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Conditionally Optimal Parallel Coloring of Forests

Authors Christoph Grunau , Rustam Latypov , Yannic Maus , Shreyas Pai , Jara Uitto

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Christoph Grunau
  • ETH Zürich, Switzerland
Rustam Latypov
  • Aalto University, Finland
Yannic Maus
  • TU Graz, Austria
Shreyas Pai
  • Aalto University, Finland
Jara Uitto
  • Aalto University, Finland

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Christoph Grunau, Rustam Latypov, Yannic Maus, Shreyas Pai, and Jara Uitto. Conditionally Optimal Parallel Coloring of Forests. In 37th International Symposium on Distributed Computing (DISC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 281, pp. 23:1-23:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We show the first conditionally optimal deterministic algorithm for 3-coloring forests in the low-space massively parallel computation (MPC) model. Our algorithm runs in O(log log n) rounds and uses optimal global space. The best previous algorithm requires 4 colors [Ghaffari, Grunau, Jin, DISC'20] and is randomized, while our algorithm are inherently deterministic. Our main technical contribution is an O(log log n)-round algorithm to compute a partition of the forest into O(log n) ordered layers such that every node has at most two neighbors in the same or higher layers. Similar decompositions are often used in the area and we believe that this result is of independent interest. Our results also immediately yield conditionally optimal deterministic algorithms for maximal independent set and maximal matching for forests, matching the state of the art [Giliberti, Fischer, Grunau, SPAA'23]. In contrast to their solution, our algorithms are not based on derandomization, and are arguably simpler.

Subject Classification

ACM Subject Classification
  • Theory of computation → Massively parallel algorithms
  • massively parallel computation
  • coloring
  • forests
  • optimal


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