BPTree: an $\ell_2$ heavy hitters algorithm using constant memory

Abstract: The task of finding heavy hitters is one of the best known and well studied problems in the area of data streams. One is given a list $i_1,i_2,\ldots,i_m\in[n]$ and the goal is to identify the items among $[n]$ that appear frequently in the list. In sub-polynomial space, the strongest guarantee available is the $\ell_2$ guarantee, which requires finding all items that occur at least $\epsilon|f|_2$ times in the stream, where the vector $f\in\mathbb{R}<sup>n$ is the count histogram of the stream with $i$th coordinate equal to the number of times~$i$ appears $f_i:=#{j\in[m]:i_j=i}$. The first algorithm to achieve the $\ell_2$ guarantee was the CountSketch of [CCF04], which requires $O(\epsilon^{-2}\log n)$ words of memory and $O(\log n)$ update time and is known to be space-optimal if the stream allows for deletions. The recent work of [BCIW16] gave an improved algorithm for insertion-only streams, using only $$O(\epsilon^{{-2}\log\epsilon^{-1}\log\log} n)$$ words of memory. In this work, we give an algorithm \bptree for $\ell_2$ heavy hitters in insertion-only streams that achieves $O(\epsilon<sup>{-2}\log\epsilon<sup>{-1})$ words of memory and $O(\log\epsilon<sup>{-1})$ update time, which is the optimal dependence on $n$ and $m$. In addition, we describe an algorithm for tracking $|f|_2$ at all times with $O(\epsilon<sup>{-2})$ memory and update time. Our analyses rely on bounding the expected supremum of a Bernoulli process involving Rademachers with limited independence, which we accomplish via a Dudley-like chaining argument that may have applications elsewhere.