An Output-Sensitive Convex Hull Algorithm for Planar Objects

TLDR: In this paper, a convex hull of a set of n planar convex objects of fixed type m is computed in O(nβ(h,m log h) time.

Abstract: A set of planar objects is said to be of type m if the convex hull of any two objects has its size bounded by 2m. In this paper, we present an algorithm based on the marriage-before-conquest paradigm to compute the convex hull of a set of n planar convex objects of fixed type m. The algorithm is output-sensitive, i.e. its time complexity depends on the size h of the computed convex hull. The main ingredient of this algorithm is a linear method to find a bridge, i.e. a facet of the convex hull intersected by a given line. We obtain an O(nβ(h,m log h)-time convex hull algorithm for planar objects. Here β(h,2)=O(1) and β(h,m) is an extremely slowly growing function. As a direct consequence, we can compute in optimal Θ(n log h) time the convex hull of disks, convex homothets, non-overlapping objects. The method described in this paper also applies to compute lower envelopes of functions. In particular, we obtain an optimal Θ(n log h)-time algorithm to compute the upper envelope of line segments.

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An Output-Sensitive Convex Hull Algorithm for Planar
Objects
Franck Nielsen, Mariette Yvinec
To cite this version:
Franck Nielsen, Mariette Yvinec. An Output-Sensitive Convex Hull Algorithm for Planar Objects.
RR-2575, INRIA. 1995. �inria-00074107�

ISSN 0249-6399
apport 

de recherche
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
An Output-Sensitive Convex Hull Algorithm for
Planar Objects
Franck Nielsen, Mariette Yvinec
N˚ 2575
Mai 1995
PROGRAMME 4

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Q1. What contributions have the authors mentioned in the paper "An output-sensitive convex hull algorithm for planar objects" ?

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