## Abstract

Let S be a set of n geometric objects of constant complexity (e.g., points, line segments, disks, ellipses) in R^{2}, and let ϱ : S × S → R_{≥0} be a distance function on S. For a parameter r ≥ 0, we define the proximity graph G(r) = (S, E) where E = {(e1, e2) ∈ S × S | e1 ≠ e2, ϱ(e1, e2) ≤ r}. Given S, s, t ∈ S, and an integer k ≥ 1, the reverse-shortest-path (RSP) problem asks for computing the smallest value r^{∗} ≥ 0 such that G(r^{∗}) contains a path from s to t of length at most k. In this paper we present a general randomized technique that solves the RSP problem efficiently for a large family of geometric objects and distance functions. Using standard, and sometimes more involved, semi-algebraic range-searching techniques, we first give an efficient algorithm for the decision problem, namely, given a value r ≥ 0, determine whether G(r) contains a path from s to t of length at most k. Next, we adapt our decision algorithm and combine it with a random-sampling method to compute r^{∗}, by efficiently performing a binary search over an implicit set of O(n^{2}) candidate values that contains r^{∗}. We illustrate the versatility of our general technique by applying it to a variety of geometric proximity graphs. For example, we obtain (i) an O^{∗}(n^{4/3}) expected-time randomized algorithm (where O^{∗}(·) hides polylog(n) factors) for the case where S is a set of pairwise-disjoint line segments in R^{2} and ϱ(e1, e2) = minx∈e1,y∈e_{2} ∥x - y∥ (where ∥ · ∥ is the Euclidean distance), and (ii) an O^{∗}(n + m^{4/3}) expected-time randomized algorithm for the case where S is a set of m points lying on an x-monotone polygonal chain T with n vertices, and ϱ(p, q), for p, q ∈ S, is the smallest value h such that the points p^{′} := p + (0, h) and q^{′} := q + (0, h) are visible to each other, i.e., all points on the segment p^{′}q^{′} lie above or on the polygonal chain T.

Original language | American English |
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Title of host publication | 33rd International Symposium on Algorithms and Computation, ISAAC 2022 |

Editors | Sang Won Bae, Heejin Park |

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

ISBN (Electronic) | 9783959772587 |

DOIs | |

State | Published - 1 Dec 2022 |

Event | 33rd International Symposium on Algorithms and Computation, ISAAC 2022 - Virtual, Online, Korea, Republic of Duration: 19 Dec 2022 → 21 Dec 2022 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 248 |

### Conference

Conference | 33rd International Symposium on Algorithms and Computation, ISAAC 2022 |
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Country/Territory | Korea, Republic of |

City | Virtual, Online |

Period | 19/12/22 → 21/12/22 |

## Keywords

- Geometric optimization
- proximity graphs
- reverse shortest path
- semi-algebraic range searching

## All Science Journal Classification (ASJC) codes

- Software