New classes of distributed time complexity

Alkida Balliu, Juho Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, Dennis Olivetti, Jukka Suomela

Research output: Chapter in Book/Report/Conference proceedingConference contributionScientificpeer-review

10 Citations (Scopus)
131 Downloads (Pure)


A number of recent papers – e.g. Brandt et al. (STOC 2016), Chang et al. (FOCS 2016), Ghaffari & Su (SODA 2017), Brandt et al. (PODC 2017), and Chang & Pettie (FOCS 2017) – have advanced our understanding of one of the most fundamental questions in theory of distributed computing: what are the possible time complexity classes of LCL problems in the LOCAL model? In essence, we have a graph problem Π in which a solution can be verified by checking all radius-O(1) neighbourhoods, and the question is what is the smallest T such that a solution can be computed so that each node chooses its own output based on its radius-T neighbourhood. Here T is the distributed time complexity of Π.

The time complexity classes for deterministic algorithms in bounded-degree graphs that are known to exist by prior work are Θ(1), Θ(log n), Θ(log n), Θ(n1/k), and Θ(n). It is also known that there are two gaps: one between ω(1) and o(log log n), and another between ω(log n) and o(log n). It has been conjectured that many more gaps exist, and that the overall time hierarchy is relatively simple – indeed, this is known to be the case in restricted graph families such as cycles and grids.

We show that the picture is much more diverse than previously expected. We present a general technique for engineering LCL problems with numerous different deterministic time complexities, including Θ(logα n) for any α ≥ 1, 2Θ(log^α n) for any α ≤ 1, and Θ(nα) for any α < 1/2 in the high end of the complexity spectrum, and Θ(logα log n) for any α ≥ 1, 2Θ(log^α log^∗ n) for any α ≤ 1, and Θ((log n)α) for any α ≤ 1 in the low end of the complexity spectrum; here α is a positive rational number.

Original languageEnglish
Title of host publicationSTOC 2018 – Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
Number of pages12
ISBN (Electronic)9781450355599
Publication statusPublished - 20 Jun 2018
MoE publication typeA4 Article in a conference publication
EventACM Symposium on Theory of Computing - Los Angeles, United States
Duration: 25 Jun 201829 Jun 2018
Conference number: 50


ConferenceACM Symposium on Theory of Computing
Abbreviated titleSTOC
CountryUnited States
CityLos Angeles
Internet address


  • distributed complexity theory
  • graph algorithms
  • locally checkable labellings
  • LOCAL model

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