Completing the Node-Averaged Complexity Landscape of LCLs on Trees
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arXiv
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| Format: | Preprint |
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2024
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| author | Balliu, Alkida Brandt, Sebastian Kuhn, Fabian Olivetti, Dennis Schmid, Gustav |
| author_facet | Balliu, Alkida Brandt, Sebastian Kuhn, Fabian Olivetti, Dennis Schmid, Gustav |
| contents | The node-averaged complexity of a problem captures the number of rounds nodes of a graph have to spend on average to solve the problem in the LOCAL model. A challenging line of research with regards to this new complexity measure is to understand the complexity landscape of locally checkable labelings (LCLs) on families of bounded-degree graphs. Particularly interesting in this context is the family of bounded-degree trees as there, for the worst-case complexity, we know a complete characterization of the possible complexities and structures of LCL problems. A first step for the node-averaged complexity case has been achieved recently [DISC '23], where the authors in particular showed that in bounded-degree trees, there is a large complexity gap: There are no LCL problems with a deterministic node-averaged complexity between $ω(\log^* n)$ and $n^{o(1)}$. For randomized algorithms, they even showed that the node-averaged complexity is either $O(1)$ or $n^{Ω(1)}$. In this work we fill in the remaining gaps and give a complete description of the node-averaged complexity landscape of LCLs on bounded-degree trees. Our contributions are threefold.
- On bounded-degree trees, there is no LCL with a node-averaged complexity between $ω(1)$ and $(\log^*n)^{o(1)}$.
- For any constants $0<r_1 < r_2 \leq 1$ and $\varepsilon>0$, there exists a constant $c$ and an LCL problem with node-averaged complexity between $Ω((\log^* n)^c)$ and $O((\log^* n)^{c+\varepsilon})$.
- For any constants $0<α\leq 1/2$ and $\varepsilon>0$, there exists an LCL problem with node-averaged complexity $Θ(n^x)$ for some $x\in [α, α+\varepsilon]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01366 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Completing the Node-Averaged Complexity Landscape of LCLs on Trees Balliu, Alkida Brandt, Sebastian Kuhn, Fabian Olivetti, Dennis Schmid, Gustav Distributed, Parallel, and Cluster Computing The node-averaged complexity of a problem captures the number of rounds nodes of a graph have to spend on average to solve the problem in the LOCAL model. A challenging line of research with regards to this new complexity measure is to understand the complexity landscape of locally checkable labelings (LCLs) on families of bounded-degree graphs. Particularly interesting in this context is the family of bounded-degree trees as there, for the worst-case complexity, we know a complete characterization of the possible complexities and structures of LCL problems. A first step for the node-averaged complexity case has been achieved recently [DISC '23], where the authors in particular showed that in bounded-degree trees, there is a large complexity gap: There are no LCL problems with a deterministic node-averaged complexity between $ω(\log^* n)$ and $n^{o(1)}$. For randomized algorithms, they even showed that the node-averaged complexity is either $O(1)$ or $n^{Ω(1)}$. In this work we fill in the remaining gaps and give a complete description of the node-averaged complexity landscape of LCLs on bounded-degree trees. Our contributions are threefold. - On bounded-degree trees, there is no LCL with a node-averaged complexity between $ω(1)$ and $(\log^*n)^{o(1)}$. - For any constants $0<r_1 < r_2 \leq 1$ and $\varepsilon>0$, there exists a constant $c$ and an LCL problem with node-averaged complexity between $Ω((\log^* n)^c)$ and $O((\log^* n)^{c+\varepsilon})$. - For any constants $0<α\leq 1/2$ and $\varepsilon>0$, there exists an LCL problem with node-averaged complexity $Θ(n^x)$ for some $x\in [α, α+\varepsilon]$. |
| title | Completing the Node-Averaged Complexity Landscape of LCLs on Trees |
| topic | Distributed, Parallel, and Cluster Computing |
| url | https://arxiv.org/abs/2405.01366 |