A Faster Fully Polynomial-Time Randomized Approximation Scheme (FPRAS) for Counting the Number of Strings Accepted by a Non-Deterministic Finite Automaton (NFA)

Core Concepts

The authors present a faster fully polynomial-time randomized approximation scheme (FPRAS) for the #NFA problem, which is to determine the size of the set of words of a given length accepted by a non-deterministic finite automaton (NFA). The new FPRAS significantly improves the time complexity compared to the previous state-of-the-art FPRAS.

Abstract

The paper focuses on the #NFA problem, which is to determine the size of the set of words of a given length accepted by a non-deterministic finite automaton (NFA). The #NFA problem is known to be #P-hard, and recently an FPRAS (fully polynomial-time randomized approximation scheme) was developed for it. However, the time complexity of the previous FPRAS was prohibitively high, limiting its practical applicability.
The authors present a new FPRAS for #NFA that has significantly lower time complexity. The key technical differences are:

The new FPRAS maintains a weaker invariant about the quality of the estimate of the number of samples for each state and length, compared to the previous FPRAS.
The new FPRAS only requires the distribution of the samples maintained to be close to the uniform distribution in total variation distance, instead of the stronger maximum norm condition used in the previous FPRAS.

These differences allow the new FPRAS to maintain significantly fewer samples per state, leading to a much lower overall time complexity. Specifically, the new FPRAS runs in time ̃O((m^2 * n^10 + m^3 * n^6) * (1/ε^4) * log^2(1/δ)), where m is the number of states in the NFA, n is the length of the strings, ε is the approximation error, and δ is the failure probability. This is a significant improvement over the previous FPRAS, which had a time complexity of ̃O(m^17 * n^17 * (1/ε^14) * log(1/δ)).
The authors also present two key subroutines - AppUnion and sample - and provide formal guarantees for their correctness and running time. The main algorithm combines these subroutines to achieve the overall FPRAS for #NFA.

Stats

The size of the NFA, m, is a key parameter that affects the time complexity.
The length of the strings, n, is another key parameter that affects the time complexity.
The approximation error parameter, ε, and the failure probability parameter, δ, also affect the time complexity.

Quotes

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Key Insights Distilled From

A faster FPRAS for #NFA

by Kuldeep S. M... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2312.13320.pdf

Deeper Inquiries

How can the insights from this work be applied to further improve the time complexity of the FPRAS for #NFA or to develop FPRASes for other related counting problems

The insights from this work can be applied to further improve the time complexity of the FPRAS for #NFA or to develop FPRASes for other related counting problems by focusing on reducing the number of samples required for each state. By systematically reducing the dependency on the number of states, length, and error tolerance, as demonstrated in the proposed FPRAS, significant improvements in time complexity can be achieved. This reduction in the number of samples per state can lead to a more efficient algorithm overall. Additionally, exploring different techniques for estimating the size of sets and optimizing the sampling process can contribute to further improvements in time complexity for FPRAS algorithms.

What are the practical implications of having a faster FPRAS for #NFA, and how can it enable the development of more efficient tools for the various applications mentioned in the introduction

Having a faster FPRAS for #NFA has several practical implications. It can pave the way for the development of more efficient tools for various applications mentioned in the introduction, such as probabilistic query evaluation, counting answers to regular path queries, and probabilistic graph homomorphism. With a faster FPRAS, approximate counting and sampling tasks in these applications can be performed more efficiently, leading to quicker results and improved performance. This can enhance the scalability and usability of tools in database systems, graph query languages, and other domains where approximate counting is essential.

Are there any other technical insights or approaches that could potentially lead to even faster FPRASes for #NFA or other counting problems in the future

There are several technical insights and approaches that could potentially lead to even faster FPRASes for #NFA or other counting problems in the future. One approach could involve further optimizing the sampling process by exploring advanced sampling techniques or leveraging parallel computing to speed up the sampling and estimation procedures. Additionally, refining the estimation algorithms to reduce the variance in the estimates and improve the accuracy of the approximations could contribute to faster FPRAS algorithms. Exploring novel data structures or algorithmic optimizations tailored to the specific characteristics of the counting problem could also lead to significant improvements in time complexity for FPRAS algorithms.