Constructing strongly convex hulls using exact or rounded arithmetic

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

One useful generalization of the convex hull of a set S of n points is the e{open}-strongly convex δ-hull. It is defined to be a convex polygon with vertices taken from S such that no point in S lies farther than δ outside and such that even if the vertices of are perturbed by as much as e{open}, remains convex. It was an open question as to whether an e{open}-strongly convex O(e{open})-hull existed for all positive e{open}. We give here an O(n log n) algorithm for constructing it (which thus proves its existence). This algorithm uses exact rational arithmetic. We also show how to construct an e{open}-strongly convex O(e{open} + μ)-hull in O(n log n) time using rounded arithmetic with rounding unit μ. This is the first rounded-arithmetic convex-hull algorithm which guarantees a convex output and which has error independent of n.

Original languageEnglish (US)
Pages (from-to)345-364
Number of pages20
JournalAlgorithmica (New York)
Volume8
Issue number1-6
DOIs
StatePublished - Dec 1992
Externally publishedYes

Fingerprint

Convex Hull
Convex polygon
Rounding
Unit
Output

Keywords

  • Approximate arithmetic
  • Convex hulls
  • Exact rational arithmetic
  • Numerical analysis
  • Robust geometry
  • Strong and weak convexity

ASJC Scopus subject areas

  • Applied Mathematics
  • Safety, Risk, Reliability and Quality
  • Software
  • Computer Graphics and Computer-Aided Design

Cite this

Constructing strongly convex hulls using exact or rounded arithmetic. / Li, Zhenyu; Milenkovic, Victor.

In: Algorithmica (New York), Vol. 8, No. 1-6, 12.1992, p. 345-364.

Research output: Contribution to journalArticle

@article{bb46fd5a9cb64d12ad1b94f318567018,
title = "Constructing strongly convex hulls using exact or rounded arithmetic",
abstract = "One useful generalization of the convex hull of a set S of n points is the e{open}-strongly convex δ-hull. It is defined to be a convex polygon with vertices taken from S such that no point in S lies farther than δ outside and such that even if the vertices of are perturbed by as much as e{open}, remains convex. It was an open question as to whether an e{open}-strongly convex O(e{open})-hull existed for all positive e{open}. We give here an O(n log n) algorithm for constructing it (which thus proves its existence). This algorithm uses exact rational arithmetic. We also show how to construct an e{open}-strongly convex O(e{open} + μ)-hull in O(n log n) time using rounded arithmetic with rounding unit μ. This is the first rounded-arithmetic convex-hull algorithm which guarantees a convex output and which has error independent of n.",
keywords = "Approximate arithmetic, Convex hulls, Exact rational arithmetic, Numerical analysis, Robust geometry, Strong and weak convexity",
author = "Zhenyu Li and Victor Milenkovic",
year = "1992",
month = "12",
doi = "10.1007/BF01758851",
language = "English (US)",
volume = "8",
pages = "345--364",
journal = "Algorithmica",
issn = "0178-4617",
publisher = "Springer New York",
number = "1-6",

}

TY - JOUR

T1 - Constructing strongly convex hulls using exact or rounded arithmetic

AU - Li, Zhenyu

AU - Milenkovic, Victor

PY - 1992/12

Y1 - 1992/12

N2 - One useful generalization of the convex hull of a set S of n points is the e{open}-strongly convex δ-hull. It is defined to be a convex polygon with vertices taken from S such that no point in S lies farther than δ outside and such that even if the vertices of are perturbed by as much as e{open}, remains convex. It was an open question as to whether an e{open}-strongly convex O(e{open})-hull existed for all positive e{open}. We give here an O(n log n) algorithm for constructing it (which thus proves its existence). This algorithm uses exact rational arithmetic. We also show how to construct an e{open}-strongly convex O(e{open} + μ)-hull in O(n log n) time using rounded arithmetic with rounding unit μ. This is the first rounded-arithmetic convex-hull algorithm which guarantees a convex output and which has error independent of n.

AB - One useful generalization of the convex hull of a set S of n points is the e{open}-strongly convex δ-hull. It is defined to be a convex polygon with vertices taken from S such that no point in S lies farther than δ outside and such that even if the vertices of are perturbed by as much as e{open}, remains convex. It was an open question as to whether an e{open}-strongly convex O(e{open})-hull existed for all positive e{open}. We give here an O(n log n) algorithm for constructing it (which thus proves its existence). This algorithm uses exact rational arithmetic. We also show how to construct an e{open}-strongly convex O(e{open} + μ)-hull in O(n log n) time using rounded arithmetic with rounding unit μ. This is the first rounded-arithmetic convex-hull algorithm which guarantees a convex output and which has error independent of n.

KW - Approximate arithmetic

KW - Convex hulls

KW - Exact rational arithmetic

KW - Numerical analysis

KW - Robust geometry

KW - Strong and weak convexity

UR - http://www.scopus.com/inward/record.url?scp=52449145571&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=52449145571&partnerID=8YFLogxK

U2 - 10.1007/BF01758851

DO - 10.1007/BF01758851

M3 - Article

VL - 8

SP - 345

EP - 364

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 1-6

ER -