H. Edelsbrunner and T. S. Tan. A Quadratic Time Algorithm for the MinMax Length Triangulation. Proc. FOCS '91, 1991, pp. 414-423.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to (n log n) by Supowit [25] Several papers dealing with relative neighborhood graphs have since been published. Among them, Agarwal and Matousek [1] proved a subquadratic upper bound for the number of edges of relative neighborhood graphs in the three dimensional space. Edelsbrunner and Tan [10] exploited the properties of relative neighborhood graphs on the plane to give a quadratic time algorithm for the minmax edge length triangulation problem. Many other proximity graphs have been studied. For example, Kirkpatrick and Radke [14] defined and studied the skeleton graphs that include ....

H. Edelsbrunner and T. S. Tan. A Quadratic Time Algorithm for the MinMax Length Triangulation. Proc. FOCS '91, 1991, pp. 414-423.

....a successively increased number of digitized point data ( 1367# Delta Delta Delta# 7222] from the same object using the TAC criterion show an exponential slowdown of the convergence velocity. However, optima with nearly the same fitness values have been found after at least 300 generations. Edelsbrunner and Tan (1993) showed that the number of different triangulations of P in IR 2 depends on the number of vertices n = jP j and on their relative location. If P is in convex position, then it admits 1 n;1 ; 2n;4 n;2 Delta 2 n;3 different triangulations. In order to choose an optimal triangulation ....

Edelsbrunner, H. and Tan, T.S. (1993). A Quadratic Time Algorithm for the Minmax Length Triangulation, In: Siam J. Comp., Vol. 22, No. 3, pp. 527-551.

.... the NP completeness of packing problems in the plane, see e.g. Johnson [10] or in view of the intrinsic complexity of Heilbronn s triangle problem, see [14] For the case of a fixed point set, minimizing the maximum edge length is known to be solvable in O(n 2 ) time; see Edelsbrunner and Tan [7]. Nooshin et al. 12] developed a potential based heuristic method for Problem 2, but did not give a theoretical guarantee for the obtained solution. In this paper, we o#er an O(n 2 log n) heuristic capable of producing constant approximations for any of the three problems stated above. ....

H. Edelsbrunner and T.S. Tan, "A quadratic time algorithm for the minmax length triangulation", SIAM Journal on Computing 22 (1993), 527-551.

....[64] gave a thorough experimental comparison of available Delaunay triangulation algorithms. On the negative side, DT (S) fails to ful ll optimization criteria similar to those mentioned above, such as minimizing the maximum angle, or minimizing the longest edge. Edelsbrunner et al. 28] [27] gave O(n 2 log n) time and O(n 2 ) time algorithms, respectively, for computing triangulations optimal in these respects. The former algorithm is based on an edge insertion paradigm which is shown in Bern et al. 7] to lead in polynomial time to triangulations with maxmin triangle height, ....

H.Edelsbrunner, T.S.Tan: `A quadratic time algorithm for the minmax length triangulation', SIAM J. Comput. 22 (1993) pp. 527-551

....data analysis, and finite element methods. Optimization criteria for which e#cient algorithms are known include maximizing the minimum angle [20, 24] minimizing the maximum angle [6] minimizing the minimum angle [7] minimizing the maximum aspect ratio [3] and minimizing the maximum edge length [5]. The most longstanding open problem in computational geometry is the complexity of another optimal triangulation problem, the minimum weight triangulation (MWT) in which the optimization criterion is the sum of the edge lengths. Indeed, this seems to have been known as the optimal ....

H. Edelsbrunner and T.S. Tan. A quadratic time algorithm for the minmax length triangulation. Report UIUCDCS-R-91-1665, Univ. Illinois, Urbana-Champaign (1991).

....(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 and T.S. Tan, A quadratic time algorithm for the minmax length triangulation, Proc. 32nd Ann. Sympos. Found. Comp. Sci. (1991), pp. 414 - 423.

.... if angles of triangles become too large, the discretization error in the finite element solution is increased and, if the angles become too small, the condition number of the element matrix is increased [1, 10, 11] Polynomial time algorithms have been developed in determining those triangulations [2, 7, 8, 15]. In computational geometry another important research object is to compute the minimum weight triangulation. The weight of a triangulation is defined to be the sum of the Euclidean lengths of the edges in the triangulation. Despite the intensive study made during the lase two decades, it remains ....

H. Edelsbrunner and T. S. Tan, "A quadratic time algorithm for the minmax length triangulation", SIAM J. Comput. 22, pp. 527-551, 1993.

