Preprocessing ambiguous imprecise points

Ivor van der Hoog, Irina Kostitsyna, Maarten Löffler, Bettina Speckmann

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Abstract

Let R = {R 1,R 2, . . . ,R n} be a set of regions and let X = {x 1, x 2, . . . , x n} be an (unknown) point set with x i ϵ R i. Region Ri represents the uncertainty region of x i. We consider the following question: how fast can we establish order if we are allowed to preprocess the regions in R? The preprocessing model of uncertainty uses two consecutive phases: a preprocessing phase which has access only to R followed by a reconstruction phase during which a desired structure on X is computed. Recent results in this model parametrize the reconstruction time by the ply of R, which is the maximum overlap between the regions in R. We introduce the ambiguity A(R) as a more fine-grained measure of the degree of overlap in R. We show how to preprocess a set of d-dimensional disks in O(n log n) time such that we can sort X (if d = 1) and reconstruct a quadtree on X (if d ≥ 1 but constant) in O(A(R)) time. If A(R) is sub-linear, then reporting the result dominates the running time of the reconstruction phase. However, we can still return a suitable data structure representing the result in O(A(R)) time. In one dimension, R is a set of intervals and the ambiguity is linked to interval entropy, which in turn relates to the well-studied problem of sorting under partial information. The number of comparisons necessary to find the linear order underlying a poset P is lower-bounded by the graph entropy of P. We show that if P is an interval order, then the ambiguity provides a constant-factor approximation of the graph entropy. This gives a lower bound of (A(R)) in all dimensions for the reconstruction phase (sorting or any proximity structure), independent of any preprocessing; hence our result is tight. Finally, our results imply that one can approximate the entropy of interval graphs in O(n log n) time, improving the O(n 2.5) bound by Cardinal et al.

Original languageEnglish
Title of host publication35th International Symposium on Computational Geometry (SoCG 2019)
EditorsGill Barequet, Yusu Wang
Place of PublicationDagstuhl, Germany
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
Pages42:1-42:16
Number of pages16
Volume129
ISBN (Electronic)9783959771047
ISBN (Print)978-3-95977-104-7
DOIs
Publication statusPublished - 1 Jun 2019
Event35th International Symposium on Computational Geometry, (SoCG2019) - Portland, United States
Duration: 18 Jun 201921 Jun 2019
http://www.wikicfp.com/cfp/servlet/event.showcfp?eventid=80745&copyownerid=35838

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume129
ISSN (Print)1868-8969

Conference

Conference35th International Symposium on Computational Geometry, (SoCG2019)
Abbreviated titleSoCG2019
CountryUnited States
CityPortland
Period18/06/1921/06/19
Internet address

Keywords

  • And phrases preprocessing
  • Entropy
  • Imprecise points
  • Proximity structures
  • Sorting

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