Please use this identifier to cite or link to this item: doi:10.22028/D291-26476
Title: New lower bounds for Hopcroft´s problem
Author(s): Erickson, Jeff
Language: English
Year of Publication: 1994
DDC notations: 004 Computer science, internet
Publikation type: Report
Abstract: We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in \mathbb{R}^{d}, is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time \Omega(n\log m+n^{2/3}m^{2/3}+m\log n) in two dimensions, or \Omega(n\log m+n^{5/6}m^{1/2}+n^{1/2}m^{5/6}+m\log n) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcrofts problem in four or more dimensions. Our planar lower bound is within a factor of 2^{O(\log*(n+m))} of the best known upper bound, due to Matousek. Previously, the best known lower bound, in any dimension, was \Omega(n\log m+m\log n). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called a monochromatic cover, and derive lower bounds on the complexity of this representation. We then show that the running time of any partitioning algorithm is bounded below by the size of some monochromatic cover.
Link to this record: urn:nbn:de:bsz:291-scidok-51801
Series name: Technischer Bericht / A / Fachbereich Informatik, Universität des Saarlandes
Series volume: 1994/04
Date of registration: 5-Apr-2013
Faculty: MI - Fakultät für Mathematik und Informatik
Department: MI - Informatik
Collections:SciDok - Der Wissenschaftsserver der Universität des Saarlandes

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