2026-07-23T01:12:43Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:13845 2024-03-06T10:52:52Z ddc:004 open_access

Sublinear Average-Case Shortest Paths in Weighted Unit-Disk Graphs Karczmarz, Adam Pawlewicz, Jakub Sankowski, Piotr unit-disk graphs shortest paths distance oracles We consider the problem of computing shortest paths in weighted unit-disk graphs in constant dimension d. Although the single-source and all-pairs variants of this problem are well-studied in the plane case, no non-trivial exact distance oracles for unit-disk graphs have been known to date, even for d = 2. The classical result of Sedgewick and Vitter [Algorithmica '86] shows that for weighted unit-disk graphs in the plane the A^* search has average-case performance superior to that of a standard shortest path algorithm, e.g., Dijkstra’s algorithm. Specifically, if the n corresponding points of a weighted unit-disk graph G are picked from a unit square uniformly at random, and the connectivity radius is r ∈ (0,1), A^* finds a shortest path in G in O(n) expected time when r = Ω(√{log n/n}), even though G has Θ((nr)²) edges in expectation. In other words, the work done by the algorithm is in expectation proportional to the number of vertices and not the number of edges. In this paper, we break this natural barrier and show even stronger sublinear time results. We propose a new heuristic approach to computing point-to-point exact shortest paths in unit-disk graphs. We analyze the average-case behavior of our heuristic using the same random graph model as used by Sedgewick and Vitter and prove it superior to A^*. Specifically, we show that, if we are able to report the set of all k points of G from an arbitrary rectangular region of the plane in O(k + t(n)) time, then a shortest path between arbitrary two points of such a random graph on the plane can be found in O(1/r² + t(n)) expected time. In particular, the state-of-the-art range reporting data structures imply a sublinear expected bound for all r = Ω(√{log n/n}) and O(√n) expected bound for r = Ω(n^{-1/4}) after only near-linear preprocessing of the point set. Our approach naturally generalizes to higher dimensions d ≥ 3 and yields sublinear expected bounds for all d = O(1) and sufficiently large r. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Adam Karczmarz and Jakub Pawlewicz and Piotr Sankowski 2021 Is Part Of LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2021.46 urn:nbn:de:0030-drops-138454 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.46 eng https://creativecommons.org/licenses/by/4.0/legalcode