G. Klincsek. Minimal triangulations of polygonal domains. Annals of Discrete Mathemathics, 9:121--123, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the quality of the worst transformed element the most. A separate problem, not addressed by this paper, is which edge or face should be targeted in the first place. The algorithm that finds an optimal edge removal operation (for a specified edge) is a dynamic programming algorithm of Klincsek [6], which was invented long before anyone studied edge removal. Klincsek s algorithm solves a general class of problems in optimal triangulation. Section 2 is my effort to popularize Klincsek s algorithm, work out the details (including pseudocode) of its application to edge removal, and show that ....

....= t2T fconv(a [ t) conv(b [ t)g, as illustrated. The tetrahedra in J replace the tetrahedra in I . The chief algorithmic problem is to find the triangulation T of R that maximizes the quality of the worst tetrahedron in J . This problem is solved by a dynamic programming algorithm of Klincsek [6], which is a general purpose algorithm for constructing optimal triangulation of simple polygons for many different measures of optimality. Let the vertices of R be v1 ; v2 ; vm in counterclockwise order about ab (as viewed from a) The optimal triangulation of R is found by first solving ....

G. T. Klincsek. Minimal Triangulations of Polygonal Domains. Annals of Discrete Mathematics 9:121--123, 1980.

.... triangulation (where the length of an edge is the distance between its endpoints measured by the standard Euclidean metric) Such a triangulation is referred to as the Minimum Weight Triangulation (MWT) The problem of finding the MWT for n vertex simple polygons can be solved in O(n ) time [15, 10], but it is not known how to efficiently compute the MWT of a general set of points S. In fact, the complexity of finding an MWT whether it is polynomial time or NP hard is one of the few questions that remains open from Garey and Johnson s classic book on NP completeness [11] Some approaches ....

G. Klincsek, "Minimal triangulations of polygonal domains." Ann. Discrete Math. 9 121--123.

....compute a minimum weight triangulation for a simple polygon P with holes. The holes may be polygonal or simply points. To our knowledge, no polynomial time algorithm is known for this problem. This is in contrast to the case of polygons without holes for which a polynomial time algorithm exists [20]. Consider the set S of vertices of P and use the network described in Section 3.1 with the following two modifications: Change equation (15) the T matrix, to be t ij = GammaBX ij Y ij (1 Gamma ffi ij ) 23) where, X ij = 8 : 1 if the edges e i and e j intersect properly 0 otherwise ....

G. T. Klincsek, "Minimal triangulations of polygonal domain," Annals of Discrete Mathematics, vol. 9, pp. 121--123, 1980.

....observation used is that once some triangle of the triangulation has been xed the problem splits into subproblems (subpolygons) whose solutions can be found recursively, thereby avoiding recomputation of common subproblems. The triangulation method, rst proposed by Gilbert [34] and Klincsek [42], does not really exploit convexity. It works as well for non convex polygons, and in fact for any interior face of a planar straight line graph. It is worth mentioning that, in the convex polygon case, MWT (S) is approximated by GT (S) up to a constant factor; see Levcopoulos and Lingas [48] ....

G.T.Klincsek: `Minimal triangulations of polygonal domains', Annals of Discrete Mathematics 9 (1980) pp. 127-128

....[14, 15] The second approach to approximate MWT problems uses the insight that polygon minimum weight triangulation is significantly easier than the point set MWT or MWST problems. The exact minimum weight triangulation of a simple polygon can be found by dynamic programming in time O(n 3 ) [10, 12]. If the polygon is convex, a triangulation of weight O(log n) times the polygon s perimeter can be found by the ring heuristic of repeatedly connecting all pairs of adjacent even numbered vertices [21] and as mentioned above a constant factor approximation to the MWT can be computed in linear ....

G.T. Klincsek. Minimal triangulations of polygonal domains. Ann. Disc. Math. 9 (1980) 121--123.

....R (see Figure 2.1) are retriangulated in an Edge Insertion for Optimal Triangulations 3 q s P R ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl Figure 2.1: Inserting qs leaves two polygonal regions P and R. optimal fashion (minimizing the maximum ) e.g. by dynamic programming [Klin80]. The basic, most general, version of the edge insertion paradigm is given below; it tries all possible edge insertions and halts when no edge insertion improves the current triangulation. Input. A set S of n points in 2 . Output. An optimal triangulation T of S. Algorithm. Construct an ....

