Optimal Tracking of Distributed Heavy Hitters and Quantiles

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 $U=\{1,\mathrm{.}\mathrm{.}\mathrm{.},u\}$. For a given $0 \le \phi \le 1$,the$\phi$-heavyhittersarethoseelementsof$A$whosefrequencyin$A$isatleast$\phi |A|$;the$\phi$-quantileof$A$isanelement$x$of$U$suchthatatmost$\phi|A|$elementsof$A$aresmallerthan$A$andatmost$(1-\phi)|A|$elementsof$A$aregreaterthan$x$\mathrm{.}Supposetheelementsof$A$arereceivedat$k$ 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 $\varphi$-heavy hitters and the $\varphi$-quantile of $A$ approximately at all times with minimum communication. We give tracking algorithms with worst-case communication cost $O(k\mathrm{/}\text{eps}\cdot \log n)$ for both problems, where $n$ is the total number of items in $A$, and $\text{eps}$ 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 $\varphi$-quantiles for all $0\le \varphi \le 1$.