Closest Substring Problems for Regular Languages

Yo Sub Han, Sang Ki Ko, Timothy Ng, Kai Salomaa

Research output: Chapter in Book/Report/Conference proceedingConference contribution


It is well known that given a finite set of strings of equal length, the Consensus String problem—the problem of deciding whether or not there exists a consensus string whose distance is at most r from every string in the given set—is proven to be NP-complete. A similar problem called the Closest Substring problem asks whether there exists a string w of length l such that each string in a given set L has a substring whose distance is at most r (called radius) from w. As the Closest Substring problem is a generalized version of the Consensus String problem, it is obvious that the problem is NP-hard for a finite set of strings. We show that the Closest Substring problem for regular languages represented by nondeterministic finite automata (NFAs) is PSPACE-complete. The main difference from the previous work is that we consider an infinite set of strings, which is recognized by an NFA as input instead of a finite set of strings. We also prove that the Closest Substring problem for acyclic NFAs lies in the second level of the polynomial-time hierarchy (formula presented) and is both NP-hard and coNP-hard.

Original languageEnglish
Title of host publicationDevelopments in Language Theory - 22nd International Conference, DLT 2018, Proceedings
EditorsMizuho Hoshi, Shinnosuke Seki
PublisherSpringer Verlag
Number of pages12
ISBN (Print)9783319986531
Publication statusPublished - 2018
Event22nd International Conference on Developments in Language Theory, DLT 2018 - Tokyo, Japan
Duration: 2018 Sept 102018 Sept 14

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11088 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other22nd International Conference on Developments in Language Theory, DLT 2018

Bibliographical note

Funding Information:
Acknowledgements. This work was supported by Institute for Information & communications Technology Promotion (IITP) grant funded by the Korea government (MSIP) (2018-0-00255, Autonomous digital companion framework and application).

Publisher Copyright:
© 2018, Springer Nature Switzerland AG.

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)


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