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

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

Thumbnail PDF


  • Filesize: 2.04 MB
  • 20 pages

Document Identifiers

Author Details

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.

Cite AsGet BibTex

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


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Susanne Albers and Sebastian Schraink. Tight bounds for online coloring of basic graph classes. Algorithmica, 83(1):337-360, 2021. URL: https://doi.org/10.1007/s00453-020-00759-7.
  2. Noga Alon, Ronitt Rubinfeld, Shai Vardi, and Ning Xie. Space-efficient local computation algorithms. In Proc. 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2012), pages 1132-1139. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.89.
  3. Sepehr Assadi, Krzysztof Onak, Baruch Schieber, and Shay Solomon. Fully dynamic maximal independent set with sublinear update time. In Proc. 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2018), pages 815-826, 2018. URL: https://doi.org/10.1145/3188745.3188922.
  4. Alkida Balliu, Sebastian Brandt, Yi-Jun Chang, Dennis Olivetti, Mikaël Rabie, and Jukka Suomela. The distributed complexity of locally checkable problems on paths is decidable. In Proc. 38th ACM Symposium on Principles of Distributed Computing (PODC 2019), pages 262-271. ACM Press, 2019. URL: https://doi.org/10.1145/3293611.3331606.
  5. Alkida Balliu, Sebastian Brandt, Yuval Efron, Juho Hirvonen, Yannic Maus, Dennis Olivetti, and Jukka Suomela. Classification of distributed binary labeling problems. In Proc. 34th International Symposium on Distributed Computing (DISC 2020), pages 17:1-17:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.DISC.2020.17.
  6. Alkida Balliu, Sebastian Brandt, Juho Hirvonen, Dennis Olivetti, Mikaël Rabie, and Jukka Suomela. Lower bounds for maximal matchings and maximal independent sets. Journal of the ACM, 68(5), 2021. URL: https://doi.org/10.1145/3461458.
  7. Alkida Balliu, Sebastian Brandt, Dennis Olivetti, Jan Studený, Jukka Suomela, and Aleksandr Tereshchenko. Locally checkable problems in rooted trees. In Proc. 40th ACM Symposium on Principles of Distributed Computing (PODC 2021), pages 263-272. ACM Press, 2021. URL: https://doi.org/10.1145/3465084.3467934.
  8. Alkida Balliu, Juho Hirvonen, Dennis Olivetti, and Jukka Suomela. Hardness of minimal symmetry breaking in distributed computing. In Proc. 38th ACM Symposium on Principles of Distributed Computing (PODC 2019), pages 369-378. ACM Press, 2019. URL: https://doi.org/10.1145/3293611.3331605.
  9. Leonid Barenboim and Tzalik Maimon. Fully dynamic graph algorithms inspired by distributed computing: Deterministic maximal matching and edge coloring in sublinear update-time. ACM Journal of Experimental Algorithmics, 24, 2019. Google Scholar
  10. Dwight R. Bean. Effective coloration. The Journal of Symbolic Logic, 41(2):469-480, 1976. URL: https://doi.org/10.2307/2272247.
  11. Sayan Bhattacharya, Deeparnab Chakrabarty, Monika Henzinger, and Danupon Nanongkai. Dynamic algorithms for graph coloring. In Proc. 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2018), pages 1-20. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.1.
  12. Maria Paola Bianchi, Hans-Joachim Böckenhauer, Juraj Hromkovič, and Lucia Keller. Online coloring of bipartite graphs with and without advice. In Computing and Combinatorics, pages 519-530, 2012. URL: https://doi.org/10.1007/978-3-642-32241-9_44.
  13. Sebastian Brandt, Juho Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, Patric R. J. Östergård, Christopher Purcell, Joel Rybicki, Jukka Suomela, and Przemysław Uznański. LCL problems on grids. In Proc. 36th ACM Symposium on Principles of Distributed Computing (PODC 2017), pages 101-110. ACM Press, 2017. URL: https://doi.org/10.1145/3087801.3087833.
