Tracking the Frequency Moments at All Times

The traditional requirement for a randomized streaming algorithm is just {\em one-shot}, i.e., algorithm should be correct (within the stated $\eps$-error bound) at the end of the stream. In this paper, we study the {\em 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 $1 < p \le 2$, 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 $\Omega(\log m \log\log m)$ 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 $\Omega(\log^2 m)$ lower bound in the turnstile model, showing that the standard approach by using the union bound is essentially optimal.