The furthest-site geodesic voronoi diagram

Aronov, Boris; Fortune, Steven; Wilfong, Gordon

doi:10.1007/bf02189321

Cited by 40 publications

(103 citation statements)

References 10 publications

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“…The nearest-and furthest-site geodesic Voronoi diagram both are a concatenation of O(n + m) straight and hyperbolic arcs, proven by Aronov [4] and Aronov et al [5], respectively.…”

Section: Voronoi Diagrammentioning

confidence: 92%

Geodesic-Preserving Polygon Simplification

Aichholzer

Hackl

Korman

et al. 2013

Algorithms and Computation

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Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon P by a polygon P such that (1) P contains P, (2) P has its reflex vertices at the same positions as P, and (3) the number of vertices of P is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both P and P , our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of P. We describe several of these applications (including linear-time post-processing steps that might be necessary).

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“…The nearest-and furthest-site geodesic Voronoi diagram both are a concatenation of O(n + m) straight and hyperbolic arcs, proven by Aronov [4] and Aronov et al [5], respectively.…”

Section: Voronoi Diagrammentioning

confidence: 92%

Geodesic-Preserving Polygon Simplification

Aichholzer

Hackl

Korman

et al. 2013

Algorithms and Computation

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“…Note that geodesic convex sets are also known in [6,8] (and in [18], respectively) as relative convex sets ( -convex sets, respectively). For all distinct points p and q of the simple polygon , U [p, q] and U [q, p] are geodesic convex.…”

Section: Geodesic Convex Setsmentioning

confidence: 99%

“…Some properties on "extreme" points and "cone" of geodesic convex sets were presented in [6]. It is well known that intersection of two convex sets is convex and the closure and interior of a convex set are convex, too (see [22]).…”

Section: Geodesic Convex Setsmentioning

confidence: 99%

“…Some computational aspects of geodesic convex hull were presented in [18] and [20]. Some properties of these sets and their applications in robotics (in computing the furthest-site Voronoi diagrams, respectively) 222 P. T. An et al were presented in [6] (and in [8], respectively). In this article, we deal with some computational aspects of geodesic convex sets.…”

Section: Introductionmentioning

confidence: 98%

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Some Computational Aspects of Geodesic Convex Sets in a Simple Polygon

Giang

Hải

2010

Numerical Functional Analysis and Optimization

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In this article, we deal with some computational aspects of geodesic convex sets. Motzkintype theorem, Radon-type theorem, and Helly-type theorem for geodesic convex sets are shown. In particular, given a finite collection of geodesic convex sets in a simple polygon and an "oracle," which accepts as input three sets of the collection and which gives as its output an intersection point or reports its nonexistence; we present an algorithm for finding an intersection point of this collection.

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“…In the case of h = 0, Aronov et al [2] in 1993 proved that there are at most O(k) faces in the diagram and the complexity of the diagram is at most O(n + k). However, any nontrivial upper bound on the geodesic farthest-site Voronoi diagram in a polygonal domain when h > 0 remains unknown afterwards.…”

mentioning

confidence: 98%

The geodesic farthest-site Voronoi diagram in a polygonal domain with holes

Bae

Chwa

2009

Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry

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We investigate the farthest-site Voronoi diagram of k point sites with respect to the geodesic distance in a polygonal domain of n corners and h (≥ 0) holes. In the case of h = 0, Aronov et al. [2] in 1993 proved that there are at most O(k) faces in the diagram and the complexity of the diagram is at most O(n + k). However, any nontrivial upper bound on the geodesic farthest-site Voronoi diagram in a polygonal domain when h > 0 remains unknown afterwards. In this paper, we show that the diagram in a polygonal domain consists of Θ(hk) faces and its total combinatorial complexity is Θ(nk) in the worst case for any h ≥ 1. Interestingly, the worst-case complexity of the diagram is independent from the number h of holes if h ≥ 1 while the maximum possible number of faces is dependent on h rather than on the complexity n of the polygonal domain. Also, we present an O(nk log 2 (n + k) log k)-time algorithm that constructs the diagram explicitly.

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The furthest-site geodesic voronoi diagram

Abstract: We present an O((n + k) log(n + k))-time, O(n + k)-space algorithm for computing the furthest-site Voronoi diagram of k point sites with respect to the geodesic metric within a simple n-sided polygon.

Cited by 40 publications

References 10 publications

Geodesic-Preserving Polygon Simplification

Geodesic-Preserving Polygon Simplification

Some Computational Aspects of Geodesic Convex Sets in a Simple Polygon

The geodesic farthest-site Voronoi diagram in a polygonal domain with holes

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