BPTree: An ℓ2 Heavy Hitters Algorithm Using Constant Memory

Vladimir Braverman, Stephen R. Chestnut, Nikita Ivkin, Jelani Nelson, Zhengyu Wang, David P. Woodruff

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 i1,i2,...,im∈[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 l2 guarantee, which requires finding all items that occur at least ε||ƒ||2 times in the stream, where the vector ƒ∈Rn is the count histogram of the stream with ith coordinate equal to the number of times i appears ƒi:=#{jε[m]:ij=i. The first algorithm to achieve the l2 guarantee was the CountSketch of [11], which requires O(ε-2log 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 [7] gave an improved algorithm for insertion-only streams, using only O(ε-2logε-1log log n) words of memory. In this work, we give an algorithm BPTree for l2 heavy hitters in insertion-only streams that achieves O(ε-2logε-1) words of memory and O(logε-1) update time, which is the optimal dependence on n and m. In addition, we describe an algorithm for tracking ||ƒ||2 at all times with O(ε-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.