H. Edelsbrunner, T. S. Tan and R. Waupotitsch, O(N 2 log N ) Time Algorithm for the Minmax Angle Triangulation, SIAM J. Sci. Statist. Comput. 13 (1992) 994-1008.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....given set of points. Because it is easily seen that the tolerance of a triangulation is half the smallest height among all the triangles in the triangulation, we conclude that the solution to this problem is the well known max min height triangulation, which can be computed in O(n 2 log n) time [7]. 5 p i p j p k p l c Figure 4: Illustration of the proof of Theorem 2 2.1 Local tolerance Given an edge e ij 2 DT (S) the tolerance of e ij is defined as the supremum of all the 0 such that for all S 0 with ffi (S; S 0 ) we have e ij 2 DT (S 0 ) If d i denotes the ....

H. Edelsbrunner, T. S. Tan and R. Waupotitsch, O(N 2 log N ) Time Algorithm for the Minmax Angle Triangulation, SIAM J. Sci. Statist. Comput. 13 (1992) 994-1008.

....(V; E) as a subset of a geometric graph G = V; E) with E E is studied by Lloyd [9] Using a reduction from 3 SAT he showed that this triangulation problem is NP complete. Given a plane geometric graph with or without constraining edges, several optimal triangulation problems have been studied [1, 2, 3]. Optimal means that the form of the triangles or the triangulations is optimized. In contrast to polynomial algorithms in [1, 2, 3] we give the rst negative result for an optimal triangulation problem. The NP completeness of a similar problem to triangulate a planar graph while minimizing the ....

....this triangulation problem is NP complete. Given a plane geometric graph with or without constraining edges, several optimal triangulation problems have been studied [1, 2, 3] Optimal means that the form of the triangles or the triangulations is optimized. In contrast to polynomial algorithms in [1, 2, 3], we give the rst negative result for an optimal triangulation problem. The NP completeness of a similar problem to triangulate a planar graph while minimizing the maximum degree has been proved by Kant and Bodlaender [7] One di erence in [7] to our considered problem and to the studied ....

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H. Edelsbrunner, T.S. Tan and R. Waupotitsch, An O(n 2 logn) time algorithm for the minmax angle triangulation, Proc. 6th Ann. Sympos. Comput. Geom. (1990), pp. 44 { 52.

....small angles is doomed to use a nonpolynomial number of Steiner points, dependent on the input geometry. There are several previous algorithms that achieve similar results (by dissimilar techniques) for polygonal input. See Bern and Eppstein [5] for a summary. Edelsbrunner, Tan, and Waupotitsch [8] shows how to generate a constrained triangulation (one where no Steiner points are allowed) of a PSLG such that the maximum angle is minimized. The technique used is edge insertion, a global strategy that is a generalization of local edge flip. Mitchell [10] shows how to generate a covering ....

....point on another edge of the triangle. We call a sequence of induced Steiner points a Steiner path. Besides being fairly intuitive, the fact that Steiner paths are sometimes necessary can be proved as a direct result of a lemma about constrained triangulations in Edelsbrunner, Tan, and Waupotitsch [8] (see Section 2.1) A variation on Paterson s example provides additional motivation for Steiner paths in Section 2. We can change the direction of propagation of a Steiner path with a sequence of triangles all having a vertex in common: We build the example of Figure 1 right by using p = ....

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H. Edelsbrunner, T.S. Tan, and R. Waupotitsch, An O(n 2 log n) Time Algorithm for the MinMax Angle Triangulation, Proc. 6th ACM Symposium on Computational Geometry, 44--52, 1990.

....Delaunay triangulations. An interesting observation is that like the present work, the Delaunay triangulation can be determined locally. In contrast a constrained triangulation that minimizes the maximum angle or maximizes the minimum height depends on global features of the input geometry (see Edelsbrunner, Tan, and Waupotitsch [1990] and Bern, Edelsbrunner, Eppstein, Mitchell, and Tan [1992] However, Mitchell and Park [1993] has now shown that a covering triangulation whose maximum angle is approximately as small as possible can be determined from the local geometry. Bern [1993] has recently shown that a covering ....

H. Edelsbrunner, T.S. Tan, and R. Waupotitsch [1990], An O(n 2 log n) Time Algorithm for the MinMax Angle Triangulation, Proc. 6th ACM Symposium on Computational Geometry, 44--52.

....inside the square. Property Expected value Maximum vertex degree ( n log log n ) Maximum length of an edge (log 1 2 n) Minimum angle of a triangle (n Gamma 1 2 ) Maximum angle of a triangle Gamma (n Gamma 1 5 ) Table 1: Properties of Delaunay triangulation [ETW90, ETW92] shows an algorithm that minimizes the maximum angle of any triangle in O(n 2 log n) time. It starts with an arbitrary triangulation and improves it iteratively by edgeinsertion . This algorithm can also be used to minimize the sorted angle vector, if the points are in general position. At ....

....This algorithm can also be used to minimize the sorted angle vector, if the points are in general position. At the edge insertion step, all old edges that intersect the new edge are deleted, and the polygons on the two sides of the new edge are retriangulated. Although a local operation, ETW90, ETW92] proves that a global optimum is reached. It is important how this new edge uv is chosen. If 6 xuy is the largest angle of the triangulation, an ear cutting procedure finds uv such that uv 2 S Gamma fx; u; yg; and xy uv 6= OE. ET91, ET93] minimizes, instead of the maximum internal angle ....

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Herbert Edelsbrunner, Tiow Seng Tan, and Roman Waupotitsch. O(N 2 log N ) time algorithm for the minmax angle triangulation. SIAM J. Sci. Statist. Comput., 13(4):994--1008, July 1992.

