The Unweighted and Weighted Reverse Shortest Path Problem for Disk Graphs
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].
Computational geometry
geometric optimization
disk graphs
BFS
Dijkstra’s algorithm
reverse shortest path
Theory of computation~Computational geometry
67:1-67:14
Regular Paper
https://doi.org/10.48550/arXiv.2307.14663
Haim
Kaplan
Haim Kaplan
School of Computer Science, Tel Aviv University, Israel
https://orcid.org/0000-0001-9586-8002
Work partially supported by Grant 1595/19 from the Israel Science Foundation and by the Blavatnik Research Foundation.
Matthew J.
Katz
Matthew J. Katz
Department of Computer Science, Ben Gurion University, Beer-Sheva, Israel
https://orcid.org/0000-0002-0672-729X
Work partially supported by Grant 2019715/CCF-20-08551 from the US-Israel Binational Science Foundation/US National Science Foundation.
Rachel
Saban
Rachel Saban
Department of Computer Science, Ben Gurion University, Beer-Sheva, Israel
Micha
Sharir
Micha Sharir
School of Computer Science, Tel Aviv University, Israel
https://orcid.org/0000-0002-2541-3763
Work partially supported by Grant 260/18 from the Israel Science Foundation.
10.4230/LIPIcs.ESA.2023.67
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Haim Kaplan, Matthew J. Katz, Rachel Saban, and Micha Sharir
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