Dynamic Treewidth in Logarithmic Time
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913788210446336 |
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| author | Korhonen, Tuukka |
| author_facet | Korhonen, Tuukka |
| contents | We present a dynamic data structure that maintains a tree decomposition of width at most $9k+8$ of a dynamic graph with treewidth at most $k$, which is updated by edge insertions and deletions. The amortized update time of our data structure is $2^{O(k)} \log n$, where $n$ is the number of vertices. The data structure also supports maintaining any ``dynamic programming scheme'' on the tree decomposition, providing, for example, a dynamic version of Courcelle's theorem with $O_{k}(\log n)$ amortized update time; the $O_{k}(\cdot)$ notation hides factors that depend on $k$. This improves upon a result of Korhonen, Majewski, Nadara, Pilipczuk, and Sokołowski [FOCS 2023], who gave a similar data structure but with amortized update time $2^{k^{O(1)}} n^{o(1)}$. Furthermore, our data structure is arguably simpler.
Our main novel idea is to maintain a tree decomposition that is ``downwards well-linked'', which allows us to implement local rotations and analysis similar to those for splay trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02790 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamic Treewidth in Logarithmic Time Korhonen, Tuukka Data Structures and Algorithms We present a dynamic data structure that maintains a tree decomposition of width at most $9k+8$ of a dynamic graph with treewidth at most $k$, which is updated by edge insertions and deletions. The amortized update time of our data structure is $2^{O(k)} \log n$, where $n$ is the number of vertices. The data structure also supports maintaining any ``dynamic programming scheme'' on the tree decomposition, providing, for example, a dynamic version of Courcelle's theorem with $O_{k}(\log n)$ amortized update time; the $O_{k}(\cdot)$ notation hides factors that depend on $k$. This improves upon a result of Korhonen, Majewski, Nadara, Pilipczuk, and Sokołowski [FOCS 2023], who gave a similar data structure but with amortized update time $2^{k^{O(1)}} n^{o(1)}$. Furthermore, our data structure is arguably simpler. Our main novel idea is to maintain a tree decomposition that is ``downwards well-linked'', which allows us to implement local rotations and analysis similar to those for splay trees. |
| title | Dynamic Treewidth in Logarithmic Time |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2504.02790 |