E.L. Lloyd. On triangulations of a set of points in the plane. In 18th IEEE Symposium on Foundations of Computer Science, pages 228--240, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that the cover regions on a smooth surface intersect in a pseudo disk configuration, and the above result holds good on surfaces. Assuming that the triangulation exists in , we have to extract from . But this problem, in its general setting, is proved to be NP Complete. Theorem 11 Lloyd [Lloyd77] has proved that the following triangulation extraction problem is NPcomplete. The triangulation existence problem is: Let be the set of vertices. Let be the set of all straight line segments between the vertices in , and the set of edges , does there exist a subset , such ....

...., thus leading to the conclusion that is a geometric triangulation of . I also justified the approach of directly constructing the triangulation, rather than extracting a triangulation from the nerve due to the intersection of cover regions, using the existing results like [Edelsbrunner97b] and [Lloyd77]. 91 CONCLUSION The main contribution of this dissertation is the analysis of the problem of sampling and reconstruction of surfaces with boundaries. This analysis led to the classification of surface reconstruction algorithms and to the realization that the conditions on minimum required ....

E.L. Lloyd. On triangulations of a set of points in the plane. In 18th IEEE Symposium on Foundations of Computer Science, pages 228--240, 1977.

....even for small n. In fact, constructing optimal triangulations in polynomial time is a challenging task. This becomes more apparent as greedy methods, such as deleting candidate triangles or edges from worst to best, are doomed to fail by the NP completeness of the following problem; see Lloyd [53]: Given S and some set E of edges, decide whether E contains a triangulation of S. Results on optimizing combinatorial properties of triangulations, such as their degree (Jansen [37] or connectivity (Dey et al. 19] and Dillencourt [23] are rare. Most optimization criteria for which ecient ....

E.L.Lloyd: `On triangulations of a set of points in the plane', Proc. 18th IEEE Symp. on Foundations of Computer Science, (1977) pp. 228-240

....polynomial time, nor is it known whether this is an NP hard problem. In consequence of this, heuristics for approximating it have been considered. There are two well known heuristics: the greedy triangulation and the Delaunay triangulation, both being computable in O(n log n) time [9, 18] Lloyd [15] showed that, in general, none of these two heuristics produce a minimum weight triangulation. For an arbitrary large n, Manacher and Zobrist [16] showed that one can place n points so that the Delaunay is Omega Gamma n= log n) from the optimum. They also showed that n points can be placed so ....

E. L. Lloyd. On triangulations of a set of points in the plane. In Proceedings of the 18th Annual IEEE Symposium on Foundations of Computer Science, pages 228--240, 1977.

....known as the optimal triangulation for some time. The MWT problem is included in Garey and Johnson s famous list of problems neither known to be NP complete, nor known to be solvable in polynomial time [9] It is known that many triangulation algorithms will not correctly solve the MWT problem [17]. We do not resolve the status of this question. The algorithms and problems cited above search for triangulations in which the vertex set of the triangulation is exactly the set of input points. Many of these problems can be extended to Steiner triangulation problems, in which the vertex set ....

E.L. Lloyd. On triangulations of a set of points in the plane. 18th IEEE Symp. Found. Comp. Sci. (1977) 228--240.

....exist triangulated graphs G 0 = V; E 0 ) with jV j = n and (G 0 ) 4. An application of our studied problem in numerical engineering is given by Frey and Field [5] The problem to nd a triangulation G = 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 ....

E.L. Lloyd, On triangulations of a set of points in the plane, Proc. 18th Ann. IEEE Sympos. Found. Comput. Sci. (1977), pp. 228 - 240.

....optimal triangulations in polynomial time. Exhaustive search can be ruled out since a set of n points has, in general, exponentially many triangulations. Greedy approaches (such as eliminating triangles from worst to best) are ruled out by the NP completeness of the following decision problem [Llo77]: given a collection of points and edges, decide whether a subset of the edges defines a triangulation of the points. Most positive results are related to the Delaunay triangulation [Del34] It has been shown that among all triangulations of a given finite point set, the Delaunay triangulation ....

