Locality in Online, Dynamic, Sequential, and Distributed Graph Algorithms

Authors Amirreza Akbari , Navid Eslami, Henrik Lievonen , Darya Melnyk, Joona Särkijärvi, Jukka Suomela

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Amirreza Akbari
  • Aalto University, Espoo, Finland
Navid Eslami
  • Aalto University, Espoo, Finland
  • Sharif University of Technology, Tehran, Iran
Henrik Lievonen
  • Aalto University, Espoo, Finland
Darya Melnyk
  • Aalto University, Espoo, Finland
  • TU Berlin, Germany
Joona Särkijärvi
  • Aalto University, Espoo, Finland
Jukka Suomela
  • Aalto University, Espoo, Finland


We would like to thank Alkida Balliu, Sameep Dahal, Chetan Gupta, Fabian Kuhn, Dennis Olivetti, Jan Studený, and Jara Uitto for useful discussions. We would also like to thank the anonymous reviewers for the very helpful feedback they have provided for previous versions of this work.

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Amirreza Akbari, Navid Eslami, Henrik Lievonen, Darya Melnyk, Joona Särkijärvi, and Jukka Suomela. Locality in Online, Dynamic, Sequential, and Distributed Graph Algorithms. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


In this work, we give a unifying view of locality in four settings: distributed algorithms, sequential greedy algorithms, dynamic algorithms, and online algorithms. We introduce a new model of computing, called the online-LOCAL model: the adversary presents the nodes of the input graph one by one, in the same way as in classical online algorithms, but for each node we get to see its radius-T neighborhood before choosing the output. Instead of looking ahead in time, we have the power of looking around in space. We compare the online-LOCAL model with three other models: the LOCAL model of distributed computing, where each node produces its output based on its radius-T neighborhood, the SLOCAL model, which is the sequential counterpart of LOCAL, and the dynamic-LOCAL model, where changes in the dynamic input graph only influence the radius-T neighborhood of the point of change. The SLOCAL and dynamic-LOCAL models are sandwiched between the LOCAL and online-LOCAL models. In general, all four models are distinct, but we study in particular locally checkable labeling problems (LCLs), which is a family of graph problems extensively studied in the context of distributed graph algorithms. We prove that for LCL problems in paths, cycles, and rooted trees, all four models are roughly equivalent: the locality of any LCL problem falls in the same broad class - O(log* n), Θ(log n), or n^Θ(1) - in all four models. In particular, this result enables one to generalize prior lower-bound results from the LOCAL model to all four models, and it also allows one to simulate e.g. dynamic-LOCAL algorithms efficiently in the LOCAL model. We also show that this equivalence does not hold in two-dimensional grids or general bipartite graphs. We provide an online-LOCAL algorithm with locality O(log n) for the 3-coloring problem in bipartite graphs - this is a problem with locality Ω(n^{1/2}) in the LOCAL model and Ω(n^{1/10}) in the SLOCAL model.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Computing methodologies → Distributed algorithms
  • Theory of computation → Dynamic graph algorithms
  • Online computation
  • spatial advice
  • distributed algorithms
  • computational complexity


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