Perfect $L_p$ Sampling with Polylogarithmic Update Time

Perfect $L_p$ sampling in a stream was introduced by Jayaram and Woodruff (FOCS 2018) as a streaming primitive which, given turnstile updates to a vector $x \in \{-\text{poly}(n), \ldots, \text{poly}(n)\}^n$, outputs an index $i^* \in \{1, 2, \ldots, n\}$ such that the probability of returning index $i$ is exactly \[\Pr[i^* = i] = \frac{|x_i|^p}{\|x\|_p^p} \pm \frac{1}{n^C},\] where $C > 0$ is an arbitrarily large constant. Jayaram and Woodruff achieved the optimal $\tilde{O}(\log^2 n)$ bits of memory for $0 < p < 2$, but their update time is at least $n^C$ per stream update. Thus an important open question is to achieve efficient update time while maintaining optimal space. For $0 < p < 2$, we give the first perfect $L_p$-sampler with the same optimal amount of memory but with only $\text{poly}(\log n)$ update time. Crucial to our result is an efficient simulation of a sum of reciprocals of powers of truncated exponential random variables by approximating its characteristic function, using the Gil-Pelaez inversion formula, and applying variants of the trapezoid formula to quickly approximate it.