S. Fortune. Stable maintenance of point-set triangulations in two dimensions. In Proceedings of the 30th IEEE Symposium on Foundations of Computer Sience, pages 494--499, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of combinatorial struc tures (the reasoning paradigm [HHK88] is generally hard. Indeed, the NP hard existential theory of reals can be reduced to such problems. In some sense, the ultimate approach to ensuring consistency is to design parsimonious algorithms in the sense of Fortune [For89] This also amounts to theorem proving as it entails deducing the consequences of all previous decisions along a computation path. STABILITY This is a metric form of topological distortion where we place a priori bounds on the amount of distortion. It is analogous to backwards error analysis in ....

....if # is a constant (resp. O(n) where n is the input size. Fortune and Milenkovic [FM91] provide both linearly stable and strongly stable algorithms for line arrangements. Stable algorithms have been achieved for two other problems on planar point sets: maintaining a triangulation of a point set [For89] and Delaunay triangulations [For92, For95a] The latter problem can be solved stably using either an incremental or a diagonal flipping algorithm that is O(n ) in the worst case. Jaromczk and Wasilkowski [JW94] presented stable algorithms for convex hulls. Stability is a stronger requirement ....

Steven J. Fortune. Stable maintenance of point-set triangulations in two dimensions. IEEE Foundations of Computer Science, 30:494--499, 1989.

....ensure that the combinatorics we compute must be part of some consistent geometric object. This amounts to saying that the decisions we make in evaluating predicates during a computation must never be inconsistent. Below we give a slightly more formal version of the consistency approach. Fortune [24] points out that in principle it is possible to make many algorithms parsimonious (meaning that we only perform conditional tests which are independent of the results of previous tests) The basis for this observation is that, assuming all predicates are polynomial sign evaluations, the ....

S. J. Fortune. Stable maintenance of point-set triangulations in two dimensions. IEEE Foundations of Computer Science, 30:494--499, 1989.

....structure. Finite precision arithmetic. The sensitivity of algorithms to approximate data structures is related in spirit to the challenging problems that arise from various types of error in numeric computations. Such errors has been studied, for example, in the context of computational geometry [8, 9, 13, 14, 21, 22, 23]. We discuss this further in Section 6. Approximate sorting. Bern, Karloff, Raghavan, and Schieber [3] introduced approz imate sorting and applied it to several geometric problems. Their results include an O( log og ,4 O im gorihm h nds ( 0 approximate Euclidean minimum spanning tree. They ....

....in studying the effect of such errors on algorithms. Of particular interest were algorithms in computational geometry. Frameworks such as the epsilon geometry of Guibas, Salesin and Stolfi [14] may be therefore relevant in our context. The robust algorithms described by Fortune and Milenkovic [8, 9, 21, 22, 23] are natural candidates for approximate data structures. Expanding the range of applications of approximate data structures is a fruitful area for further research. Other possible candidates include algorithms in computational geometry that use the well known sweeping technique, provided that ....

S. Fortune. Stable maintenance of point-set triangula- tion in two dimensions. In Proc. of the 30th IEEE Annual Syrup. on Foundation of Computer Science, 1989.

....uses two terms, robustness and stability. The former usually means that the program handles all input occurring in practice, whereas the latter often refers to how the implementation copes with numerical errors. Here, both notions are used synonymously, mostly following Fortune s definition [35]: A robust or stable implementation of a geometric algorithm is a program whose output is always correct in the sense that, although it might not produce the exact result for the given input 5 , it guarantees the correct result for a set 5 very close to the original. One school of researchers ....

S Fortune. Stable maintenance of point-set triangulations in two dimensions. Unpublished manuscript, 1989.

....have been shown to have serious shortcomings, since they may cause not only numerical errors but also fatal membership errors (such as inclusion of a point in an interval to which it does not belong, etc. This diOEculty has received some deserved attention in recent years (see, e.g. [For89, GY86, HHK89, Mil88a, Mil88b, Mil89, GSS89, SI88]) and several approaches have been proposed on how to obviate the shortcoming. The common objective is to produce robust algorithms, namely algorithms whose answer is a (small) perturbation of the cor rect answer (as produced by the innite precision algorithm) As noted by Fortune [For89] there ....

....GSS89, SI88] and several approaches have been proposed on how to obviate the shortcoming. The common objective is to produce robust algorithms, namely algorithms whose answer is a (small) perturbation of the cor rect answer (as produced by the innite precision algorithm) As noted by Fortune [For89], there are basically two categories of approaches to this objective: The most common one resorts to approximate (i.e. rounded) computations, and uses properties of the assumed primitives to establish the topological correctness of the results (i.e. robustness) see, e.g. For89, For92, FM91, ....

