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# On the Node-Averaged Complexity of Locally Checkable Problems on Trees

conference contribution

posted on 2024-03-05, 12:14 authored by Balliu, Alkida, Sebastian BrandtSebastian Brandt, Kuhn, Fabian, Olivetti, Dennis, Schmid, GustavOver the past decade, a long line of research has investigated the distributed complexity landscape of locally checkable labeling (LCL) problems on bounded-degree graphs, culminating in an almost-complete classification on general graphs and a complete classification on trees. The latter states that, on bounded-degree trees, any LCL problem has deterministic worst-case time complexity O(1), Θ(log* n), Θ(log n), or Θ(n^(1/k)) for some positive integer k, and all of those complexity classes are nonempty. Moreover, randomness helps only for (some) problems with deterministic worst-case complexity Θ(log n), and if randomness helps (asymptotically), then it helps exponentially.
In this work, we study how many distributed rounds are needed on average per node in order to solve an LCL problem on trees. We obtain a partial classification of the deterministic node-averaged complexity landscape for LCL problems. As our main result, we show that every problem with worst-case round complexity O(log n) has deterministic node-averaged complexity O(log* n). We further establish bounds on the node-averaged complexity of problems with worst-case complexity Θ(n^(1/k)): we show that all these problems have node-averaged complexity Ω^~(n^(1 / (2^k - 1))), and that this lower bound is tight for some problems.

## History

## Preferred Citation

Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Dennis Olivetti, Gustav Schmid. On the Node-Averaged Complexity of Locally Checkable Problems on Trees. In: 37th International Symposium on Distributed Computing. 2023.## Primary Research Area

- Algorithmic Foundations and Cryptography

## Name of Conference

DISC International Symposium on Distributed Computing (DISC)## Legacy Posted Date

2023-09-08## Open Access Type

- Gold

## BibTeX

@inproceedings{cispa_all_4032, author = {Alkida Balliu AND Sebastian Brandt AND Fabian Kuhn AND Dennis Olivetti AND Gustav Schmid}, title = {On the Node-Averaged Complexity of Locally Checkable Problems on Trees}, booktitle = {37th International Symposium on Distributed Computing}, year = {2023} }## Usage metrics

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