| Project/Area Number |
19K11815
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 60010:Theory of informatics-related
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| Research Institution | Akita University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Project Status |
Completed (Fiscal Year 2023)
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| 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)
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| Keywords | distinct 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 |
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| 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.
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| 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.
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| 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.
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