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Parallel Reachability and Shortest Paths on Non-Sparse Digraphs: Near-Linear Work and Sub-Square-Root Depth

Authors Vikrant Ashvinkumar, Aaron Bernstein , Maximilian Probst Gutenberg , Thatchaphol Saranurak

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • shortcut set
  • parallel reachability
  • hopset
  • parallel SSSP

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Abstract

We present parallel algorithms for computing single-source reachability and shortest paths on directed n-vertex m-edge graphs using near-linear Õ(m) work and o(√n) depth whenever m ≥ n^{1+o(1)}. At the extreme of m = Ω(n²), our reachability and shortest path algorithms have depth only n^0.136 and n^{0.25+o(1)}, respectively. The state-of-the-art parallel algorithms with near-linear work for both problems [Jambulapati et al., 2019; Cao et al., 2020; Rozhoň et al., 2023; Cao and Fineman, 2023; Brand et al., 2025] require Ω(√n) depth in all density regimes.

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Vikrant Ashvinkumar, Aaron Bernstein, Maximilian Probst Gutenberg, and Thatchaphol Saranurak. Parallel Reachability and Shortest Paths on Non-Sparse Digraphs: Near-Linear Work and Sub-Square-Root Depth. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 15:1-15:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.15

Author Details

Vikrant Ashvinkumar
  • Rutgers University, Piscataway, NJ, USA
Aaron Bernstein
  • New York University, NY, USA
Maximilian Probst Gutenberg
  • ETH Zurich, Switzerland
Thatchaphol Saranurak
  • University of Michigan, Ann Arbor, MI, USA

Funding

  • Bernstein, Aaron: Supported by Sloan Fellowship, Google Research Fellowship, NSF Grant 1942010, and Charles S. Baylis endowment at NYU.
  • Gutenberg, Maximilian Probst: The research leading to these results has received funding from grant no. 200021 204787 of the Swiss National Science Foundation.
  • Saranurak, Thatchaphol: Supported by NSF Grant CCF-2238138 and a Sloan Fellowship.

Acknowledgements

We would like to thank Shang-En Huang, who declined co-authorship despite being involved in our discussions.

References

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