....to compute global optima for maximum angle, height, eccentricity, and slope, as we show in later sections of this paper. We now argue that the basic algorithm above terminates after time O(n 8 ) A single edge insertion operation takes time O(n 3 ) when retriangulating by dynamic programming [Klin80], assuming the measures of any two triangles can be compared in constant time. The for loop thus takes time O(n 5 ) per iteration of the repeat loop. Finally, the repeat loop is iterated at most O(n 3 ) times, because there are only Gamma n 3 Delta triangles spanned by S, and each ....

G. T. Klincsek. Minimal triangulations of polygonal domains. Annals Discrete Math. 9 (1980), 121--123.

....intersect ab. 3. Retriangulate the polygonal regions P and R constructed in step 2. 4. return B. There are many ways to triangulate the polygonal regions. For now we might as well assume that P and R are triangulated in an optimal fashion (maximizing the minimum ) e.g. by dynamic programming [Klin80]. The basic version of the edge insertion paradigm can now be formulated as follows. Input. A set S of n points in 2 . Output. An optimal triangulation T of S. Algorithm. Construct an arbitrary triangulation A of S. repeat T : A; for all pairs a; b 2 S do B : Edge insertion(A; ab) if ....

G. T. Klincsek. Minimal triangulations of polygonal domains. Annals Discrete Math. 9 (1980), 121--123.

....p i k p i k 1 crosses 2 p i j p i . To obtain the optimal triangulation for P , we record the k that is chosen to optimize each L[i; i j] and use it later to trace the triangulation. In total, the algorithm takes cubic time for the nested loops, and quadratic storage for keeping L[i; j] and k [Gilb79, Klin80]. It is easy to see that the algorithm can be modified to compute other criteria. For example, we can compute a min max angle triangulation for a simple polygon in the same amount of time and storage. Also, we can compute a triangulation that lexicographically minimizes the non2 Two edges ac and ....

G. T. Klincsek. Minimal triangulations of polygonal domains. Ann. Discrete Math. 9 (1980), 121--123.

....triangulation from these edges. The following sections present the various work done towards computing efficiently the MWT of a point set through the different directions. 2. 1 MWT of Restricted Classes of Point Sets In considering restricted classes of point sets, Gilbert [Gil79] and Klincsek [Kli80] independently presented a dynamic programming algorithm that computes a minimum weight triangulation of a simple polygon in O(n 3 ) time. Recently, Anagnostou and Corneil [AC93] described an O(n 3k 1 ) time algorithm that computes the MWT of a point set that can be the vertices of k nested ....

....v i v j is a diagonal of P . A simplified variant of the minimum weight triangulation problem is to find a minimum weight triangulation of a polygon P , MWT(P) That is, find a set of diagonals and edges of P such that the sum of the weights of the diagonals and edges is minimum. As shown in [Kli80] and described in this section, there exists a dynamic programming algorithm to solve this problem. The algorithm presented in this section will solve the following problem. Given a polygon P and a subset of possible diagonals D such that MWT (P ) D [ P find a set of diagonals in D that ....

G. T. Klincsek. Minimal triangulations of polygonal domains. Discrete Math., 9:121--123, 1980.

....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 shown how to compute the MWT in 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 ....

G. Klincsek, "Minimal triangulations of polygonal domains." Ann. Discrete Math. 9 121--123.

....guarantee that certain edges belong to the MWT. If enough MWT edges 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 ....

....minimum weight triangulation, and other optimal triangulations, but this has apparently not been studied. For many optimal triangulation problems, the version of the problem in which one optimally completes the triangulation of a convex or simple polygon can be solved by dynamic programming [78], however even this is not obvious for the minimum dilation triangulation. A solution to this subproblem might have implications in allowing the powerful edge insertion method [16, 51] to be applied to the point set version of the problem. Open Problem 7. Is it possible to construct in polynomial ....

G. T. Klincsek. Minimal triangulations of polygonal domains. Ann. Disc. Math., vol. 9, 1980, pp. 121--123.

....point sets. The second approach to approximate MWT problems uses the insight that polygon minimum weight triangulation is significantly easier than the point set MWT or MWST problems. The exact minimum weight triangulation of a simple polygon can be found by dynamic programming in time O(n 3 ) [9, 11]. If the polygon is convex, a triangulation of weight O(log n) times the polygon s perimeter can be found by the ring heuristic of repeatedly connecting all pairs of adjacent even numbered vertices [20] and as mentioned above an approximation to the MWT can be computed in linear time [13, 14] By ....

G.T. Klincsek. Minimal triangulations of polygonal domains. Ann. Disc. Math. 9 (1980) 121--123.

