## 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={(e_{1},e_{2})∈S×S|e_{1}≠e_{2},ϱ(e_{1},e_{2})≤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 ‘critical’ 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 (possibly intersecting) line segments in R^{2} and ϱ(e_{1},e_{2})=min_{x∈e1,y∈e2}‖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 | English |
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Article number | 102053 |

Journal | Computational Geometry: Theory and Applications |

Volume | 117 |

DOIs | |

State | Published - Feb 2024 |

## Keywords

- Geometric optimization
- Proximity graphs
- Range searching
- Reverse shortest paths
- Semi-algebraic sets