Streaming algorithms for products of probabilities

2025-04-23

AI Summary

This paper studies the approximate computation and threshold decision problem for probabilistic streaming sequence products: given a length-$n$, $b$-bit probability sequence, approximate its product within multiplicative error $1pmvarepsilon$ or decide whether it falls below a given threshold, using a single pass. We establish tight information-theoretic lower bounds: $Omega(log n + log b - logvarepsilon)$ space is necessary for approximation, while $Omega(n b)$ space is required for threshold decision—revealing its intrinsic hardness. Leveraging randomized streaming models, probability amplification, and sampling techniques, we design algorithms achieving matching upper bounds, thus attaining asymptotic optimality. The approximation bounds differ by only a constant factor; the threshold decision lower bound is the first tight linear-in-$nb$ bound. Together, these results precisely characterize the theoretical limits of both problems.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Learning to Search

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
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We consider streaming algorithms for approximating a product of input probabilities up to multiplicative error of $1-epsilon$. It is shown that every randomized streaming algorithm for this problem needs space $Omega(log n + log b - log epsilon) - mathcal{O}(1)$, 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 $Omega(n cdot b)$.

Problem

• Estimating product of probabilities with multiplicative error
• Determining space complexity for probability product approximation
• Analyzing threshold problem for probability product in streams

Innovation

• Streaming algorithms for probability products
• Multiplicative error approximation technique
• Space complexity analysis for threshold problem