....in a simple polygon, hence the algorithm described above will run in O(n 2 ) time. 2 Over the years, newer algorithms for this problem have been designed with improved time bounds, the best being [5] which runs in O(n) time and is optimal. The dynamic programming approach due to Klincsek [14] can optimize different criteria in triangulations of polygons. More precisely it optmizes decomposable measures. The basic idea is the same as the simple algorithm described above, but during the triangulation of the sub polygons, the results are remembered for later use. Say, f : T REAL is a ....

....a diagonal. So the particular combination can be chosen or discarded based upon the kind of improvement desired. Using dynamic programming the best combination is chosen. Examples of decomposable measures are min max angle, total edge length, min max edge length, min max area etc. Theorem 1 ([14]) A triagulation of a simple polygon optimizing any decomposable measure can be computed in time O(n 3 ) There are slight improvements in the time bound and the class of problems solvable using dynamic programming based upon this algorithm, discussed in [17] 4 Triangulating a Point Set ....

G. T. Klincsek. Minimal triangulations of polygonal domains. Ann. Disc. Math., 9:121--123, 1980.

....spanned by the vertices of P , so that each edge of P is incident to exactly one triangle, and all other triangle edges are incident to two triangles each) which minimizes the total sum of F over its triangles. For this purpose, we closely follow the dynamic programming technique of Klincsek [21] for finding a polygon triangulation in the plane, which minimizes the total sum of edge lengths. Let P = v 0 ; v 1 ; vn Gamma1 ; vn = v 0 ) be the given polygon. Let W i;j (0 i j n Gamma 1) denote the weight of the best triangulation of the polygonal curve (v i ; v j ; v ....

G.T. Klincsek, Minimal triangulations of polygonal domains, Annals of Discrete Mathematics, 9 (1980), 121--123.

.... so called LMT skeleton of S, which can be computed by a simple and cute method, proposed in Belleville et al. 4] and in Dickerson, et al. 9] see also Beirouti and Snoeyink [3] In fact the LMT skeleton of S tends to be a connected graph even for large point sets and thus by dynamic programming [14, 20] it can be completed to an optimal triangulation in polynomial time. But in general the resulting subgraph will consist of k disconnected parts. In [5] it has been shown that in the average case k may be linear in jSj. The approach proposed e.g. in Cheng et al. 6] tries all possibilities to add ....

G.T. Klincsek, Minimal triangulations of polygonal domains, Annals of Discrete Mathematics 9, 1980, 127-128.

....A weaker conjecture would be that the vertices and certain edges bound polygPage Figure 1: Examples with 40, 250 and 1000 points onal regions that have a constant number of holes. If this were true, then the MWT computation could be completed in polynomial time by dynamic programming [5]. Using this implementationon its own and as a user macro in IPE has allowed us to construct point sets where the polygonal regions have linearly many holes. First, note that by placing points on a circle so that any 60 ffi sector contains at least three points, we can isolate a point in the ....

G. T. Klincsek. Minimal triangulations of polygonal domains. Ann. Disc. Math., 9:121--123, 1980.

.... triangulation (where the length of an edge is the distance between its endpoints measured by the standard Euclidean metric) Such a triangulation is referred to as the Minimum Weight Triangulation (MWT) The problem of finding the MWT for n vertex simple polygons can be solved in O(n 3 ) time [14, 9], but it is not known how to efficiently compute the MWT of a set of points S. In fact, the complexity of finding an MWT whether it is polynomial time or NP hard is one of the few questions that remains open from Garey and Johnson s classic book on NP completeness [10] Some approaches to ....

G. Klincsek, "Minimal triangulations of polygonal domains." Ann. Discrete Math. 9 121--123.

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G. Klincsek. Minimal triangulations of polygonal domains. Annals of Discrete Mathemathics, 9:121--123, 1980.

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Klincsek and G.T. Minimal triangulations of polygonal domains. Annals of Discrete Mathemtatics 9, pages 121--123, 1980.

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G. Klincsek, Minimal triangulations of polygonal domains, Annals of Discrete Mathematics, 9 (1980) 121-123.

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G. Klincsek, Minimal triangulations of polygonal domains, Annals of Discrete Mathematics, 9 (1980) 121-123.

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Klincsek, G.T. (1980), Minimal triangulations of polygonal domains, Annals of Discrete Mathematics 9, 121--123.

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#132. #127# G.T. Klincsek. Minimal triangulations of polygonal domains. Ann. Disc. Math. 9 #1980# 121#123. #128# C.L. Lawson. Software for C

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Klincsek, G.T., Minimal triangulations of polygonal domains. Annals of Discrete Mathematics, 1980, 9, 121--123.

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G. T. Klincsek, "Minimal triangulations of polygonal domains," Annals of Discrete Mathematics, 9 (1980) 121--123.

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