2020 Fiscal Year Research-status Report
Clusters of repetition roots
| Project/Area Number |
19K11815
|
|---|
| Research Institution | Akita University |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2019-04-01 – 2022-03-31
|
| Keywords | combinatorics on words / repetitions / squares / distinct squares / upper bounds / square network / distinguished positions |
|---|
| Outline of Annual Research Achievements |
During FY2020, I made the following progress on the project.
1. In collaboration with Robert Mercas, we laid down the foundations for studying the clusters of repetition roots in a paper published in the proceedings of SOFSEM 2021. We explained and formalized the approach and proved that our conjecture holds for prefix chains of square roots, a result generalizable to larger exponents. We have looked at applications of the method to other types of repetitions, such as runs, and obtained some partial results when the suffix root of the runs are totally ordered by the prefix relation.
2. In collaboration with Shinnosuke Seki, we developed the idea of 'square networks'. We began to integrate results on squares sharing their start/center/end positions with other squares, studying what happens when multiple such positions are related by some participating square. When multiple shared position are related, in some cases this imposes special binary structures on the factors in question, making it easier to establish local upper bounds. The analysis also made it possible to conclude that in some cases a square has a unique occurrence in the word, the first result of this type that we are aware of. We also introduced new types of positions in the square network; these positions generalize the start/center/end for squares by considering overlaps instead of full squares. The manuscript describing the results has been submitted to IJFCS.
|
| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
As detailed above regarding the approach focusing on the cluster of repetition roots, we published the proof at SOFSEM 2021 that the approach works in the special case of totally prefix-ordered sets of roots. With Mercas, we are preparing a journal submission in which we generalize the bound for larger exponents and describe some further properties of the clusters. I planned a research visit to the UK, but due to the pandemic, this visit was impossible. We pivoted to online meetings, which is less than ideal, but we are making progress.
As mentioned before I also attacked another angle with Seki. Further developing that square network framework has a lot of potential because - just as the cluster of roots approach - it considers both local and global properties of square packing.
|
| Strategy for Future Research Activity |
I will continue according to the plan laid out in the project proposal, including:
- journal submission on more general bounds (larger exponents) and properties of nested cluster structures;
- considering different 'anchors' for repetitions; the one used to prove the partial result on prefix chains may need to be changed to deal with overlapping chains and we have several ideas to explore.
Additionally, I will try to further develop the work on square networks. We showed that when a special position is shared by two squares and one of the participating squares shares another special position with a third, this imposes special binary structures on the word. Proving bounds on words with this structure is much easier than general, so we will analyze more such multiple sharing scenarios.
|
| Causes of Carryover |
The main reason for incurring budget to be used next year is the ongoing coronavirus pandemic which makes research visits impossible. Most of the funds have been designated to travel related expenses, this explains the surplus.
The plan to use the incurring amount next year and beyond is to simply postpone research related travel until the public health situation allows to conduct them safely.
In particular, if travel restrictions are lifted, I plan to visit the following collaborators: Robert Mercas (Loughborough University, United Kingdom), Hwee Kim (Incheon National University, South Korea), Shinnosuke Seki (University of Electro-Communications, Tokyo).
In addition to these visits, I intend to participate at a number of relevant conferences, such as WORDS, ISAAC and STACS. Situation allowing, these participations would preferably be in-person, but if not possible, I will participate online.
|
Research Products
(4 results)
