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, . . . , u}. For a given 0 ≤ φ ≤ 1, the φ-heavy hitters are those elements of A whose frequency in A is at least φ|A|; the φ-quantile of A is an element x of U such that at most φ|A| elements of A are smaller than A and at most (1 − φ)|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 φ-heavy hitters and the φ-quantile of A approximately at all times with minimum communication. We give tracking algorithms with worst-case communication cost O(k/ · log n) 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 φ-quantiles for all 0 ≤ φ ≤ 1. A preliminary version of this article was presented at
Yi and Zhang REF provided a deterministic algorithm with communication cost O((m/ε) log N ) and O(1/ε) space at each site to continuously track φ-heavy hitters and the φquantiles.
Optimal Tracking of Distributed Heavy Hitters and Quantiles
Algorithmica
Algorithmica
Mathematics Computer Science