Perfect $L_p$ Sampling with Polylogarithmic Update Time

2025-11-29

AI Summary

For perfect $L_p$ sampling ($0 < p < 2$) in the turnstile streaming model—i.e., sampling coordinate $i$ with exact probability $|x_i|^p / |x|_p^p$, up to error $leq n^{-C}$—existing algorithms suffer from superpolynomial update time $Omega(n^C)$. This work presents the first perfect $L_p$ sampler achieving both optimal space complexity $ ilde{O}(log^2 n)$ and $mathrm{poly}(log n)$ update time. Our method leverages characteristic function simulation of the reciprocal power of truncated exponential random variables, combined with the Gil-Pelaez inversion formula and an improved trapezoidal integration scheme, to efficiently approximate the cumulative distribution function. This approach breaks a long-standing update-time bottleneck, enabling practical deployment in streaming spectral analysis, sparse recovery, and related applications.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Time-Series/Data Streams

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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 $x in {- ext{poly}(n), ldots, ext{poly}(n)}^n$, 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 $ ilde{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 $ ext{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.

Problem

• Achieves perfect Lp sampling with polylogarithmic update time
• Improves prior work with exponential update time to poly(log n)
• Simulates sum of reciprocals using characteristic function approximation

Innovation

• Perfect Lp sampling with polylogarithmic update time
• Simulating sum of reciprocals using characteristic function approximation
• Applying Gil-Pelaez inversion and trapezoid formula variants