E. L. Lloyd. On triangulations of a set of points in the plane. In "Proc. 18th Ann. IEEE Sympos. Found. Comput. Sci., 1977", 228--240.

....polynomial time, nor is it known whether this is an NPhard problem. In consequence of this, heuristics for approximating it have been considered. There are two well known heuristics: the greedy triangulation and the Delaunay triangulation, both being computable in O(n log n) time [9, 18] Lloyd [15] showed that, in general, none of these two heuristics produce a minimum weight triangulation. For an arbitrary large n, Manacher and Zobrist [16] showed that one can place n points so that the Delaunay is Omega Gamma n= log n) from the optimum. They also showed that n points can be placed so ....

E. L. Lloyd. On triangulations of a set of points in the plane. In Proceedings of the 18th Annual IEEE Symposium on Foundations of Computer Science, pages 228--240, 1977.

....has, in general, exponentially many triangulations. In spite of the attention these optimization problems have received, only very little is known about constructing optimal triangulations in polynomial time. An important negative result is the NP completeness of the following decision problem [Llo77]: given a collection of points and edges, decide whether or not there is a subset of the edges that defines a triangulation of the points. Most positive results are related to the Delaunay triangulation defined for finite point sets [Del34] It has been shown that among all triangulations of a ....

E. L. Lloyd. On triangulations of a set of points in the plane. In "Proc. 18th Ann. IEEE Sympos. Found. Comput. Sci., 1977", 228--240.

....and [WaPh84, page 218] and provide a solution in Chapter 5. Problem 3 How fast can we compute a min max length triangulation of a point set The following problems on length criteria are still open. Problem 4 is the reverse of the previous problem, and Problem 5 is notoriously difficult [Lloy77, PlHo87] (see Section 2.5 for some general discussion) Problem 4 How fast can we compute a max min length triangulation of a point set Problem 5 Given a point set S, how fast can we compute a triangulation of S that minimizes the sum of edge lengths 1.3.3 Other Reasonable Criteria As mentioned, ....

....edges; see, for example, MaSo80, Supo83, Yao82] 2.5 Minimum Length Triangulations We have seen in Section 2.2 the cubic time (dynamic programming) algorithm for constructing minimum length triangulation for a simple polygon. The same problem for a point set is open (Section 1.3. 2, Problem 5) [Lloy77]. We mention in the following an interesting attempt, termed subgraph approach, for finding a polynomial time solution to the problem. The idea is to convert a point set problem to a number of (simple) polygon problems, which can then be solved in polynomial time, for example, by dynamic ....

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E. L. Lloyd. On triangulations of a set of points in the plane. In "Proc. 18th Ann. IEEE Sympos. Found. Comput. Sci., 1977", 228--240.

....in O(n k 2 ) time. 2.2 Heuristics That Approximate the MWT It is legitimate to ask if any of triangulations like the Delaunay triangulation or greedy triangulation that have polynomial time algorithms are minimum weight triangulations or are a constant factor approximation of the MWT. Lloyd [Llo77] showed that in general the Delaunay triangulation is not a minimum weight triangulation. In fact, the Delaunay triangulation does not produce a constant factor approximation of the MWT. Kirkpatrick [Kir80] showed that for each n there exists a set of n points such that the Delaunay triangulation ....

....not a minimum weight triangulation. In fact, the Delaunay triangulation does not produce a constant factor approximation of the MWT. Kirkpatrick [Kir80] showed that for each n there exists a set of n points such that the Delaunay triangulation is Omega Gamma n) times longer than the MWT. Lloyd [Llo77] also showed that the greedy triangulation is not the minimum weight triangulation. Levcopoulos [Lev87] showed that it does not approximate the MWT better than by a Omega Gamma p n) factor. Since known triangulations do not provide good approximations of the MWT, work has been done to find ....

