Streaming algorithms for products of probabilities

TL;DR

This paper studies streaming algorithms for approximating the product of probabilities $\prod_{i=1}^n q_i$ within a multiplicative factor $(1-\epsilon)$ and shows a randomized space lower bound of $\Omega(\log n + \ log b - \log \epsilon)$ that matches the best known upper bound up to a constant; it also proves a separate lower bound $\Omega(n b)$ for the threshold problem. The lower bounds are derived via reductions to one-way randomized communication complexity, leveraging classic problems such as the greater-than problem to establish tightness under the stated parameter regimes. Together, these results clarify fundamental space-usage limits for probabilistic streaming of products and threshold decisions, informing the design and evaluation of streaming algorithms in probabilistic settings.

Abstract

We consider streaming algorithms for approximating a product of input probabilities up to multiplicative error of $1-ε$. It is shown that every randomized streaming algorithm for this problem needs space $Ω(\log n + \log b - \log ε) - \mathcal{O}(1)$, where $n$ is length of the input stream and $b$ is the bit length of the input numbers. This matches an upper bound from Alur et al.~up to a constant multiplicative factor. Moreover, we consider the threshold problem, where it is asked whether the product of the input probabilities is below a given threshold. It is shown that every randomized streaming algorithm for this problem needs space $Ω(n \cdot b)$.

Theorems & Definitions (4)

• theorem thmcountertheorem
• theorem thmcountertheorem: AlurCJK20
• theorem thmcountertheorem
• proof