Convex hull of imprecise points in o(n log n) time after preprocessing

Esther Ezra, Wolfgang Mulzer

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

3 Scopus citations

Abstract

Motivated by the desire to cope with data imprecision [29], we study methods for preprocessing a set of line-segments (or just lines) in the plane such that whenever we are given a set of points, each of which lies on a distinct object, we can compute their convex hull more eficiently than in "standard settings" (that is, without preprocessing). In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon receiving a set P of n points, each of which lies on a distinct line of L, we can construct the convex hull of P eficiently. We show that in quadratic time and space it is possible to construct a data structure on L that enables us to compute the convex hull of any such point set P in O(nα(n) log* n) expected time. If we further assume that the points are "oblivious" with respect to the data structure, the running time improves to O(nα(n)). The analysis applies almost verbatim when L is a set of line-segments, and yields similar asymptotic bounds. We present several extensions, including a trade-o- between space and query time and an output-sensitive algorithm. We also study the "dual problem" where we show how to efficiently compute the (≤k)-level of n lines in the plane, each of which lies on a distinct point (given in advance). We complement our results by Ω(n log n) lower bounds under the algebraic computation tree model for several related problems, including sorting a set of points (according to, say, their x-order), each of which lies on a given line known in advance. Therefore, the convex hull problem under our setting is easier than sorting, contrary to the "stan-dard" convex hull and sorting problems, in which the two problems require Θ(n log n) steps in the worst case (under the algebraic computation tree model).

Original languageEnglish
Title of host publicationProceedings of the 27th Annual Symposium on Computational Geometry, SCG'11
Pages11-20
Number of pages10
DOIs
StatePublished - 2011
Externally publishedYes
Event27th Annual ACM Symposium on Computational Geometry, SCG'11 - Paris, France
Duration: 13 Jun 201115 Jun 2011

Publication series

NameProceedings of the Annual Symposium on Computational Geometry

Conference

Conference27th Annual ACM Symposium on Computational Geometry, SCG'11
Country/TerritoryFrance
CityParis
Period13/06/1115/06/11

Keywords

  • Convex hull
  • Data imprecision
  • Geometric data structures
  • Planar arrangements
  • Randomized constructions

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