Space-optimal Heavy Hitters with Strong Error Bounds Citation

The problem of finding heavy hitters and approximating the frequencies of items is at the heart of many problems in data stream analysis. It has been observed that several proposed solutions to this problem can outperform their worst-case guarantees on real data. This leads to the question of whether some stronger bounds can be guaranteed. We answer this in the positive by showing that a class of “counter-based algorithms” (including the popular and very space-efficient FREQUENT and SPACESAVING algorithms) provide much stronger approximation guarantees than previously known. Specifically, we show that errors in the approximation of individual elements do not depend on the frequencies of the most frequent elements, but only on the frequency of the remaining “tail.” This shows that counter-based methods are the most spaceefficient (in fact, space-optimal) algorithms having this strong error bound. This tail guarantee allows these algorithms to solve the “sparse recovery” problem. Here, the goal is to recover a faithful representation of the vector of frequencies, f . We prove that using space O(k), the algorithms construct an approximation f∗ to the frequency vector f so that the L1 error ‖f − f∗‖1 is close to the best possible error minf ′ ‖f ′ − f‖1, where f ′ ranges over all vectors with at most k non-zero entries. This improves the previously best known space bound of about O(k log n) for streams without element deletions (where n is the size of the domain from which stream elements are drawn). Other consequences of the tail guarantees are results for skewed (Zipfian) data, and guarantees for accuracy of merging multiple summarized streams. ∗Supported in part by David and Lucille Packard Fellowship and by MADALGO (Center for Massive Data Algorithmics, funded by the Danish National Research Association) and by NSF grant CCF0728645. †Supported by NSF CAREER award CCF 0743372 and DARPA/ONR N66001-08-1-2065 Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. PODS’09, June 29–July 2, 2009, Providence, Rhode Island, USA. Copyright 2009 ACM 978-1-60558-553-6 /09/06 ...$5.00.