Near-Optimal Algorithms for Shortest Paths in Weighted Unit-Disk Graphs

Authors Haitao Wang, Jie Xue

Thumbnail PDF


  • Filesize: 0.56 MB
  • 13 pages

Document Identifiers

Author Details

Haitao Wang
  • Department of Computer Science, Utah State University, Logan, UT 84322, USA
Jie Xue
  • Department of Computer Science and Engineering, University of Minnesota - Twin Cities, Minneapolis, MN 55455, USA


The authors would like to thank Timothy Chan for the discussion, and in particular, for suggesting the algorithm for Lemma 7. Jie Xue would like to thank his advisor Ravi Janardan for his consistent advice and support.

Cite AsGet BibTex

Haitao Wang and Jie Xue. Near-Optimal Algorithms for Shortest Paths in Weighted Unit-Disk Graphs. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 60:1-60:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We revisit a classical graph-theoretic problem, the single-source shortest-path (SSSP) problem, in weighted unit-disk graphs. We first propose an exact (and deterministic) algorithm which solves the problem in O(n log^2 n) time using linear space, where n is the number of the vertices of the graph. This significantly improves the previous deterministic algorithm by Cabello and Jejčič [CGTA'15] which uses O(n^{1+delta}) time and O(n^{1+delta}) space (for any small constant delta>0) and the previous randomized algorithm by Kaplan et al. [SODA'17] which uses O(n log^{12+o(1)} n) expected time and O(n log^3 n) space. More specifically, we show that if the 2D offline insertion-only (additively-)weighted nearest-neighbor problem with k operations (i.e., insertions and queries) can be solved in f(k) time, then the SSSP problem in weighted unit-disk graphs can be solved in O(n log n+f(n)) time. Using the same framework with some new ideas, we also obtain a (1+epsilon)-approximate algorithm for the problem, using O(n log n + n log^2(1/epsilon)) time and linear space. This improves the previous (1+epsilon)-approximate algorithm by Chan and Skrepetos [SoCG'18] which uses O((1/epsilon)^2 n log n) time and O((1/epsilon)^2 n) space. Because of the Omega(n log n)-time lower bound of the problem (even when approximation is allowed), both of our algorithms are almost optimal.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Single-source shortest paths
  • Weighted unit-disk graphs
  • Geometric graph algorithms


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


  1. Jou L. Bentley. Decomposable searching problems. Information Processing Letters, 8:244-251, 1979. Google Scholar
  2. Sergio Cabello and Miha Jejčič. Shortest paths in intersection graphs of unit disks. Computational Geometry: Theory and Applications, 48:360-367, 2015. Google Scholar
  3. Timothy M. Chan and Dimitrios Skrepetos. All-pairs shortest paths in unit-disk graphs in slightly subquadratic time. In Proceedings of the 27th International Symposium on Algorithms and Computation (ISAAC), pages 24:1-24:13, 2016. Google Scholar
  4. Timothy M. Chan and Dimitrios Skrepetos. All-Pairs Shortest Paths in Geometric Intersection Graphs. In Proceedings of the 15th Algorithms and Data Structures Symposium (WADS), pages 253-264, 2017. Google Scholar
  5. Timothy M Chan and Dimitrios Skrepetos. Approximate Shortest Paths and Distance Oracles in Weighted Unit-Disk Graphs. In Proceedings of the 34th International Symposium on Computational Geometry (SoCG), pages 24:1-24:13, 2018. Google Scholar
  6. Brent N. Clark, Charles J. Colbourn, and David S. Johnson. Unit disk graphs. Discrete Mathematics, 86:165-177, 1990. Google Scholar
  7. Mark de Berg, Kevin Buchin, Bart M.P. Jansen, and Gerhard Woeginger. Fine-Grained Complexity Analysis of Two Classic TSP Variants. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP), pages 5:1-5:14, 2016. Google Scholar
  8. Steven Fortune. A sweepline algorithm for Voronoi diagrams. Algorithmica, 2:153-174, 1987. Google Scholar
  9. Jie Gao and Li Zhang. Well-separated pair decomposition for the unit-disk graph metric and its applications. SIAM Journal on Computing, 35:151-169, 2005. Google Scholar
  10. Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic planar Voronoi diagrams for general distance functions and their algorithmic applications. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2495-2504, 2017. Google Scholar
  11. Tomomi Matsui. Approximation algorithms for maximum independent set problems and fractional coloring problems on unit disk graphs. In Proceedings of Japanese Conference on Discrete and Computational Geometry (JCDCG), pages 194-200, 1998. Google Scholar
  12. Liam Roditty and Michael Segal. On bounded leg shortest paths problems. Algorithmica, 59:583-600, 2011. Google Scholar
  13. Haitao Wang and Jie Xue. Near-Optimal Algorithms for Shortest Paths in Weighted Unit-Disk Graphs. arXiv preprint, 2019. URL:
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