Heavy hitters via cluster-preserving clustering

Abstract: In turnstile $\ell_p$ $\varepsilon$-heavy hitters, one maintains a high-dimensional $x\in\mathbb{R}<sup>n$ subject to $\texttt{update}(i,\Delta)$ causing $x_i\leftarrow x_i + \Delta$, where $i\in[n]$, $\Delta\in\mathbb{R}$. Upon receiving a query, the goal is to report a small list $L\subset[n]$, $|L| = O(1/\varepsilon<sup>p)$, containing every "heavy hitter" $i\in[n]$ with $$|x_i| \ge \varepsilon |x_{\overline{1/\varepsilon<sup>p}}|_p$$, where $x_{\overline{k}}$ denotes the vector obtained by zeroing out the largest $k$ entries of $x$ in magnitude. For any $p\in(0,2]$ the CountSketch solves $\ell_p$ heavy hitters using $O(\varepsilon^{-p}\log n)$ words of space with $O(\log n)$ update time, $O(n\log n)$ query time to output $L$, and whose output after any query is correct with high probability (whp) $1 - 1/poly(n)$. Unfortunately the query time is very slow. To remedy this, the work [CM05] proposed for $p=1$ in the strict turnstile model, a whp correct algorithm achieving suboptimal space $O(\varepsilon^{-1}\log² n)$, worse update time $O(\log² n)$, but much better query time $O(\varepsilon^{{-1}poly(\log} n))$. We show this tradeoff between space and update time versus query time is unnecessary. We provide a new algorithm, ExpanderSketch, which in the most general turnstile model achieves optimal $O(\varepsilon^{-p}\log n)$ space, $O(\log n)$ update time, and fast $O(\varepsilon^{{-p}poly(\log} n))$ query time, and whp correctness. Our main innovation is an efficient reduction from the heavy hitters to a clustering problem in which each heavy hitter is encoded as some form of noisy spectral cluster in a much bigger graph, and the goal is to identify every cluster. Since every heavy hitter must be found, correctness requires that every cluster be found. We then develop a "cluster-preserving clustering" algorithm, partitioning the graph into clusters without destroying any original cluster.