A Fast Algorithm Finding the Shortest Reset Words

TL;DR

This paper introduces a faster exponential algorithm for finding minimal reset words in finite automata, utilizing bidirectional BFS and radix tries, enabling analysis of larger automata and providing refined estimates of reset word lengths.

Contribution

The paper presents a novel, more efficient algorithm for computing shortest reset words, improving practical performance and scalability over previous methods.

Findings

Able to analyze automata with up to 300 states

Provides a new estimate for the expected shortest reset word length

Demonstrates the algorithm's practical efficiency through experiments

Abstract

In this paper we present a new fast algorithm finding minimal reset words for finite synchronizing automata. The problem is know to be computationally hard, and our algorithm is exponential. Yet, it is faster than the algorithms used so far and it works well in practice. The main idea is to use a bidirectional BFS and radix (Patricia) tries to store and compare resulted subsets. We give both theoretical and practical arguments showing that the branching factor is reduced efficiently. As a practical test we perform an experimental study of the length of the shortest reset word for random automata with $n$ states and 2 input letters. We follow Skvorsov and Tipikin, who have performed such a study using a SAT solver and considering automata up to $n=100$ states. With our algorithm we are able to consider much larger sample of automata with up to $n=300$ states. In particular, we obtain a new more precise estimation of the expected length of the shortest reset word $\approx 2.5\sqrt{n-5}$.