2026-07-22T08:54:50Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:17327 2024-03-06T10:59:28Z ddc:004 open_access

On Reverse Shortest Paths in Geometric Proximity Graphs Agarwal, Pankaj K. Katz, Matthew J. Sharir, Micha Geometric optimization proximity graphs semi-algebraic range searching reverse shortest path Let S be a set of n geometric objects of constant complexity (e.g., points, line segments, disks, ellipses) in ℝ², 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(n²) 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 ℝ² and ϱ(e₁,e₂) = 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Pankaj K. Agarwal and Matthew J. Katz and Micha Sharir 2022 Is Part Of LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ISAAC.2022.42 urn:nbn:de:0030-drops-173277 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.42 eng https://creativecommons.org/licenses/by/4.0/legalcode