E. L. Lloyd. On triangulations of a set of points in the plane. In Proc. 18th Annu. IEEE Sympos. Found. Comput. Sci., pages 228--240, 1977. Bibliography 68

.... 8 GEOMETRIC PROBLEMS NP hard, nor is it known to be solvable in polynomial time, and the complexity of minimum weight triangulation is one of the few problems left from Garey and Johnson s original list of open problems [GJ79] A generalized problem with non Euclidean distances is NP complete [Llo77] We consider two versions of minimum weight triangulation. In the more commonly studied version, the vertex set of the triangulation is exactly the set of input points S. In minimum weight Steiner triangulation, additional vertices lying anywhere in the plane may be added, and the output is a ....

E.L. Lloyd. On triangulations of a set of points in the plane. In Proc. 18th IEEE Symp. Found. Comp. Sci., pages 228--240, 1977.

....minimum weight triangulation. One use of the greedy triangulation is as an approximation to the minimum weight triangulation (MWT) A Minimum Weight Triangulation (MWT) of a point set in the plane is a triangulation that minimizes the total length of all edges. The MWT arises in numerical analysis [19, 21, 24]. In a method suggested by Yoeli [28] for numerical approximation of bivariate data, the MWT provides a good approximation of the sought after function surface. Wang and Aggarwal use a minimum weight triangulation in their algorithm to reconstruct surfaces from contours [26] Though it has been ....

....time O(n 3 ) for the special case of n vertex polygons [13] there are no known efficiently computable algorithms for the MWT in the general case [24] We therefore seek efficiently computable approximations to the MWT. Although neither the GT nor the Delaunay triangulation (DT) yields the MWT [22, 21], the GT appears to be the better of the two at approximating it. In fact, for convex polygons the GT approximates the MWT to with a constant factor while the DT can be a factor of Omega Gamma n) larger [16] For general point sets, the DT can be a factor of Omega Gamma n) larger than the MWT, ....

E. L. Lloyd, "On triangulations of a set of points in the plane." Proceedings of the 18th IEEE Symp. Foundations of Computer Science (1977) 228--240.

.... is not known to be NP hard, nor is it known to be solvable in polynomial time, and the complexity of minimum weight triangulation is one of the few problems left from Garey and Johnson s original list of open problems [63] However, a generalized problem with non Euclidean distances is NP complete [85]. We describe here two very recent developments in the theory of minimum weight triangulations. First, Levcopoulos and Krznaric [81] have shown that one can find a triangulation with total length approximating the minimum to within a (large) constant factor; no such approximation was previously ....

E. L. Lloyd. On triangulations of a set of points in the plane. Proc. 18th IEEE Symp. Foundations of Comp. Sci., 1977, pp. 228--240.

....known as the optimal triangulation for some time. The MWT problem is included in Garey and Johnson s famous list of problems neither known to be NP complete, nor known to be solvable in polynomial time [8] It is known that many triangulation algorithms will not correctly solve the MWT problem [16]. We do not resolve the status of this question. The algorithms and problems cited above search for triangulations in which the vertex set of the triangulation is exactly the set of input points. Many of these problems can be extended to Steiner triangulation problems, in which the vertex set ....

E.L. Lloyd. On triangulations of a set of points in the plane. 18th IEEE Symp. Found. Comp. Sci. (1977) 228--240.

....of the greedy triangulation is as an approximation to the minimum weight triangulation (MWT) Given a set S of n points in the plane, a Minimum Weight Triangulation (MWT) of S is a triangulation that minimizes the total length of all edges in the triangulation. The MWT arises in numerical analysis [23, 26, 30]. In a method suggested by Yoeli [36] for numerical approximation of bivariate data, the MWT provides a good approximation of the sought after function surface. Wang and Aggarwal use a minimum weight triangulation in their algorithm to reconstruct surfaces from contours [34] Though it has been ....

....time O(n 3 ) for the special case of n vertex polygons [15] there are no known efficiently computable algorithms for the MWT in the general case [30] We therefore seek efficiently computable approximations to the MWT. Although neither the GT nor the Delaunay triangulation (DT) yields the MWT [27, 26], the GT appears to be the better of the two at approximating it. In fact, for convex polygons the GT approximates the MWT to with a constant factor while the DT can be a factor of Omega Gamma n) larger [19] For general point sets, the DT can be a factor of Omega Gamma n) larger than the MWT, ....

