Closest Substring Problems for Regular Languages

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

doi:10.1007/978-3-319-98654-8_32

Closest Substring Problems for Regular Languages

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

Department of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

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 language	English
Title of host publication	Developments in Language Theory - 22nd International Conference, DLT 2018, Proceedings
Editors	Mizuho Hoshi, Shinnosuke Seki
Publisher	Springer Verlag
Pages	392-403
Number of pages	12
ISBN (Print)	9783319986531
DOIs	https://doi.org/10.1007/978-3-319-98654-8_32
Publication status	Published - 2018
Event	22nd International Conference on Developments in Language Theory, DLT 2018 - Tokyo, Japan Duration: 2018 Sep 10 → 2018 Sep 14

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	11088 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Other

Other	22nd International Conference on Developments in Language Theory, DLT 2018
Country/Territory	Japan
City	Tokyo
Period	18/9/10 → 18/9/14

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)

Access to Document

10.1007/978-3-319-98654-8_32

Cite this

Han, Y. S., Ko, S. K., Ng, T., & Salomaa, K. (2018). Closest Substring Problems for Regular Languages. In M. Hoshi, & S. Seki (Eds.), Developments in Language Theory - 22nd International Conference, DLT 2018, Proceedings (pp. 392-403). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11088 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-98654-8_32

Han, Yo Sub ; Ko, Sang Ki ; Ng, Timothy ; Salomaa, Kai. / Closest Substring Problems for Regular Languages. Developments in Language Theory - 22nd International Conference, DLT 2018, Proceedings. editor / Mizuho Hoshi ; Shinnosuke Seki. Springer Verlag, 2018. pp. 392-403 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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Han, YS, Ko, SK, Ng, T & Salomaa, K 2018, Closest Substring Problems for Regular Languages. in M Hoshi & S Seki (eds), Developments in Language Theory - 22nd International Conference, DLT 2018, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11088 LNCS, Springer Verlag, pp. 392-403, 22nd International Conference on Developments in Language Theory, DLT 2018, Tokyo, Japan, 18/9/10. https://doi.org/10.1007/978-3-319-98654-8_32

Closest Substring Problems for Regular Languages. / Han, Yo Sub; Ko, Sang Ki; Ng, Timothy; Salomaa, Kai.

Developments in Language Theory - 22nd International Conference, DLT 2018, Proceedings. ed. / Mizuho Hoshi; Shinnosuke Seki. Springer Verlag, 2018. p. 392-403 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11088 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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N2 - 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.

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Han YS, Ko SK, Ng T, Salomaa K. Closest Substring Problems for Regular Languages. In Hoshi M, Seki S, editors, Developments in Language Theory - 22nd International Conference, DLT 2018, Proceedings. Springer Verlag. 2018. p. 392-403. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-98654-8_32

Closest Substring Problems for Regular Languages

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