Tracking the Frequency Moments at All Times

Abstract

The traditional requirement for a randomized streaming algorithm is just one-shot, i.e., algorithm should be correct (within the stated ε-error bound) at the end of the stream. In this paper, we study the tracking problem, where the output should be correct at all times. The standard approach for solving the tracking problem is to run O(log m) independent instances of the one-shot algorithm and apply the union bound to all m time instances. In this paper, we study if this standard approach can be improved, for the classical frequency moment problem. We show that for the F p problem for any 1 < p ≤ 2, we actually only need O(log log m + log n) copies to achieve the tracking guarantee in the cash register model, where n is the universe size. Meanwhile, we present a lower bound of Ω(log m log log m) bits for all linear sketches achieving this guarantee. This shows that our upper bound is tight when n = (log m) O(1) . We also present an Ω(log 2 m) lower bound in the turnstile model, showing that the standard approach by using the union bound is essentially optimal.

Beating CountSketch for heavy hitters in insertion streams

“…Since there are only O(ǫ −2 log 2 (n)) intervals we can apply Chernoff's Inequality to guarantee the tracking on every interval, which gives us the tracking at all times. This is a direct improvement over the F 2 tracking algorithm of [24] which for constant ε requires O(log n(log n + log log m)) bits.…”

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

“…The paper [7] also introduced a (1±ε)-relative error F 2 tracking scheme based on a linear sketch that uses only O(log m log log m) bits of memory, for constant ε. This is known to be optimal when n = (log m) O (1) by a lower bound of [19].…”