2023 Fiscal Year Annual Research Report
Clusters of repetition roots
| Project/Area Number |
19K11815
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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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| Keywords | repetitions in strings / distinct squares / upper bound / combinatorics on words / stringology |
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| Outline of Annual Research Achievements |
In the final year of the grant we worked on the extension of Brlek and Li's result concerning the upper bound on distinct repetitions in strings. They proved the upper bound equal to the length of the string divided by the exponent minus one, using an approach with Rauzy graphs.
We managed to extend the approach to prove our conjecture regarding the clusters of distinct repetition roots: the cluster of each root U is strictly larger than the number of distinct repetition roots that have U as a prefix. This is a stronger result, which implies the upper bound on the number of distinct repetitions proved by Brlek and Li. The method used is an extension of the Rauzy graph approach, introducing x-prefixed Rauzy graphs for each distinct root x.
The manuscript with the final results is in preparation.
Our result will open up new directions for investigating repetitions in strings by considering the nested cluster structures of the repetition roots, and studying what structures allow for high repetition density in the strings. We showed earlier that our lower bound on cluster sizes is optimal when the roots are linearly ordered by the prefix relation. An interesting question to pursue is whether the lower bound is optimal when the roots form a non-linear partial order under the prefix relation.
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