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Sublinear Time Shortest Path in Expander Graphs

Authors Noga Alon , Allan Grønlund, Søren Fuglede Jørgensen, Kasper Green Larsen

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LIPIcs.MFCS.2024.8.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
Keywords
  • Shortest Path
  • Expanders
  • Breadth First Search
  • Graph Algorithms

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Abstract

Computing a shortest path between two nodes in an undirected unweighted graph is among the most basic algorithmic tasks. Breadth first search solves this problem in linear time, which is clearly also a lower bound in the worst case. However, several works have shown how to solve this problem in sublinear time in expectation when the input graph is drawn from one of several classes of random graphs. In this work, we extend these results by giving sublinear time shortest path (and short path) algorithms for expander graphs. We thus identify a natural deterministic property of a graph (that is satisfied by typical random regular graphs) which suffices for sublinear time shortest paths. The algorithms are very simple, involving only bidirectional breadth first search and short random walks. We also complement our new algorithms by near-matching lower bounds.

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Noga Alon, Allan Grønlund, Søren Fuglede Jørgensen, and Kasper Green Larsen. Sublinear Time Shortest Path in Expander Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.8

Author Details

Noga Alon
  • Princeton University, NJ, USA
Allan Grønlund
  • Kvantify, Aarhus, Denmark
Søren Fuglede Jørgensen
  • Kvantify, Aarhus, Denmark
Kasper Green Larsen
  • Aarhus University, Denmark
  • Kvantify, Aarhus, Denmark

Funding

  • Alon, Noga: Supported by NSF grant DMS-2154082.
  • Larsen, Kasper Green: Supported by a DFF Sapere Aude Research Leader Grant No. 9064-00068B.

References

  1. Noga Alon. Eigenvalues and expanders. Combinatorica, 6(2):83-96, 1986. Google Scholar
  2. Noga Alon. Explicit expanders of every degree and size. Combinatorica, 41, February 2021. Google Scholar
  3. Noga Alon, Uriel Feige, Avi Wigderson, and David Zuckerman. Derandomized graph products. Comput. Complex., 5(1):60-75, January 1995. Google Scholar
  4. Noga Alon, Shirshendu Ganguly, and Nikhil Srivastava. High-girth near-ramanujan graphs with localized eigenvectors. Israel Journal of Mathematics, 246(1), 2021. Google Scholar
  5. Edward A. Bender. The asymptotic number of non-negative integer matrices with given row and column sums. Discret. Math., 10:217-223, 1974. Google Scholar
  6. Thomas Bläsius, Cedric Freiberger, Tobias Friedrich, Maximilian Katzmann, Felix Montenegro-Retana, and Marianne Thieffry. Efficient shortest paths in scale-free networks with underlying hyperbolic geometry. ACM Trans. Algorithms, 18(2), March 2022. URL: https://doi.org/10.1145/3516483.
  7. Thomas Bläsius and Marcus Wilhelm. Deterministic performance guarantees for bidirectional bfs on real-world networks. In Combinatorial Algorithms: 34th International Workshop, IWOCA 2023, Tainan, Taiwan, June 7–10, 2023, Proceedings, pages 99-110. Springer-Verlag, 2023. Google Scholar
  8. Béla Bollobás. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European Journal of Combinatorics, 1:311-316, 1980. Google Scholar
  9. Michele Borassi, Pierluigi Crescenzi, and Luca Trevisan. An axiomatic and an average-case analysis of algorithms and heuristics for metric properties of graphs. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '17, pages 920-939. Society for Industrial and Applied Mathematics, 2017. Google Scholar
  10. Michele Borassi and Emanuele Natale. Kadabra is an adaptive algorithm for betweenness via random approximation. ACM J. Exp. Algorithmics, 24, February 2019. URL: https://doi.org/10.1145/3284359.
  11. Fan R. K. Chung. Diameters and eigenvalues. Journal of the American Mathematical Society, 2:187-196, 1989. Google Scholar
  12. Dennis de Champeaux. Bidirectional heuristic search again. J. ACM, 30(1):22-32, January 1983. Google Scholar
  13. Joel Friedman. A Proof of Alon’s Second Eigenvalue Conjecture and Related Problems. American Mathematical Society, 2008. Google Scholar
  14. Dorit S. Hochbaum. An exact sublinear algorithm for the max-flow, vertex disjoint paths and communication problems on random graphs. Operations Research, 40(5):923-935, 1992. Google Scholar
  15. Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc., 43(04):439-562, August 2006. Google Scholar
  16. J. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, FOCS '96, page 86. IEEE Computer Society, 1996. Google Scholar
  17. Eyal Lubetzky and Yuval Peres. Cutoff on all ramanujan graphs. Geometric and Functional Analysis, 26:1190-1216, 2015. Google Scholar
  18. Michael Luby and Prabhakar Ragde. A bidirectional shortest-path algorithm with good average-case behavior. Algorithmica, 4(1–4):551-567, March 1989. Google Scholar
  19. Ira Sheldon Pohl. Bi-Directional and Heuristic Search in Path Problems. PhD thesis, Stanford University, Stanford, CA, USA, 1969. Google Scholar
  20. Lenie Sint and Dennis de Champeaux. An improved bidirectional heuristic search algorithm. J. ACM, 24(2):177-191, April 1977. Google Scholar
  21. N. C. Wormald. Models of Random Regular Graphs, pages 239-298. London Mathematical Society Lecture Note Series. Cambridge University Press, 1999. Google Scholar
  22. Nicholas C. Wormald. Some problems in the enumeration of labelled graphs. Bulletin of the Australian Mathematical Society, 21(1):159-160, 1980. Google Scholar
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