randomisation, use a large number of states and need high communication bandwidth, or have suboptimal resilience. In particular, we achieve an exponential improvement in both state complexity and message size for deterministic algorithms. Moreover, we present two complementary approaches for reducing the number of bits communicated during and after stabilisation. 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. Consider a complete communication network of @math nodes, where the nodes receive a common clock pulse. We study the synchronous @math -counting problem: given any starting state and up to @math faulty nodes with arbitrary behavior, the task is to eventually have all correct nodes labeling the pulses with increasing values modulo @math in agreement. Thus, we are considering algorithms that are self-stabilizing despite Byzantine failures. In this work, we give new algorithms for the synchronous counting problem that (1) are deterministic, (2) have optimal resilience, (3) have a linear stabilization time in @math (asymptotically optimal), (4) use a small number of states, and, consequently, (5) communicate a small number of bits per round. Prior algorithms either resort to randomization, use a large number of states and need high communication bandwidth, or have suboptimal resilience. In particular, we achieve an exponential improvement in both state complexity and message size for deterministic algorithms. Moreover, ...