2026-09-20T20:58:17Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:25810 2026-09-05T19:27:44Z ddc:004 open_access

Dynamic Nearest-Neighbor Searching Under General Metrics in ℝ³ and Its Applications Agarwal, Pankaj K. Katz, Matthew J. Sharir, Micha Homothets Minkowski metric Shallow cuttings Nearest-neighbor searching Intersection and proximity graphs Reverse-shortest-path problem Let K be a compact, centrally-symmetric, strictly-convex region in ℝ³, which is a semi-algebraic set of constant complexity, i.e. the unit ball of a corresponding metric, denoted as ‖⋅‖_K. Let 𝒦 be a set of n homothetic copies of K. This paper contains two main sets of results: (i) For a storage parameter s ∈ [n,n³], 𝒦 can be preprocessed in O^*(s) expected time into a data structure of size O^*(s), so that for a query homothet K₀ of K, an intersection-detection query (determine whether K₀ intersects any member of 𝒦, and if so, report such a member) or a nearest-neighbor query (return the member of 𝒦 whose ‖⋅‖_K-distance from K₀ is smallest) can be answered in O^*(n/s^{1/3}) time; all k homothets of 𝒦 intersecting K₀ can be reported in additional O(k) time. In addition, the data structure supports insertions/deletions in O^*(s/n) amortized expected time per operation. Here the O^*(⋅) notation hides factors of the form n^ε, where ε > 0 is an arbitrarily small constant, and the constant of proportionality depends on ε. (ii) Let 𝒢(𝒦) denote the intersection graph of 𝒦. Using the above data structure, breadth-first or depth-first search on 𝒢(𝒦) can be performed in O^*(n^{3/2}) expected time. Combining this result with the so-called shrink-and-bifurcate technique, the reverse-shortest-path problem in a suitably defined proximity graph of 𝒦 can be solved in O^*(n^{62/39}) expected time. Dijkstra’s shortest-path algorithm, as well as Prim’s MST algorithm, on a ‖⋅‖_K-proximity graph on n points in ℝ³, with edges weighted by ‖⋅‖_K, can also be performed in O^*(n^{3/2}) time. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Pankaj K. Agarwal and Matthew J. Katz and Micha Sharir 2026 Is Part Of LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2026.4 urn:nbn:de:0030-drops-258102 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.4 eng https://creativecommons.org/licenses/by/4.0/legalcode