[Article contains additional citation context not shown here]

S. Fortune. Stable maintenance of point set triangulations in two dimensions. In Proc. 30th Annu. IEEE Sympos. Found. Comput. Sci., pages 494505, 1989.

....with the known combinatorial facts . A correct algorithm (that computes a purely combinatorial result) will produce a meaningful result if its test results are wrong but are consistent with each other, because there exists an input for which those test results are correct. Following Fortune [6], an algorithm is robust if it always produces the correct output under the real RAM model, and under approximate arithmetic always produces an output that is consistent with some hypothetical input that is a perturbation of the true input; it is stable if this perturbation is small. Typically, ....

....that the point b lay between the lines ac and ad, but an incorrect incircle test claimed that a lay inside the circle dcb. 34 Jonathan Richard Shewchuk One might wonder if my triangulation program can be made robust by avoiding any test whose result can be inferred from previous tests. Fortune [6] explains that [a]n algorithm is parsimonious if it never performs a test whose outcome has already been determined as the formal consequence of previous tests. A parsimonious algorithm is clearly robust, since any path through the algorithm must correspond to some geometric input; making an ....

Steven Fortune. Stable Maintenance of Point Set Triangulations in Two Dimensions. 30th Annual Symposium on Foundations of Computer Science, pages 494--499. IEEE Computer Society Press, 1989. 52 Jonathan Richard Shewchuk

....triangulation algorithm that is asymptotically optimal in the worst case. The algorithm uses the quad edge data structure and only two geometric primitives, a CCW orientation test and an in circle test. These primitives are defined in terms of 3 by 3 and 4 by 4 determinants, respectively. Fortune [12, 13] shows how to compute these accurately with finite precision. Dwyer [8] showed that a simple modification of this algorithm runs in O(n log log n) expected time on uniformly distributed sites. Dwyer s algorithm splits the set of sites into vertical strips with p n= log n sites per strip, ....

S. Fortune. Stable maintenance of point-set triangulations in two dimensions. IEEE Symposium on Foundations of Computer Science, pages 494--499, 1989.

....may be just above both facets yet remain distant from the precise convex hull. Later, the coplanar point may be far above a new facet. If this occurs, Quickhull generates a warning and reports a wide facet. In R 2 , there are several robust convex hull and Delaunay triangulation algorithms [Fortune 1989] [Guibas et al. 1993] Li and Milenkovic 1990] In R 3 , Sugihara and Dey et al. produce a topologically robust convex hull and Delaunay triangulation [Dey et al. 1992] Sugihara 1992] Their algorithms are a variation of BeneathBeyond with steps to prevent topological anomalies such as in ....

Fortune, S. 1989. Stable maintenance of point-set triangulation in two dimensions. In 30th Annual Symposium on the Foundations of Computer Science (1989). IEEE.

....has proved unrealistic and needs to be replaced with a realistic finite precision model where geometric computations can be carried out either exactly or with a guaranteed error bound. This has motivated a great deal of research on the subject of robust computational geometry (see, e.g. [4, 11, 10, 18, 26, 27, 30, 35, 33, 38, 47, 53, 56, 20, 29, 31]) Also, efficiency must be evaluated in a finer framework than the conventional big Oh analysis. In particular, constant factors dependent on the precision requirement of the numerical computations should be taken into account. For an early survey of the different approaches to robust ....

S. Fortune. Stable maintenance of point set triangulations in two dimensions. In Proc. 30th Annu. IEEE Sympos. Found. Comput. Sci., pages 494--505, 1989.

....with the known combinatorial facts. A correct algorithm (that computes a purely combinatorial result) will produce a meaningful result if its test results are wrong but are consistent with each other, because there exists an input for which those test results are correct. Following Fortune [6], an algorithm is robust if it always produces the correct output under the real RAM model, and under approximate arithmetic always produces an output that is consistent with some hypothetical input that is a perturbation of the true input; it is stable if this perturbation is small. Typically, ....

....is difficult to discern which side of the edge the vertex falls on. Only exact arithmetic can prevent the possibility of creating an inverted triangle. One might wonder if my triangulation program can be made robust by avoiding any test whose result can be inferred from previous tests. Fortune [6] explains that [a]n algorithm is parsimonious if it never performs a test whose outcome has already been determined as the formal consequence of previous tests. A parsimonious algorithm is clearly robust, 3 b a c d Figure 1: Top left: A Delaunay triangulation. Top right: An invalid ....

Steven Fortune. Stable Maintenance of Point Set Triangulations in Two Dimensions. 30th Annual Symposium on Foundations of Computer Science, pages 494--499. IEEE Computer Society Press, 1989. 10

....have been shown to have serious shortcomings, since they may cause not only numerical errors but also fatal membership errors (such as inclusion of a point in an interval to which it does not belong, etc. This difficulty has received some deserved attention in recent years (see, e.g. [For89, GY86, HHK89, Mil88a, Mil88b, Mil89, GSS89, SI88]) and several approaches have been proposed on how to obviate the shortcoming. The common objective is to produce robust algorithms, namely algorithms whose answer is a (small) perturbation of the correct answer (as produced by the infinite precision algorithm) As noted by Fortune [For89] there ....