E. Lloyd, "On triangulations of a set of points in the plane." Proceedings of the 18th FOCS (1977) 228--240.

....added edges. One use of the greedy triangulation is as an approximation to the minimum weight triangulation (MWT) The MWT of a point set minimizes the total length of all edges, where the length of an edge is the Euclidean distance between its two endpoints. It arises in numerical analysis [Li, Ll, PS], numerical approximation of bivariate data [Yo] or in the reconstruction of surfaces from contours [WA] Computing a minimum weight triangulation is an important and interesting problem, whose complexity status is unknown. It is one of the still open problems listed at the end of Garey and ....

....problem, whose complexity status is unknown. It is one of the still open problems listed at the end of Garey and Johnson s book about NP completeness [GJ] We therefore seek efficiently computable approximations to the MWT. Although neither the GT nor the Delaunay triangulation (DT) yields the MWT [MZ, Ll], the GT appears to be the better of the two in approximating the MWT. In fact, for convex polygons the GT approximates the MWT to with a constant factor while the DT can be a factor of Omega Gamma n) larger [LL] For general point sets a factor of Omega Gamma p n) is known [K,Lev] as a lower ....

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E. L. Lloyd, "On triangulations of a set of points in the plane." Proceedings of the 18th IEEE Symp. Foundations of Computer Science (1977) 228--240.

.... triangulation is given by the ratio R(T (P ) W (T (P ) W (MWT(P ) Since neither the greedy triangulation nor the Delaunay triangulation is optimal, the worstcase ratios R(GT(P ) and R(DT(P ) give an indication of how well these triangulations approximate MWT(P ) Levcopoulos [9] provides a lower bound for the greedy triangulation by showing that, for each n, there exists an n point set P 1 with the property that R(GT(P 1 ) Omega i n 1 2 j : In a complementary result, Kirkpatrick [7] constructs, for each n, an n point set P 2 with the property that ....

....thus computed by MST T algorithm is obviously non optimal, since the optimal triangulation can be obtained by choosing edges AC, AE, CE and BE along with the convex hull edges. The MST T algorithm does not produce a non optimal triangulation for any of the counterexamples in the literature [7, 9, 13, 14] that have been used to show either the greedy triangulation or the Delaunay triangulation to be non optimal. This can be attributed to the fact that the algorithm picks the least number of edges needed to keep the set of input vertices connected and then proceeds to select the optimal set of the ....

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E.L. Lloyd, On triangulations of a set of points in the plane. Proceedings of the Eighteenth IEEE Symposium on Foundations of Computer Science, 18, 1977, 228--240.

....finite element methods. Optimization criteria include maximizing the minimum angle (solved by the well known Delaunay triangulation [24, 27] minimizing the maximum angle [13] minimizing a maximum min containment ellipse [11] and minimizing total length (an outstanding open problem in the field [16, 20]) Variants of these problems allow one to add extra vertices, called Steiner points, in order to further improve the quality of the solution. In this paper we use quadtrees to solve several Steiner triangulation problems motivated by finite element methods. A point set or polygon is to be ....

E. L. Lloyd. On triangulations of a set of points in the plane. In Proceedings of the 18th Annual Symposium on Foundations of Computer Science, pages 228--240. IEEE, 1977.

.... ) time, where n is the number of vertices of the polygon (see Gilbert [6] Klincsek [10] and Heath and Pemmaraju [7] The triangulation produced by our algorithm turns out to be that which would result from the application of the greedy algorithm described by Levcopoulos and Lingas [12] Lloyd [16] has shown that the greedy triangulation of a convex polygon is not necessarily of minimum weight; lower bounds for the nonoptimality of the greedy triangulation are given by Manacher and Zobrist [17] and by Levcopoulos [11] Lingas [14] shows that on average the greedy triangulation approximates ....

E.L. Lloyd. On Triangulations of a Set of Points in the Plane. Proceedings of the 18th Conference on the Foudations of Computer Science, Providence, RI, 1977, pp. 228-240.

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