Optimal Tracking of Distributed Heavy Hitters and Quantiles

TL;DR

This paper presents optimal algorithms for tracking heavy hitters and quantiles in distributed data streams with minimal communication, significantly improving previous methods and establishing matching lower bounds.

Contribution

The authors introduce the first optimal algorithms for distributed heavy hitters and quantiles tracking, achieving worst-case communication costs that match lower bounds.

Findings

Achieved $O(k/ ext{ε} imes ext{log} n)$ communication complexity

Provided matching lower bounds proving optimality

Extended results to tracking all quantiles simultaneously

Abstract

We consider the 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 $0 \le \phi \le 1$, the $\phi$-heavy hitters are those elements of $A$ whose frequency in $A$ is at least $\phi |A|$; the $\phi$-quantile of $A$ is an element $x$ of $U$ such that at most $\phi|A|$ elements of $A$ are smaller than $A$ and at most $(1-\phi)|A|$ elements of $A$ are greater than $x$. Suppose the elements of $A$ are received at $k$ remote {\em sites} over time, and each of the sites has a two-way communication channel to a designated {\em coordinator}, whose goal is to track the set of $\phi$-heavy hitters and the $\phi$-quantile of $A$ approximately at all times with minimum communication. We give tracking algorithms with worst-case communication cost $O(k/\eps \cdot \log n)$ for both problems, where $n$ is the total number of items in $A$, and $\eps$ 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 $\phi$-quantiles for all $0 \le \phi \le 1$.