2026-07-25T20:49:14Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:23179 2025-10-02T12:42:07Z ddc:004 open_access

Computing Oriented Spanners and Their Dilation Buchin, Kevin Kalb, Antonia Maheshwari, Anil Odak, Saeed Rehs, Carolin Smid, Michiel Wong, Sampson spanner oriented graph dilation orientation well-separated pair decomposition minimum-perimeter triangle Given a point set P in a metric space and a real number t ≥ 1, an oriented t-spanner is an oriented graph G = (P, E), where for every pair of distinct points p and q in P, the shortest oriented closed walk in G that contains p and q is at most a factor t longer than the perimeter of the smallest triangle in P containing p and q. The oriented dilation of a graph G is the minimum t for which G is an oriented t-spanner. For arbitrary point sets of size n in ℝ^d, where d ≥ 2 is a constant, the only known oriented spanner construction is an oriented 2-spanner with binom(n,2) edges. Moreover, there exists a set P of four points in the plane, for which the oriented dilation is larger than 1.46, for any oriented graph on P. We present the first algorithm that computes, in Euclidean space, a sparse oriented spanner whose oriented dilation is bounded by a constant. More specifically, for any set of n points in ℝ^d, where d is a constant, we construct an oriented (2+ε)-spanner with 𝒪(n) edges in 𝒪(n log n) time and 𝒪(n) space. Our construction uses the well-separated pair decomposition and an algorithm that computes a (1+ε)-approximation of the minimum-perimeter triangle in P containing two given query points in 𝒪(log n) time. While our algorithm is based on first computing a suitable undirected graph and then orienting it, we show that, in general, computing the orientation of an undirected graph that minimises its oriented dilation is NP-hard, even for point sets in the Euclidean plane. We further prove that even if the oriented graph is already given, computing its oriented dilation is APSP-hard for points in a general metric space. We complement this result with an algorithm that approximates the oriented dilation of a given graph in subcubic time for point sets in ℝ^d, where d is a constant. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Kevin Buchin and Antonia Kalb and Anil Maheshwari and Saeed Odak and Carolin Rehs and Michiel Smid and Sampson Wong 2025 Is Part Of LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2025.27 urn:nbn:de:0030-drops-231792 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.27 eng https://creativecommons.org/licenses/by/4.0/legalcode