....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 of the triangles, the maximum length of the any triangle. This is done in O(n 2 ) time. It starts by finding the convex hull ch(S) and the relative neighbourhood graph rng(S) of S. An edge uv belongs to rng(S) if juvj min x2S Gammafa;bg ....

....lower bound of Omega Gamma n log n) is also shown. ETW90, ETW92] can be used to minimize the maximum angle, while minimizes the maximum edge length. LL92] shows an alternate scheme that takes only O(n) time for simple polygon and O(n log n) for planar straight line graph, as opposed to [ET91, ET93] s O(n 2 ) but it does not guarantee minimality. It can be up to 3 times more than the minimum. In general, any c sensitive triangulation of a planar point set approximates the minmax triangulation within a factor of 2(c 1) LL92] also proves that the greedy triangulation and the Delaunay ....

H. Edelsbrunner and T. S. Tan. A quadratic time algorithm for the minmax length triangulation. SIAM J. Comput., 22:527--551, 1993.

....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 of the triangles, the maximum length of the any triangle. This is done in O(n 2 ) time. It starts by finding the convex hull ch(S) and the relative neighbourhood graph rng(S) of S. An edge uv belongs to rng(S) if juvj min x2S ....

....tight lower bound of Omega Gamma n log n) is also shown. ETW90, ETW92] can be used to minimize the maximum angle, while minimizes the maximum edge length. LL92] shows an alternate scheme that takes only O(n) time for simple polygon and O(n log n) for planar straight line graph, as opposed to [ET91, ET93] s O(n 2 ) but it does not guarantee minimality. It can be up to 3 times more than the minimum. In general, any c sensitive triangulation of a planar point set approximates the minmax triangulation within a factor of 2(c 1) LL92] also proves that the greedy triangulation and the ....

H. Edelsbrunner and T. S. Tan. A quadratic time algorithm for the minmax length triangulation. In Proc. 32nd Annu. IEEE Sympos. Found. Comput. Sci., pages 414--423, 1991. 16

....candidate edges in a certain order and saving old partial triangulations, the running time can be improved to O(n 2 log n) for minmax angle [53] and maxmin triangle height. We close with some results for two other optimization criteria: maximum edge length and total length. Edelsbrunner and Tan [51] showed that a triangulation of a point set that minimizes the maximum edge length must contain the edges of a minimum spanning tree. The tree divides the input into simple polygons, which can be filled in by dynamic 21 programming, giving an O(n 3 ) time algorithm (improvable to O(n 2 ) ....

H. Edelsbrunner and T.-S. Tan. A quadratic time algorithm for the minmax length triangulation. In Proc. 32nd IEEE Symp. Foundations of Comp. Science, pages 414--423, 1991.

.... could be found in this way, so that the resulting subgraph of the MWT connected all the vertices, the remaining regions of the plane could be treated as simple polygons and triangulated in polynomial time by dynamic programming [78] This approach gained in credibility when Edelsbrunner and Tan [50] used it to solve a closely related problem, the min max weight triangulation. In this problem, the quality of a triangulation is measured by the length of its longest edge; the min max weight triangulation is the one minimizing this quantity. Lemma 16 (Edelsbrunner and Tan) There exists some min ....

H. Edelsbrunner and T. S. Tan. A quadratic time algorithm for the minmax length triangulation. Proc. 32nd IEEE Symp. Foundations of Comp. Sci., 1991, pp. 414--423. 30

....data analysis, and finite element methods. Optimization criteria for which e#cient algorithms are known include maximizing the minimum angle [19, 23] minimizing the maximum angle [6] minimizing the minimum angle [7] minimizing the maximum aspect ratio [3] and minimizing the maximum edge length [5]. The most longstanding open problem in computational geometry is the complexity of another optimal triangulation problem, the minimum weight triangulation (MWT) in which the optimization criterion is the sum of the edge lengths. Indeed, this seems to have been known as the optimal ....

H. Edelsbrunner and T.S. Tan. A quadratic time algorithm for the minmax length triangulation. Report UIUCDCS-R-91-1665, Univ. Illinois, Urbana-Champaign (1991). 25

.... if angles of triangles become too large, the discretization error in the finite element solution is increased and, if the angles become too small, the condition number of the element matrix is increased [1, 15] Polynomial time algorithms have been developed for determining those triangulations [5, 12, 13, 21]. In computational geometry computing the minimum weight triangulation is another important research topic. The weight of a triangulation is defined to be the sum of the Euclidean lengths of the edges in the triangulation. Despite the intensive study made during the lase two decades, it remains ....

