An Optimal Algorithm for l1-Heavy Hitters in Insertion Streams and Related Problems

We give the first optimal bounds for returning the $\ell_1$-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 $\{1, 2, \dots, n\}$ and parameters $0 < \epsilon < \phi \leq 1$, let $f_i$ 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 $f_i \geq \phi m$, returns no items $j$ for which $f_j \leq (\phi -\epsilon)m$, and returns approximations $\tilde{f}_i$ with $|\tilde{f}_i - f_i| \leq \epsilon m$ for each item $i$ that it returns. Our algorithm uses $O(\epsilon^{-1} \log\phi^{-1} + \phi^{-1} \log n + \log \log m)$ 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 $\epsilon m$ error in the above amount of space, resolving Question 3 in the IITK 2006 Workshop on Algorithms for Data Streams for the case of $\ell_1$-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.