| 研究課題/領域番号 |
19K11815
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| 研究機関 | 秋田大学 |
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研究代表者 |
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| 研究期間 (年度) |
2019-04-01 – 2023-03-31
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| キーワード | combinatorics on words / repetitions / squares / distinct squares / upper bounds / square network / distinguished positions |
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| 研究実績の概要 |
During FY2021, I made the following progress on the project.
1. In collaboration with Seki we published a paper in the journal Theoretical Computer Science, introducing the idea of 'square networks' which attempts to integrate results on squares sharing their distinguished positions with other squares. The square network is a special type of bipartite graph with the vertices representing squares and distinguished positions, respectively. Formulating the existing results on such positions in terms of the graph allows more precise descriptions of said results and the leads to new questions which would be difficult to even pose without the framework.
2. In collaboration with Mercas, we developed the idea of clusters of repetition roots which we started with last year's paper "Clusters of repetition roots: single chains" at SOFSEM 2021. We proved that our tools work for prefix chains of k-power roots for any exponent k>1 and obtained some promising first results regarding runs and lower bound constructions for arbitrary cluster size sequences. In particular we showed lower bounds on the number of shared prefixes in prefix chains of roots of so called 'run-ending' squares in runs. Moreover, we showed that our results for single chains of square roots are optimal in the sense that for any sequence of cluster sizes one can construct a word and squares in it whose root clusters have those sizes. The manuscript summarizing the above findings is being prepared for submission to conference.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
Continuing our ongoing online meetings with Seki we managed to complete the manuscript on square network and got it published as mentioned above.
With Mercas, we implemented our plan from last year to generalize our approach of root clusters to higher exponents and other types of repetitions. Due to the impossibility of personal meetings and the time zone difference, the collaboration was harder than expected, but we still made progress and prepared a manscript which will be submitted in the following weeks.
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| 今後の研究の推進方策 |
We obtained some preliminary results which allow the application of our root cluster techniques in some cases when the roots are not linearly ordered by the prefix relation. So far the setting is rather restricted, with two partially incomparable prefix chains sharing a special type of prefix. We are planning on finding a suitable assignment of `anchor' positions which would allow the generalization of our tools to arbitrary numbers of overlapping prefix chains.
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| 次年度使用額が生じた理由 |
The main reason for incurring the amount to be used next fiscal year is still the international situation related to the coronavirus pandemic, more precisely the strict travel restrictions imposed by the national government and Akita University. Due to the restrictions in place I was not able to implement my travel plans and also could not invite overseas researchers.
I expect that the situation will improve in fiscal year 2022. I am planning on visiting two international conferences and make a research visit to the UK. I also plan on inviting my domestic and overseas collaborators to Akita University.
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