H. Edelsbrunner and T. S. Tan, "A quadratic time algorithm for the minmax length triangulation ", SIAM Journal on Computing 22 (1993) 527-551.

....Hence, after a finite number of steps we obtain a minmax Hamiltonian cycle which is a subgraph of the k Gamma RNG. A related and important problem of constructing a minmax length triangulation, i.e. a triangulation which minimizes the longest edge, has been studied by Edelsbrunner and Tan [ET91] They have proved that every finite point set V in R 2 has a minmax length triangulation which is a supergraph of the RNG(V ) The lemma suggests that construction of a minmax length triangulation can start from the RNG(V ) It leads immediately to a cubic time construction when existing ....

....length triangulation can start from the RNG(V ) It leads immediately to a cubic time construction when existing dynamic programming algorithms for triangulations are utilized; see [Kli80] It also provides the first polynomial time algorithm for this problem. Furthermore, as demonstrated in [ET91] an even faster quadratic time algorithm can be developed. The algorithm works for a general class of metrics. k Gabriel graphs: In a similar fashion k Gabriel graphs can be introduced; see [SC90] Let Gamma p;q = B( p q 2 ; ffi(p;q) 2 ) i.e. U p;q is a diameter sphere. The edges of ....

H. Edelsbrunner and T. S. Tan. A quadratic time algorithm for the MinMax length triangulation. Technical report, Dept. of Comp. Sci. at University of Illinois at Urbana Champaign, February 1991.

....is of fundamental importance, we describe in Section 5.2.4 an optimal algorithm due to Chew[16] that computes the constrained Delaunay triangulation for a planar straight line graph G(V; E) with n vertices in O(n log n) time. Triangulations that minimize the maximum angle or maximum edge length [24] were also studied. But if the constraints are on the measure of the triangles, for instance, each triangle in the triangulation must be non obtuse, then Steiner points must be introduced. See Bern and Eppstein (in [21] pp. 23 90) for a survey of triangulations satisfying different criteria and ....

H. Edelsbrunner and T. S. Tan, "A Quadratic Time Algorithm for the Minmax Length Triangulation," SIAM J. Comput., 22 (1993), 527-551.

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H. Edelsbrunner and T. S. Tan. A quadratic time algorithm for the minmax length triangulation. In "Proc. 32nd IEEE Sympos. Found. Comput. Sci. 1991", 414--423.

.... 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 ....

H. Edelsbrunner and T. S. Tan. A quadratic time algorithm for the minmax length triangulation. In "Proc. 32nd IEEE Sympos. Found. Comput. Sci. 1991", 414--423.

....metric, including the Euclidean distance and the more general l p metrics. It is currently the only (nontrivial) length criterion that can be computed efficiently. Our solution also provides additional insight into optimal triangulations under edge length criteria. This chapter also appears in [EdTa91]. Chapter 6 considers conforming Delaunay triangulations. We show that, for every plane geometric graph G with n vertices and m edges, there is a conforming Delaunay triangulation for G with 4m 2 n 10mn 4n vertices. The result also implies an efficient algorithm to compute these vertices and ....

....and c and d both lie outside the circle with radius jabj and center at a. 1 Incidentally, retriangulation is also the main idea in proving the correctness of the edge insertion paradigm. 83 Acknowledgments The results in this chapter are jointly developed by Herbert Edelsbrunner and myself [EdTa91]. We thank Jerzy Jaromczyk (University of Kentucky) for pointing out various references on relative neighborhood graph, in particular, the paper by Su and Chang [SuCh91] that implies a quadratic time algorithm for the constrained min max length triangulation problem in the Euclidean case. 84 ....

H. Edelsbrunner and T. S. Tan. A quadratic time algorithm for the minmax length triangulation. In "Proc. 32nd Ann. IEEE Sympos. Found. Comput. Sci., 1991", 414--423. Also to appear in SIAM J. Comput. 107

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H. Edelsbrunner and T. S. Tan, "A quadratic time algorithm for the minmax length triangulation ", SIAM Journal on Computing 22 (1993) 527-551.

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H. Edelsbrunner and T. S. Tan. A quadratic time algorithm for the minmax length triangulation. In Proc. 32nd IEEE Symp. Foundations of Comp. Science, pages 414--423, 1991.

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