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Clusters of repetition roots

Research Project

Project/Area Number 19K11815
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 60010:Theory of informatics-related
Research InstitutionAkita University

Principal Investigator

Fazekas Szilard Zsolt  秋田大学, 理工学研究科, 准教授 (70725382)

Project Period (FY) 2019-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2021: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywordsdistinct repetitions / combinatorics / compressibility / repetitions in strings / distinct squares / upper bound / combinatorics on words / stringology / repetitions / squares / upper bounds / square network / distinguished positions / Repetitions / Squares / Combinatorics on words / Distinct repetitions
Outline of Research at the Start

The main questions in this research project are: how many repetition occurrences and how many unique repetition types (distinct repetitions) can there be in a word (sequence)? I aim to improve existing bounds for distinct repetitions, in particular, tackling a famous conjecture by Fraenkel and Simpson on the number of squares. I propose a fresh approach to understand the structure of distinct repetitions through clusters of their roots, which is expected to lead to improvements in the bounds and provide an easy, visually informative way of presenting their proofs.

Outline of Final Research Achievements

The research goal was to obtain better upper bounds on the number of distinct repetitions of the form xx...x that can occur in a sequence. We introduced a new approach to study the number of such repetitions through the set of positions their root x occurs in the sequence, called the cluster of the repetition. We aimed to show that each cluster must be larger than the number of other clusters included in it.
During the project we first proved that our conjecture about clusters in some special cases. In the final year we worked on extending a recent result by Brlek and Li that proved the upper bound on such repetitions equal to the length of the string divided by the exponent minus one, using Rauzy graphs. We managed to extend the approach to prove our conjecture regarding the clusters of distinct repetition roots. Our result opens up new directions for investigating repetitions in strings by considering the nested cluster structures of the repetition roots.

Academic Significance and Societal Importance of the Research Achievements

The significance of our results is that now we have better tools to study sequences containing many repetitions, which can lead to a better understanding of compression and pattern matching algorithms, which are of critical importance to our web infrastructure and computing in general.

Report

(6 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (14 results)

All 2023 2022 2021 2020 2019 Other

All Int'l Joint Research (1 results) Journal Article (7 results) (of which Int'l Joint Research: 7 results,  Peer Reviewed: 6 results,  Open Access: 2 results) Presentation (6 results)

  • [Int'l Joint Research] Loughborough University(英国)

    • Related Report
      2022 Research-status Report
  • [Journal Article] Clusters of Repetition Roots Forming Prefix Chains2022

    • Author(s)
      Fazekas Szilard Zsolt、Mercas Robert
    • Journal Title

      Lecture Notes in Computer Science

      Volume: 13439 Pages: 43-56

    • DOI

      10.1007/978-3-031-13257-5_4

    • ISBN
      9783031132568, 9783031132575
    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On Algorithmic Self-Assembly of Squares by Co-Transcriptional Folding2022

    • Author(s)
      Fazekas Szilard Zsolt、Kim Hwee、Matsuoka Ryuichi、Seki Shinnosuke, Takeuchi Hinano
    • Journal Title

      Leibniz International Proceedings in Informatics (LIPIcs)

      Volume: 248

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Square network on a word2021

    • Author(s)
      Fazekas Szilard Zsolt、Seki Shinnosuke
    • Journal Title

      Theoretical Computer Science

      Volume: 894 Pages: 121-134

    • DOI

      10.1016/j.tcs.2021.08.004

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Linear Bounds on the Size of Conformations in Greedy Deterministic Oritatami2021

    • Author(s)
      Fazekas Szilard Zsolt、Kim Hwee、Matsuoka Ryuichi、Morita Reoto、Seki Shinnosuke
    • Journal Title

      International Journal of Foundations of Computer Science

      Volume: 32 Issue: 05 Pages: 575-596

    • DOI

      10.1142/s0129054121410082

    • Related Report
      2021 Research-status Report 2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Clusters of Repetition Roots: Single Chains2021

    • Author(s)
      Fazekas Szilard Zsolt, Mercas Robert
    • Journal Title

      Lecture Notes in Computer Science

      Volume: 12607 Pages: 400-409

    • DOI

      10.1007/978-3-030-67731-2_29

    • ISBN
      9783030677305, 9783030677312
    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Two-way deterministic automata with jumping mode2021

    • Author(s)
      Fazekas Szilard Zsolt, Hoshi Kaito, Yamamura Akihiro
    • Journal Title

      Theoretical Computer Science

      Volume: 864 Pages: 92-102

    • DOI

      10.1016/j.tcs.2021.02.030

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Clusters of repetition roots: single chains2020

    • Author(s)
      Szilard Zsolt Fazekas, Robert Mercas
    • Journal Title

      RIMS Kokyuroku

      Volume: TBA

    • Related Report
      2019 Research-status Report
    • Open Access / Int'l Joint Research
  • [Presentation] The general case of the clusters conjecture2023

    • Author(s)
      Szilard Fazekas
    • Organizer
      RIMS workshop on "Group, Ring, Language and Related Areas in Computer Science"
    • Related Report
      2022 Research-status Report
  • [Presentation] Clusters of Repetition Roots Forming Prefix Chains2022

    • Author(s)
      Szilard Fazekas
    • Organizer
      24th IFIP WG 1.02 International Conference on Descriptional Complexity of Formal Systems
    • Related Report
      2022 Research-status Report
  • [Presentation] Clusters of Repetition Roots: Single Chains2021

    • Author(s)
      Szilard Zsolt Fazekas
    • Organizer
      SOFSEM 2021: 47th International Conference on Current Trends in Theory and Practice of Informatics
    • Related Report
      2020 Research-status Report
  • [Presentation] Chains of repetition roots2020

    • Author(s)
      Szilard Zsolt Fazekas
    • Organizer
      Algebraic system, Logic, Language and Related Areas in Computer Science II
    • Related Report
      2019 Research-status Report
  • [Presentation] Clusters of repetition roots: generalization and optimality2020

    • Author(s)
      Szilard Zsolt Fazekas
    • Organizer
      Nagaokakyo Seminar
    • Related Report
      2019 Research-status Report
  • [Presentation] New upper bounds on chains of square root clusters2019

    • Author(s)
      Szilard Zsolt Fazekas
    • Organizer
      Nagaokakyo Seminar
    • Related Report
      2019 Research-status Report

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Published: 2019-04-18   Modified: 2025-01-30  

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