## Abstract

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.^{1} 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 [4].

Original language | English |
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Title of host publication | 31st Annual European Symposium on Algorithms, ESA 2023 |

Editors | Inge Li Gortz, Martin Farach-Colton, Simon J. Puglisi, Grzegorz Herman |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772952 |

DOIs | |

State | Published - Sep 2023 |

Event | 31st Annual European Symposium on Algorithms, ESA 2023 - Amsterdam, Netherlands Duration: 4 Sep 2023 → 6 Sep 2023 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 274 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 31st Annual European Symposium on Algorithms, ESA 2023 |
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Country/Territory | Netherlands |

City | Amsterdam |

Period | 4/09/23 → 6/09/23 |

### Funding

Funders | Funder number |
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Blavatnik Research Foundation | 2019715/CCF-20-08551 |

National Science Foundation | 260/18 |

National Science Foundation | |

United States-Israel Binational Science Foundation | |

Israel Science Foundation |

## Keywords

- BFS
- Computational geometry
- Dijkstra’s algorithm
- disk graphs
- geometric optimization
- reverse shortest path