2026-08-17T14:17:46Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:5928 2024-03-06T10:37:18Z ddc:004 open_access

On the Number of Maximum Empty Boxes Amidst n Points Dumitrescu, Adrian Jiang, Minghui Maximum empty box Davenport-Schinzel sequence approximation algorithm data mining. We revisit the following problem (along with its higher dimensional variant): Given a set S of n points inside an axis-parallel rectangle U in the plane, find a maximum-area axis-parallel sub-rectangle that is contained in U but contains no points of S. 1. We prove that the number of maximum-area empty rectangles amidst n points in the plane is O(n log n 2^alpha(n)), where alpha(n) is the extremely slowly growing inverse of Ackermann's function. The previous best bound, O(n^2), is due to Naamad, Lee, and Hsu (1984). 2. For any d at least 3, we prove that the number of maximum-volume empty boxes amidst n points in R^d is always O(n^d) and sometimes Omega(n^floor(d/2)). This is the first superlinear lower bound derived for this problem. 3. We discuss some algorithmic aspects regarding the search for a maximum empty box in R^3. In particular, we present an algorithm that finds a (1-epsilon)-approximation of the maximum empty box amidst n points in O(epsilon^{-2} n^{5/3} log^2{n}) time. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Adrian Dumitrescu and Minghui Jiang 2016 Is Part Of LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2016.36 urn:nbn:de:0030-drops-59281 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.36 eng https://creativecommons.org/licenses/by/3.0/legalcode