The Unweighted and Weighted Reverse Shortest Path Problem for Disk Graphs

Authors Haim Kaplan , Matthew J. Katz , Rachel Saban, Micha Sharir

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Author Details

Haim Kaplan
  • School of Computer Science, Tel Aviv University, Israel
Matthew J. Katz
  • Department of Computer Science, Ben Gurion University, Beer-Sheva, Israel
Rachel Saban
  • Department of Computer Science, Ben Gurion University, Beer-Sheva, Israel
Micha Sharir
  • School of Computer Science, Tel Aviv University, Israel

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Haim Kaplan, Matthew J. Katz, Rachel Saban, and Micha Sharir. The Unweighted and Weighted Reverse Shortest Path Problem for Disk Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 67:1-67:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We study the reverse shortest path problem on disk graphs in the plane. In this problem we consider the proximity graph of a set of n disks in the plane of arbitrary radii: In this graph two disks are connected if the distance between them is at most some threshold parameter r. The case of intersection graphs is a special case with r = 0. We give an algorithm that, given a target length k, computes the smallest value of r for which there is a path of length at most k between some given pair of disks in the proximity graph. Our algorithm runs in O^*(n^{5/4}) randomized expected time, which improves to O^*(n^{6/5}) for unit disk graphs, where all the disks have the same radius. Our technique is robust and can be applied to many variants of the problem. One significant variant is the case of weighted proximity graphs, where edges are assigned real weights equal to the distance between the disks or between their centers, and k is replaced by a target weight w. In other variants, we want to optimize a parameter different from r, such as a scale factor of the radii of the disks. The main technique for the decision version of the problem (determining whether the graph with a given r has the desired property) is based on efficient implementations of BFS (for the unweighted case) and of Dijkstra’s algorithm (for the weighted case), using efficient data structures for maintaining the bichromatic closest pair for certain bicliques and several distance functions. The optimization problem is then solved by combining the resulting decision procedure with enhanced variants of the interval shrinking and bifurcation technique of [R. Ben Avraham et al., 2015].

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Computational geometry
  • geometric optimization
  • disk graphs
  • BFS
  • Dijkstra’s algorithm
  • reverse shortest path


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  1. P. K. Agarwal. Simplex range searching and its variants: A review. In Journey through Discrete Mathematics: A Tribute to Jiří Matoušek, pages 1-30. Springer Verlag, Berlin-Heidelberg, 2017. Google Scholar
  2. P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri. Selecting distances in the plane. Algorithmica, 9:495-514, 1993. Google Scholar
  3. P. K. Agarwal, M. J. Katz, and M. Sharir. On reverse shortest paths in geometric proximity graphs. In 33rd Int. Sympos. on Algorithms and Computation (ISAAC), pages 42:1-42:19, 2022. Google Scholar
  4. R. Ben Avraham, O. Filtser, H. Kaplan, M. J. Katz, and M. Sharir. The discrete and semicontinuous Fréchet distance with shortcuts via approximate distance counting and selection. ACM Trans. Algorithms, 11, 2015, Art. 29. Google Scholar
  5. S. Cabello and M. Jejčič. Shortest paths in intersection graphs of unit disks. Comput. Geom. Theory Appls., 48:360-367, 2015. Google Scholar
  6. T. M. Chan. Dynamic generalized closest pair: Revisiting Eppstein’s technique. In Proc. 3rd Sympos. Simplicity in Algorithms (SOSA), pages 33-37, 2020. Google Scholar
  7. T. M. Chan and D. Skrepetos. All-pairs shortest paths in unit disk graphs in slightly subquadratic time. In Proc. 27th Int. Sympos on Algorithms and Computation (ISAAC), pages 24:1-24:13, 2016. Google Scholar
  8. H. Kaplan, W. Mulzer, L. Roditty, P. Seiferth, and M. Sharir. Dynamic planar voronoi diagrams for general distance functions and their algorithmic applications. Discrete Comput. Geom., 64:838-904, 2020. Google Scholar
  9. M. J. Katz and M. Sharir. Efficient algorithms for optimization problems involving semi-algebraic range searching. in URL:
  10. M. J. Katz and M. Sharir. An expander-based approach to geometric optimization. SIAM J. Comput., 26:1384-1408, 1997. Google Scholar
  11. C.-H. Liu. Nearly optimal planar k nearest neighbors queries under general distance functions. SIAM J. Comput., 51(3):723-765, 2022. Google Scholar
  12. N. Megiddo. Applying parallel computation algorithms in the design of serial algorithms. J. of the ACM, 30:852-865, 1983. Google Scholar
  13. H. Wang and J. Xue. Near-optimal algorithms for shortest paths in weighted unit-disk graphs. Discrete Comput. Geom., 64:1141-1166, 2020. Google Scholar
  14. H. Wang and Y. Zhao. Reverse shortest path problem for unit-disk graphs. In Proc. 17th Algorithms and Data Structures Sympos. (WADS), pages 655-668, 2021. Google Scholar
  15. H. Wang and Y. Zhao. Reverse shortest path problem in weighted unit-disk graphs. In Proc. 16th Int. Conf. and Workshops on Algorithms and Computation (WALCOM), pages 135-146, 2022. Google Scholar
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