Sublinear Average-Case Shortest Paths in Weighted Unit-Disk Graphs

Authors Adam Karczmarz , Jakub Pawlewicz , Piotr Sankowski

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Author Details

Adam Karczmarz
  • Institute of Informatics, University of Warsaw, Poland
Jakub Pawlewicz
  • Institute of Informatics, University of Warsaw, Poland
Piotr Sankowski
  • Institute of Informatics, University of Warsaw, Poland

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Adam Karczmarz, Jakub Pawlewicz, and Piotr Sankowski. Sublinear Average-Case Shortest Paths in Weighted Unit-Disk Graphs. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We consider the problem of computing shortest paths in weighted unit-disk graphs in constant dimension d. Although the single-source and all-pairs variants of this problem are well-studied in the plane case, no non-trivial exact distance oracles for unit-disk graphs have been known to date, even for d = 2. The classical result of Sedgewick and Vitter [Algorithmica '86] shows that for weighted unit-disk graphs in the plane the A^* search has average-case performance superior to that of a standard shortest path algorithm, e.g., Dijkstra’s algorithm. Specifically, if the n corresponding points of a weighted unit-disk graph G are picked from a unit square uniformly at random, and the connectivity radius is r ∈ (0,1), A^* finds a shortest path in G in O(n) expected time when r = Ω(√{log n/n}), even though G has Θ((nr)²) edges in expectation. In other words, the work done by the algorithm is in expectation proportional to the number of vertices and not the number of edges. In this paper, we break this natural barrier and show even stronger sublinear time results. We propose a new heuristic approach to computing point-to-point exact shortest paths in unit-disk graphs. We analyze the average-case behavior of our heuristic using the same random graph model as used by Sedgewick and Vitter and prove it superior to A^*. Specifically, we show that, if we are able to report the set of all k points of G from an arbitrary rectangular region of the plane in O(k + t(n)) time, then a shortest path between arbitrary two points of such a random graph on the plane can be found in O(1/r² + t(n)) expected time. In particular, the state-of-the-art range reporting data structures imply a sublinear expected bound for all r = Ω(√{log n/n}) and O(√n) expected bound for r = Ω(n^{-1/4}) after only near-linear preprocessing of the point set. Our approach naturally generalizes to higher dimensions d ≥ 3 and yields sublinear expected bounds for all d = O(1) and sufficiently large r.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • unit-disk graphs
  • shortest paths
  • distance oracles


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  1. Ittai Abraham, Daniel Delling, Amos Fiat, Andrew V. Goldberg, and Renato F. Werneck. Highway dimension and provably efficient shortest path algorithms. J. ACM, 63(5), 2016. URL:
  2. Ittai Abraham and Cyril Gavoille. Object location using path separators. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Principles of Distributed Computing, PODC '06, page 188–197, New York, NY, USA, 2006. Association for Computing Machinery. URL:
  3. Hannah Bast, Daniel Delling, Andrew Goldberg, Matthias Müller-Hannemann, Thomas Pajor, Peter Sanders, Dorothea Wagner, and Renato F. Werneck. Route Planning in Transportation Networks, pages 19-80. Springer International Publishing, Cham, 2016. URL:
  4. Sergio Cabello and Miha Jejcic. Shortest paths in intersection graphs of unit disks. Comput. Geom., 48(4):360-367, 2015. URL:
  5. Timothy M. Chan. Optimal partition trees. Discret. Comput. Geom., 47(4):661-690, 2012. URL:
  6. Timothy M. Chan and Dimitrios Skrepetos. All-pairs shortest paths in unit-disk graphs in slightly subquadratic time. In Seok-Hee Hong, editor, 27th International Symposium on Algorithms and Computation, ISAAC 2016, December 12-14, 2016, Sydney, Australia, volume 64 of LIPIcs, pages 24:1-24:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL:
  7. Timothy M. Chan and Dimitrios Skrepetos. Approximate shortest paths and distance oracles in weighted unit-disk graphs. J. Comput. Geom., 10(2):3-20, 2019. URL:
  8. Panagiotis Charalampopoulos, Paweł Gawrychowski, Shay Mozes, and Oren Weimann. Almost optimal distance oracles for planar graphs. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, page 138–151, New York, NY, USA, 2019. Association for Computing Machinery. URL:
  9. Martin Dietzfelbinger and Friedhelm Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In Mike Paterson, editor, Automata, Languages and Programming, 17th International Colloquium, ICALP90, Warwick University, England, UK, July 16-20, 1990, Proceedings, volume 443 of Lecture Notes in Computer Science, pages 6-19. Springer, 1990. URL:
  10. Richard Durrett. Oriented percolation in two dimensions. The Annals of Probability, 12(4):999-1040, 1984. URL:
  11. P. Gupta and P. R. Kumar. Critical power for asymptotic connectivity. In Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171), volume 1, pages 1106-1110 vol.1, 1998. URL:
  12. P. Gupta and P. R. Kumar. The capacity of wireless networks. IEEE Transactions on Information Theory, 46(2):388-404, 2000. URL:
  13. Peter E. Hart, Nils J. Nilsson, and Bertram Raphael. A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. Syst. Sci. Cybern., 4(2):100-107, 1968. URL:
  14. Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic planar voronoi diagrams for general distance functions and their algorithmic applications. Discret. Comput. Geom., 64(3):838-904, 2020. URL:
  15. Ken-Ichi Kawarabayashi, Philip N. Klein, and Christian Sommer. Linear-space approximate distance oracles for planar, bounded-genus and minor-free graphs. In Proceedings of the 38th International Colloquim Conference on Automata, Languages and Programming - Volume Part I, ICALP'11, page 135–146, Berlin, Heidelberg, 2011. Springer-Verlag. Google Scholar
  16. Jirí Matousek. Efficient partition trees. Discret. Comput. Geom., 8:315-334, 1992. URL:
  17. Mathew D. Penrose. On k-connectivity for a geometric random graph. Random Structures & Algorithms, 15(2):145-164, 1999. Google Scholar
  18. Robert Sedgewick and Jeffrey Scott Vitter. Shortest paths in euclidean graphs. Algorithmica, 1(1):31-48, 1986. URL:
  19. Mikkel Thorup and Uri Zwick. Approximate distance oracles. J. ACM, 52(1):1–24, 2005. URL:
  20. Haitao Wang and Jie Xue. Near-optimal algorithms for shortest paths in weighted unit-disk graphs. Discret. Comput. Geom., 64(4):1141-1166, 2020. URL:
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