  14. Elisabet Burjons, Juraj Hromkovič, Xavier Muñoz, and Walter Unger. Online graph coloring with advice and randomized adversary. In Proc. 42nd International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2016), pages 229-240. Springer, 2016. URL: https://doi.org/10.1007/978-3-662-49192-8_19.
  15. Yi-Jun Chang. The complexity landscape of distributed locally checkable problems on trees. In Proc. 34th International Symposium on Distributed Computing (DISC 2020). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.DISC.2020.18.
  16. Yi-Jun Chang, Tsvi Kopelowitz, and Seth Pettie. An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model. In Proc. 57th IEEE Symposium on Foundations of Computer Science (FOCS 2016), pages 615-624. IEEE, 2016. URL: https://doi.org/10.1109/FOCS.2016.72.
  17. Yi-Jun Chang and Seth Pettie. A Time Hierarchy Theorem for the LOCAL Model. SIAM Journal on Computing, 48(1):33-69, 2019. URL: https://doi.org/10.1137/17M1157957.
  18. Yi-Jun Chang, Jan Studený, and Jukka Suomela. Distributed graph problems through an automata-theoretic lens. In Proc. 28th International Colloquium on Structural Information and Communication Complexity (SIROCCO 2021), pages 31-49. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-79527-6_3.
  19. Richard Cole and Uzi Vishkin. Deterministic coin tossing with applications to optimal parallel list ranking. Information and Control, 70(1):32-53, 1986. URL: https://doi.org/10.1016/S0019-9958(86)80023-7.
  20. Stefan Dobrev, Rastislav Královič, and Richard Královič. Independent set with advice: The impact of graph knowledge. In Proc. 10th Workshop on Approximation and Online Algorithms (WAOA 2012). Springer, 2013. URL: https://doi.org/10.1007/978-3-642-38016-7_2.
  21. Yuhao Du and Hengjie Zhang. Improved algorithms for fully dynamic maximal independent set, 2018. URL: https://arxiv.org/abs/1804.08908.
  22. Yuval Emek, Pierre Fraigniaud, Amos Korman, and Adi Rosén. Online computation with advice. In Proc. 36th edition of the International Colloquium on Automata, Languages and Programming (ICALP 2009), pages 427-438. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-02927-1_36.
  23. Yuval Emek, Shay Kutten, and Roger Wattenhofer. Online matching: Haste makes waste! In Proc. 48th Annual ACM Symposium on Theory of Computing (STOC 2016), pages 333-344, 2016. URL: https://doi.org/10.1145/2897518.2897557.
  24. Guy Even, Moti Medina, and Dana Ron. Deterministic stateless centralized local algorithms for bounded degree graphs. In Proc. 22nd European Symposium on Algorithms (ESA 2014), pages 394-405. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-44777-2_33.
  25. Mohsen Ghaffari, David G. Harris, and Fabian Kuhn. On derandomizing local distributed algorithms. In Proc. 59th IEEE Annual Symposium on Foundations of Computer Science (FOCS 2018), pages 662-673. IEEE, 2018. URL: https://doi.org/10.1109/FOCS.2018.00069.
  26. Mohsen Ghaffari, Fabian Kuhn, and Yannic Maus. On the complexity of local distributed graph problems. In Proc. 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2017), pages 784-797. ACM Press, 2017. URL: https://doi.org/10.1145/3055399.3055471.
  27. Mika Göös, Juho Hirvonen, Reut Levi, Moti Medina, and Jukka Suomela. Non-local probes do not help with many graph problems. In Proc. 30th International Symposium on Distributed Computing (DISC 2016). Springer, 2016. URL: https://doi.org/10.1007/978-3-662-53426-7_15.
  28. Manoj Gupta and Shahbaz Khan. Simple dynamic algorithms for maximal independent set and other problems, 2018. URL: https://arxiv.org/abs/1804.01823.
  29. András Gyárfás and Jenő Lehel. On-line and first fit colorings of graphs. Journal of Graph Theory, 12(2):217-227, 1988. URL: https://doi.org/h10.1002/jgt.3190120212.
