Faster Update Time for Turnstile Streaming Algorithms

Abstract: In this paper, we present a new algorithm for maintaining linear sketches in turnstile streams with faster update time. As an application, we show that $\log n$ \texttt{Count} sketches or \texttt{CountMin} sketches with a constant number of columns (i.e., buckets) can be implicitly maintained in \emph{worst-case} $O(\log^{0.582} n)$ update time using $O(\log n)$ words of space, on a standard word RAM with word-size $w=\Theta(\log n)$. The exponent $0.582\approx 2\omega/3-1$, where $\omega$ is the current matrix multiplication exponent. Due to the numerous applications of linear sketches, our algorithm improves the update time for many streaming problems in turnstile streams, in the high success probability setting, without using more space, including $\ell_2$ norm estimation, $\ell_2$ heavy hitters, point query with $\ell_1$ or $\ell_2$ error, etc. Our algorithm generalizes, with the same update time and space, to maintaining $\log n$ linear sketches, where each sketch partitions the coordinates into $k<\log^{o(1)} n$ buckets using a $c$-wise independent hash function for constant $c$, and maintains the sum of coordinates for each bucket. Moreover, if arbitrary word operations are allowed, the update time can be further improved to $O(\log^{0.187} n)$, where $0.187\approx \omega/2-1$. Our update algorithm is adaptive, and it circumvents the non-adaptive cell-probe lower bounds for turnstile streaming algorithms by Larsen, Nelson and Nguy{^{e}}n (STOC'15). On the other hand, our result also shows that proving unconditional cell-probe lower bound for the update time seems very difficult, even if the space is restricted to be (nearly) the optimum. If $\omega=2$, the cell-probe update time of our algorithm would be $\log^{o(1)} n$. Hence, proving any higher lower bound would imply $\omega>2$.