New results on stabbing segments with a polygon

We consider a natural variation of the concept of stabbing a set of segments with a simple polygon: a segment s is stabbed by a simple polygon P if at least one endpoint of s is contained in P, and a segment set S is stabbed by P if P stabs every element of S. Given a segment set S, we study the pro...

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Bibliographic Details
Published in:Computational geometry : theory and applications Vol. 48; no. 1; pp. 14 - 29
Main Authors: Díaz-Báñez, José Miguel, Korman, Matias, Pérez-Lantero, Pablo, Pilz, Alexander, Seara, Carlos, Silveira, Rodrigo I.
Format: Journal Article Publication
Language:English
Published: Elsevier B.V 01.01.2015
Subjects:
12 Field theory and polynomials
12Y05 Computational aspects of field theory and polynomials
65 Numerical analysis
65D Numerical approximation and computational geometry
Algorithms
Anàlisi numèrica
Classificació AMS
Computational geometry
Convexity
Hulls (structures)
Imprecise points
Matemàtiques i estadística
Modelització matemàtica
Numerical analysis
Polinomis
Polygons
Polynomials
Polypropylenes
Segments
Stabber
Teoria de cossos i polinomis
Transversal
Àlgebra
Àrees temàtiques de la UPC
Transversal
Imprecise points
Stabber
Segments
ISSN:0925-7721
Online Access:Get full text
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Summary:We consider a natural variation of the concept of stabbing a set of segments with a simple polygon: a segment s is stabbed by a simple polygon P if at least one endpoint of s is contained in P, and a segment set S is stabbed by P if P stabs every element of S. Given a segment set S, we study the problem of finding a simple polygon P stabbing S in a way that some measure of P (such as area or perimeter) is optimized. We show that if the elements of S are pairwise disjoint, the problem can be solved in polynomial time. In particular, this solves an open problem posed by Löffler and van Kreveld [Algorithmica 56(2), 236–269 (2010)] [16] about finding a maximum perimeter convex hull for a set of imprecise points modeled as line segments. Our algorithm can also be extended to work for a more general problem, in which instead of segments, the set S consists of a collection of point sets with pairwise disjoint convex hulls. We also prove that for general segments our stabbing problem is NP-hard.
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ISSN:0925-7721
DOI:10.1016/j.comgeo.2014.06.002