....entirely inside the square. Property Expected value Maximum vertex degree ( n log log n ) Maximum length of an edge (log 1 2 n) Minimum angle of a triangle (n Gamma 1 2 ) Maximum angle of a triangle Gamma (n Gamma 1 5 ) Table 1: Properties of Delaunay triangulation [ETW90, ETW92] shows an algorithm that minimizes the maximum angle of any triangle in O(n 2 log n) time. It starts with an arbitrary triangulation and improves it iteratively by edgeinsertion . This algorithm can also be used to minimize the sorted angle vector, if the points are in general ....

.... . This algorithm can also be used to minimize the sorted angle vector, if the points are in general position. At the edge insertion step, all old edges that intersect the new edge are deleted, and the polygons on the two sides of the new edge are retriangulated. Although a local operation, ETW90, ETW92] proves that a global optimum is reached. It is important how this new edge uv is chosen. If 6 xuy is the largest angle of the triangulation, an ear cutting procedure finds uv such that uv 2 S Gamma fx; u; yg; and xy uv 6= OE. ET91, ET93] minimizes, instead of the maximum internal ....

[Article contains additional citation context not shown here]

H. Edelsbrunner, T. S. Tan, and R. Waupotitsch. An O(n 2 log n) time algorithm for the minmax angle triangulation. In Proc. 6th Annu. ACM Sympos. Comput. Geom., pages 44--52, 1990.

....with an arbitrary triangulation T and iterates until no improvement is possible. A single iteration adds a new edge to the triangulation. All edges that intersect this new edge must of course be deleted. The resulting polygons are now retriangulated. Delicate details of this scheme can be found in [2, 12]. Companion to video 3 9 The implementation supports the case where the original triangulation contains constraining edges. Furthermore, it lexicographically optimizes the entire vector of measures, not just the worst one. Because of this property it usually computes a unique optimum. For the ....

H. Edelsbrunner, T. S. Tan, and R. Waupotitsch. An O(n 2 log n) time algorithm for the minmax angle triangulation. In Proc. 6th Ann. Sympos. Comput. Geom., pages 44--52, 1990. REFERENCES 30

....divide and conquer sweep line algorithm for the solution of one version of this problem for any simple polygon. Perhaps the most prevalent and widely applicable form of polygon decomposition is triangulation. Algorithms abound for creating triangulations of polygons with various characteristics ([3, 4, 8, 10, 11, 12], e.g. Polynomial time algorithms have also been presented for decomposing polygons into trapezoids [2] convex polygons [6, 15, 17, 19, 21, 31] star shaped or monotone polygons [19] and rectangles [21, 23] In [1] an algorithm is presented for decomposing a polygon into regions based on ....

H. Edelsbrunner, T. S. Tan, and R. Waupotitsch. An O(n 2 log n) time algorithm for the minmax angle triangulation. In Proc. 6th Annu. ACM Sympos. Comput. Geom., pages 44--52, 1990.

.... paradigms as edge flipping [Laws72, Laws77] divide and conquer [ShHo75, GuSt85] geometric transformation [Brow79] plane sweep [For87] and randomized incrementation [GuKS90] Recently, Edelsbrunner, Tan, and Waupotitsch devised a polynomial time algorithm that minimizes the maximum angle [EdTW92]. This algorithm constructs a minmax angle triangulation by iteratively inserting a new edge, removing old edges crossed by the new edge, and then retriangulating the polygonal holes on either side of the new edge. This paper presents an abstraction of the minmax angle algorithm, which we call ....

....edges ab that lie in the interior of the restricting polygonal region. As a consequence, a triangulation that lexicographically minimizes the decreasing vector of triangle measures can be constructed in the non degenerate case, that is, when (abc) 6= xyz) unless abc = xyz. Details can be found in [EdTW92]. Edge Insertion for Optimal Triangulations 4 3 Two Sufficient Conditions We now formulate two conditions on measures , sufficient to show that the edge insertion paradigm computes a global optimum (i.e. minmax ) They are also sufficient to imply algorithms much faster than O(n 8 ) these ....

[Article contains additional citation context not shown here]

H. Edelsbrunner, T. S. Tan and R. Waupotitsch. An O(n 2 log n) time algorithm for the minmax angle triangulation. SIAM J. Stat. Sci. Comput. 13 (1992), 994-1008.

.... paradigms as edge flipping [Laws72, Laws77] divide andconquer [ShHo75, GuSt85] geometric transformation [Brow79] plane sweep [For87] and randomized incrementation [GuKS90] Recently, polynomial time algorithms have also been found for the minmax angle and the minmax edge length criteria [EdTW92, EdTa91]. The method of [EdTW92] is most relevant to this paper. It constructs a minmax angle triangulation by iterative application of the so called edge insertion operation. This paper presents an abstraction of this method, termed the edge insertion paradigm, and applies it to get polynomial time ....

.... [Laws72, Laws77] divide andconquer [ShHo75, GuSt85] geometric transformation [Brow79] plane sweep [For87] and randomized incrementation [GuKS90] Recently, polynomial time algorithms have also been found for the minmax angle and the minmax edge length criteria [EdTW92, EdTa91] The method of [EdTW92] is most relevant to this paper. It constructs a minmax angle triangulation by iterative application of the so called edge insertion operation. This paper presents an abstraction of this method, termed the edge insertion paradigm, and applies it to get polynomial time algorithms for other optimal ....

[Article contains additional citation context not shown here]

H. Edelsbrunner, T. S. Tan and R. Waupotitsch. An O(n 2 log n) time algorithm for the minmax angle triangulation. SIAM J. Stat. Sci. Comput. 13 (1992), 994-1008.

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