Regular Languages meet Prefix Sorting

Jarno Alanko, Nicola Prezza, Alberto Policriti, Giovanna D’Agostino

Research output: Chapter in Book/Report/Conference proceedingConference contributionScientificpeer-review

Abstract

Indexing strings via prefix (or suffix) sorting is, arguably, one of the most successful algorithmic techniques developed in the last decades. Can indexing be extended to languages? The main contribution of this paper is to initiate the study of the sub-class of regular languages accepted by an automaton whose states can be prefix-sorted. Starting from the recent notion of Wheeler graph [Gagie et al., TCS 2017]-which extends naturally the concept of prefix sorting to labeled graphs-we investigate the properties of Wheeler languages, that is, regular languages admitting an accepting Wheeler finite automaton. Interestingly, we characterize this family as the natural extension of regular languages endowed with the co-lexicographic ordering: when sorted, the strings belonging to a Wheeler language are partitioned into a finite number of co-lexicographic intervals, each formed by elements from a single Myhill-Nerode equivalence class. Moreover: (i) We show that every Wheeler NFA (WNFA) with n states admits an equivalent Wheeler DFA (WDFA) with at most 2n−1−|Σ| states that can be computed in O(n^3) time. This is in sharp contrast with general NFAs. (ii) We describe a quadratic algorithm to prefix-sort a proper superset of the WDFAs, a O(nlogn)-time online algorithm to sort acyclic WDFAs, and an optimal linear-time offline algorithm to sort general WDFAs. By contribution (i), our algorithms can also be used to index any WNFA at the moderate price of doubling the automaton's size. (iii) We provide a minimization theorem that characterizes the smallest WDFA recognizing the same language of any input WDFA. The corresponding constructive algorithm runs in optimal linear time in the acyclic case, and in O(nlogn) time in the general case. (iv) We show how to compute the smallest WDFA equivalent to any acyclic DFA in nearly-optimal time.
Original languageEnglish
Title of host publicationSODA '20: Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithm
Place of PublicationPhiladelphia
PublisherSociety for Industrial and Applied Mathematics
Publication date2020
Pages911–930
ISBN (Electronic)978-1-61197-599-4
DOIs
Publication statusPublished - 2020
MoE publication typeA4 Article in conference proceedings
EventACM-SIAM Symposium on Discrete Algorithms - Salt Lake City, United States
Duration: 5 Jan 20208 Jan 2020

Fields of Science

  • 113 Computer and information sciences
  • Data Structures

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