Largest and Smallest Convex Hulls for Imprecise Points

Löffler, Maarten; Kreveld, Marc van

doi:10.1007/s00453-008-9174-2

Cited by 95 publications

(98 citation statements)

References 25 publications

(23 reference statements)

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“…The problem MPIP is related to two other geometric optimization problems called largest and smallest convex hulls for imprecise points [10]. Given a set R of n regions that model n imprecise points in the plane, the problem largest (resp.…”

Section: Discussionmentioning

confidence: 99%

“…Note that MPIP is equivalent to smallest-perimeter convex hull for imprecise points as segments. The dual problem of largest convex hull for imprecise points as segments has been recently shown to be NP-hard [10] for both area and perimeter measures, and to admit a PTAS [9] for the area measure. We note that the core-set technique used in obtaining the PTAS for largest-area convex hull [9] cannot be used for MPIP because, for minimization, there could be many optimal or near-optimal solutions that are far from each other.…”

Section: Discussionmentioning

confidence: 99%

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Minimum-Perimeter Intersecting Polygons

Dumitrescu

Jiang

2011

Algorithmica

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Given a set S of segments in the plane, a polygon P is an intersecting polygon of S if every segment in S intersects the interior or the boundary of P . The problem MPIP of computing a minimum-perimeter intersecting polygon of a given set of n segments in the plane was first considered by Rappaport in 1995. This problem is not known to be polynomial, nor it is known to be NP-hard. Rappaport (1995) gave an exponential-time exact algorithm for MPIP. Hassanzadeh and Rappaport (2009) gave a polynomial-time approximation algorithm with ratio π 2 ≈ 1.57. In this paper, we present two improved approximation algorithms for MPIP: a 1.28-approximation algorithm by linear programming, and a polynomial-time approximation scheme by discretization and enumeration. Our algorithms can be generalized for computing an approximate minimum-perimeter intersecting polygon of a set of convex polygons in the plane. From the other direction, we show that computing a minimum-perimeter intersecting polygon of a set of (not necessarily convex) simple polygons is NP-hard.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Minimum-Perimeter Intersecting Polygons

Dumitrescu

Jiang

2011

Algorithmica

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“…Khanban and co-authors [13,15] developed a theory for returning partial Delaunay or Voronoi diagrams, consisting of the portion of the diagram that is certain. Van Kreveld and Löffler [18,17] consider the problem of determining the smallest and largest possible values for geometric extent measures-such as the diameter or convex hull area-of a set of imprecise points.…”

Section: Related Workmentioning

confidence: 99%

Preprocessing Imprecise Points and Splitting Triangulations

Kreveld¹,

Löffler²,

Mitchell³

2010

SIAM J. Comput.

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Traditional algorithms in computational geometry assume that the input points are given precisely. In practice, data is usually imprecise, but information about the imprecision is often available. In this context, we investigate what the value of this information is. We show here how to preprocess a set of disjoint regions in the plane of total complexity n in O(n log n) time so that if one point per set is specified with precise coordinates, a triangulation of the points can be computed in linear time. In our solution, we solve another problem which we believe to be of independent interest. Given a triangulation with red and blue vertices, we show how to compute a triangulation of only the blue vertices in linear time.

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“…We prove this by reduction from SAT. The construction is similar to the one used to prove NP-hardness of computing the largest possible convex hull of a set of line segments [8], but the nature of the width measure requires some new ideas.…”

Section: Line Segmentsmentioning

confidence: 99%