# Almost global problems in the LOCAL model

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openAccess

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Conference article in proceedings

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##### Date

2018

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##### Mcode

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##### Language

en

##### Pages

1-16

##### Series

32nd International Symposium on Distributed Computing (DISC 2018), Leibniz International Proceedings in Informatics (LIPIcs), Volume 121

##### Abstract

The landscape of the distributed time complexity is nowadays well-understood for subpolynomial complexities. When we look at deterministic algorithms in the LOCAL model and locally checkable problems (LCLs) in bounded-degree graphs, the following picture emerges: - There are lots of problems with time complexities Theta(log^* n) or Theta(log n). - It is not possible to have a problem with complexity between omega(log^* n) and o(log n). - In general graphs, we can construct LCL problems with infinitely many complexities between omega(log n) and n^{o(1)}. - In trees, problems with such complexities do not exist. However, the high end of the complexity spectrum was left open by prior work. In general graphs there are problems with complexities of the form Theta(n^alpha) for any rational 0 < alpha <=1/2, while for trees only complexities of the form Theta(n^{1/k}) are known. No LCL problem with complexity between omega(sqrt{n}) and o(n) is known, and neither are there results that would show that such problems do not exist. We show that: - In general graphs, we can construct LCL problems with infinitely many complexities between omega(sqrt{n}) and o(n). - In trees, problems with such complexities do not exist. Put otherwise, we show that any LCL with a complexity o(n) can be solved in time O(sqrt{n}) in trees, while the same is not true in general graphs.##### Description

##### Keywords

Distributed complexity theoryDistributed complexity theory, Locally checkable labellings, LOCAL model

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##### Citation

Balliu, A, Brandt, S, Olivetti, D & Suomela, J 2018, Almost global problems in the LOCAL model . in 32nd International Symposium on Distributed Computing (DISC 2018) . vol. 121, 9, Leibniz International Proceedings in Informatics (LIPIcs), vol. 121, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 1-16, International Symposium on Distributed Computing, New Orleans, Louisiana, United States, 15/10/2018 . https://doi.org/10.4230/LIPIcs.DISC.2018.9