  30. Magnús M. Halldórsson, Kazuo Iwama, Shuichi Miyazaki, and Shiro Taketomi. Online independent sets. Theoretical Computer Science, 289(2):953-962, 2002. URL: https://doi.org/10.1016/S0304-3975(01)00411-X.
  31. Magnús M. Halldórsson. Parallel and on-line graph coloring. Journal of Algorithms, 23(2):265-280, 1997. URL: https://doi.org/10.1006/jagm.1996.0836.
  32. Magnús M. Halldórsson. Online coloring known graphs. Electronic Journal of Combinatorics, 7, 2000. URL: https://doi.org/10.37236/1485.
  33. Magnús M. Halldórsson and Mario Szegedy. Lower bounds for on-line graph coloring. Theoretical Computer Science, 130(1):163-174, 1994. URL: https://doi.org/10.1016/0304-3975(94)90157-0.
  34. Zoran Ivković and Errol L. Lloyd. Fully dynamic maintenance of vertex cover. In Proc. 19th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 1993). Springer, 1994. Google Scholar
  35. Richard M. Karp, Umesh V. Vazirani, and Vijay V. Vazirani. An optimal algorithm for on-line bipartite matching. In Proc. 22nd Annual ACM Symposium on Theory of Computing (STOC 1990), pages 352-358, 1990. URL: https://doi.org/10.1145/100216.100262.
  36. Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. Local computation: Lower and upper bounds. Journal of the ACM, 63(2):1-44, 2016. URL: https://doi.org/10.1145/2742012.
  37. Nathan Linial. Locality in distributed graph algorithms. SIAM Journal on Computing, 21(1):193-201, 1992. URL: https://doi.org/10.1137/0221015.
  38. László Lovász, Michael Saks, and W.T. Trotter. An on-line graph coloring algorithm with sublinear performance ratio. Discrete Mathematics, 75(1):319-325, 1989. URL: https://doi.org/10.1016/0012-365X(89)90096-4.
  39. Yishay Mansour, Aviad Rubinstein, Shai Vardi, and Ning Xie. Converting online algorithms to local computation algorithms. In Proc. 39th International Colloquium on Automata, Languages and Programming (ICALP 2012), pages 653-664. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-31594-7_55.
  40. Yishay Mansour and Shai Vardi. A local computation approximation scheme to maximum matching. In Proc. 16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2013) and 17th International Workshop on Randomization and Computation (RANDOM 2013), pages 260-273. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-40328-6_19.
  41. Darya Melnyk, Jukka Suomela, and Neven Villani. Mending partial solutions with few changes. In Proc. 25th International Conference on Principles of Distributed Systems (OPODIS 2022), Leibniz International Proceedings in Informatics (LIPIcs). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  42. Moni Naor and Larry Stockmeyer. What can be computed locally? SIAM Journal on Computing, 24(6):1259-1277, 1995. URL: https://doi.org/10.1137/S0097539793254571.
  43. Ofer Neiman and Shay Solomon. Simple deterministic algorithms for fully dynamic maximal matching. ACM Transactions on Algorithms, 12(1), 2015. URL: https://doi.org/10.1145/2700206.
  44. David Peleg. Distributed Computing: A Locality-Sensitive Approach. SIAM, 2000. URL: https://doi.org/10.1137/1.9780898719772.
  45. Will Rosenbaum and Jukka Suomela. Seeing far vs. seeing wide: volume complexity of local graph problems. In Proc. 39th ACM Symposium on Principles of Distributed Computing (PODC 2020), pages 89-98. ACM Press, 2020. URL: https://doi.org/10.1145/3382734.3405721.
  46. Ronitt Rubinfeld, Gil Tamir, Shai Vardi, and Ning Xie. Fast local computation algorithms. In Proc. 2nd Symposium on Innovations in Computer Science (ICS 2011), pages 223-238, 2011. Google Scholar
  47. Sundar Vishwanathan. Randomized online graph coloring. Journal of Algorithms, 13(4):657-669, 1992. URL: https://doi.org/10.1016/0196-6774(92)90061-G.
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail