An Optimal Algorithm for ℓ ₁ -Heavy Hitters in Insertion Streams and Related Problems

Abstract

We give the first optimal bounds for returning the ℓ1-heavy hitters in a data stream of insertions, together with their approximate frequencies, closing a long line of work on this problem. For a stream of m items in {1, 2, . . . , n} and parameters 0 < ε < ϕ 1, let fi denote the frequency of item i, i.e., the number of times item i occurs in the stream. With arbitrarily large constant probability, our algorithm returns all items i for which fi ϕm, returns no items j for which fj (ϕ − ε)m, and returns approximationsfi with |fi − fi| εm for each item i that it returns. Our algorithm uses O(ε −1 log ϕ −1 + ϕ −1 log n + log log m) bits of space, processes each stream update in O(1) worst-case time, and can report its output in time linear in the output size. We also prove a lower bound, which implies that our algorithm is optimal up to a constant factor in its space complexity. A modification of our algorithm can be used to estimate the maximum frequency up to an additive εm error in the above amount of space, resolving Question 3 in the IITK 2006 Workshop on Algorithms for Data Streams for the case of ℓ1-heavy hitters. We also introduce several variants of the heavy hitters and maximum frequency problems, inspired by rank aggregation and voting schemes, and show how our techniques can be applied in such settings. Unlike the traditional heavy hitters problem, some of these variants look at comparisons between items rather than numerical values to determine the frequency of an item.

A high-performance algorithm for identifying frequent items in data streams

“…Trivially, one can treat an update (i, ∆) as ∆ updates of the form (i, 1), but this does not maintain the time bound (even in an amortized sense) if ∆ = ω(1) on average. We now sketch an adaptation that has the same accuracy guarantees as the simple algorithm of Bhattacharyya et al [3], and runs in constant amortized time per stream update with high probability, assuming that the number of stream updates n is Ω(ε −2 log(1/δ)).…”

“…If t samples are required before the sum of the variables is more than ∆, then the algorithm feeds update (i, t) into an instance of any counter-based algorithm capable handling weighted updates. The remainder of the details, and the error analysis, are similar to [3], and we omit them for brevity.…”

“…The above adaptation allows our efficiency improvements relative to prior work for weighted streams and mergeability to carry over in a black-box manner to the simple algorithm of Bhattacharyya et al [3]. One simply feeds the (weighted) sampled stream updates into our new optimized algorithms for weighted updates.…”

“…are two of the most basic computational tasks performed on data streams. Due to their practical importance, streaming algorithms for these tasks have been studied intensely [2][3][4]6,7,9,[13][14][15][16][17]. It may therefore seem as though streaming frequency approximation is well-understood, with little room for further insight or improvement.…”

Optimal bounds for approximate counting

“…For example, Jayaram and Woodruff showed that for p ∈ (0, 1] an approximate counter can be used effectively as a subroutine in an algorithm for approximating the pth moment of an insertion-only stream up to 1 + ε in Õ(1/ε 2 + log n) bits of space [JW19], improving over a derandomization of an algorithm of Indyk that uses O(ε −2 log n) bits [Ind06,KNW10]. Approximate counting also finds use in approximating large frequency moments [AMS99, GS09], approximate reservoir sampling [GS09], approximating the number of inversion when streaming over a permutation [AJKS02], and 1 heavy hitters in insertion-only streams [BDW19].…”