Fast Identification of Heavy Hitters by Cached and Packed Group Testing

Abstract

The $\epsilon$-approximate $\phi$-heavy hitters problem is, for any element from some universe $\mathbb {U}=[0..n)$, to maintain its frequency under an arbitrary data stream of form $(x_i, \varDelta _i)\in \mathbb {U}\times \mathbb {Z}$ that changes the frequency of $x_i$ by $\varDelta _i$, such that one can output every element with frequency more than $\phi {N}$ and no element with frequency no more than $(\phi -\epsilon ){N}$ for ${N}=\sum _i \varDelta _i$ and prespecified parameters $\epsilon , \phi \in \mathbb {R}$. To solve this problem in small space, Cormode and Muthukrishnan (ACM TODS, 2005) have proposed an ${O}(\rho \epsilon ^{-1}\lg {n})$-space probabilistic data structure with good practical performance, where $\rho =\lg {(1/(\delta \phi ))}$ for any failure probability $\delta \in \mathbb {R}$. In this paper, we improve its output time from ${O}(\rho \epsilon ^{-1}(\lg {n}+\rho ))$ to ${O}(\rho ^2\epsilon ^{-1})$ for arbitrary updates ($\varDelta _i\in \mathbb {Z}$) and its update time from ${O}(\rho \lg {n})$ to amortized ${O}(\rho )$ for constant updates ($\varDelta _i\in {O}(1)$) with the same space and output guarantee by removing application-specific $\lg {n}$ terms that are not tunable, unlike other parameters $\delta$, $\epsilon$, and $\phi$.