Structural tolerance and Delaunay triangulation

Abellanas, Manuel; Hurtado, Ferrán; Ramos, Pedro

doi:10.1016/s0020-0190(99)00107-6

Cited by 45 publications

(39 citation statements)

References 12 publications

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“…Abellanas et al [1] studied this structure with respect to the largest perturbation of a set of planar points that keeps the Delaunay triangulation unchanged. Conversely, problems of determining a necessary perturbation in order to achieve a desired change have also been studied; e.g., see Arkin et al [3] for deciding whether a given set of neighborhoods has a convex stabber, which amounts to deciding whether a given set can be moved into a convex position.…”

Section: Related Workmentioning

confidence: 99%

“…We call such a tree a minimum solution tree T . 1 We call a choice of points on segments that results in a minimum solution tree T a best point set for T . In a minimum solution tree T , not only are longest edges as short as possible, but also the number of longest edges is minimum.…”

Section: Note That This Defines a Linear Ordering On Lists In Generalmentioning

confidence: 99%

“…Then there are two locally longest edges incident to p laying in different half-planes delimited by a line perpendicular to the segment passing through p. We observe that moving p in any direction would increase the length of one of these two edges. 1 We remark that for lists L and L ′ for two different spanning trees with two different sets of points on the segments, it is possible that L = L ′ , so that optimum solutions that permit minimum solutions trees are in general not unique.…”

Section: Note That This Defines a Linear Ordering On Lists In Generalmentioning

confidence: 99%

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Connectivity Graphs of Uncertainty Regions

Chambers

Erickson

Fekete

et al. 2010

Algorithms and Computation

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We study connectivity relations among points, where the precise location of each input point lies in a region of uncertainty. We distinguish two fundamental scenarios under which uncertainty arises. In the favorable Best-Case Uncertainty (BU), each input point can be chosen from a given set to yield the best possible objective value. In the unfavorable Worst-Case Uncertainty (WU), the input set has worst possible objective value among all possible point locations, which are uncertain due, for example, to imprecise data.We consider these notions of uncertainty for the bottleneck spanning tree problem, giving rise to the following Best-Case Connectivity with Uncertainty (BCU) problem: Given a family of geometric regions, choose one point per region, such that the longest edge length of an associated geometric spanning tree is minimized. We show that this problem is NP-hard even for very simple scenarios in which the regions are line segments or squares. On the other hand, we give an exact solution for the case in which there are n + k regions, where k of the regions are line segments and n of the regions are fixed points. We then give approximation algorithms for cases where the regions are either all line segments or all unit discs. We also provide approximation methods for the corresponding Worst-Case Connectivity with Uncertainty (WCU) problem: Given a set of uncertainty regions, find the minimal distance r such that for any choice of points, one per region, there is a spanning tree among the points with edge length at most r.A preliminary extended abstract summarizing parts of this paper appears in [5].

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Section: Related Workmentioning

confidence: 99%

Section: Note That This Defines a Linear Ordering On Lists In Generalmentioning

confidence: 99%

Section: Note That This Defines a Linear Ordering On Lists In Generalmentioning

confidence: 99%

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Connectivity Graphs of Uncertainty Regions

Chambers

Erickson

Fekete

et al. 2010

Algorithms and Computation

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“…Guibas et al [10] introduce the notion of epsilon geometry, a framework for robust computations on imprecise points. Abellanas et al [1] and Weller [22] study the tolerance of a geometric structure: the largest perturbation of the vertices such that the combinatorial structure remains the same. Bandyopadhyay and Snoeyink [2] compute the set of "almost-Delaunay simplices", which are the tuples of points that can define a Delaunay simplex if the entire point set is perturbed by at most ε > 0.…”

Section: Related Workmentioning

confidence: 99%

Preprocessing Imprecise Points and Splitting Triangulations

Kreveld

Löffler

Mitchell

2008

Algorithms and Computation

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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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“…Guibas et al [6] introduce the notion of espilon geometry, a framework for robust computations on imprecise points. Abellanas et al [1] study the tolerance of a geometric structure: the largest perturbation of the vertices such that the topology of the structure remains the same.…”

Section: Related Workmentioning

confidence: 99%