....GSS89, SI88] and several approaches have been proposed on how to obviate the shortcoming. The common objective is to produce robust algorithms, namely algorithms whose answer is a (small) perturbation of the correct answer (as produced by the infinite precision algorithm) As noted by Fortune [For89], there are basically two categories of approaches to this objective: The most common one resorts to approximate (i.e. rounded) computations, and uses properties of the assumed primitives to establish the topological correctness of the results (i.e. robustness) see, e.g. For89, For92, FM91, ....

[Article contains additional citation context not shown here]

S. Fortune. Stable maintenance of point set triangulations in two dimensions. In Proc. 30th Annu. IEEE Sympos. Found. Comput. Sci., pages 494--505, 1989.

....a (finite precision) floating point arithmetic is used. In floating point arithmetic, which is widely used for implementing algorithms, the computation of a convex hull is a much less explored problem. Relatively few numerically stable and time complexity optimal algorithms are known; see e.g. [2] for computing a convex hull of a point set, and [5] for computing the convex hull of a simple polygon. These numerically stable algorithms provide, as the output, only an approximate hull. That is, the constructed approximate hull need not be convex; it is the convex hull only for slightly ....

....algorithms might be insufficient. Therefore, one needs an algorithm that is numerically stable in a stronger sense. We will present such an algorithm, called Convex, that constructs a truly convex polygon. More precisely, for a given output from a numerically stable convex hull algorithm (such as [2, 5]) Convex returns in optimal Theta(n) time a subset of points that are the vertices of a convex Partially supported by the National Science Foundation under Grant CCR 91 14042 1 polygon. Furthermore, all rejected points are very close to this polygon; see Section 2 for a precise statement. In ....

[Article contains additional citation context not shown here]

S. Fortune, "Stable maintenance of point-set triangulations in two dimension," Proceedings of the 27th IEEE Symposium on Foundations of Computer Science, 494-499, 1989.

....and division is replaced by a single corresponding floating point operation. Clearly, robust algorithms are of great practical interest. Robust algorithms have been devised for constructing arrangements of lines [6, 7] or algebraic curves [5] convex hulls and triangulations of planar point sets [1], and Voronoi diagrams [8] As one would expect, these algorithms generate objects which are near but not equal to objects constructed using exact arithmetic. As one does not expect, the robustly generated objects are not actually line or algebraic curve arrangements, convex hulls, triangulations, ....

....vertex that is not 2ffl strongly convex (see Lemma 1) generates a 2nffl hull. If one sets ffl to zero, the algorithm of Section 4 becomes the first robust geometric algorithm to construct a convex O( hull faster than the best exact arithmetic algorithm for the same problem. Fortune s algorithm [1] is slightly faster and has a little less error (6 ) but it only generates a 6 weakly convex polygon. Guibas, Salesin, and Stolfi [4] have developed a rounded arithmetic algorithm for constructing strongly convex hulls, but it has running time O(n 2 log n) 2 Constructing A Strongly Convex ....

Steven Fortune. Stable Maintenance of Point-Set Triangulation in Two Dimensions. In 30th Annual Symposium on the Foundations of Computer Science, IEEE, October 1989.

....why this was necessary. Strictly robust algorithms were devised for the construction of arrangements of lines [13] line segments [15] planes [13] and algebraic curves [14] With Li [17] this author also devised a strictly robust convex hull algorithm for points in two dimensions. Fortune [5] has also given a strictly robust algorithm for constructing convex hulls. Hoffman and Hopcroft [8] show how to prove a strong form of feasibility for the intersection of no more than two polygonal regions. For this construction, the Pappus configuration (Section 2.4) cannot arise, and thus it is ....

....is no subtle infeasibility. A number of algorithms set aside strict accuracy in favor of good error bounds in all but pathological cases. This author [12] gives such an algorithm for constructing unions and intersections of polygonal regions. Segal and Sequin [20] have a similar result. Fortune [5] gives a robust algorithm for maintaining point set triangulations which has optimal time in all cases and good error bounds in non pathological cases. Fortune and this author [6] give optimal running time algorithms for constructing arrangements of lines and prove an error bound linear in the ....

[Article contains additional citation context not shown here]

Steven Fortune. Stable Maintenance of Point-Set Triangulations in Two Dimensions. In 30th Annual Symposium on the Foundations of Computer Science, IEEE, October 1989.

....of software is of vital importance. In geometric algorithms, many decisions are based on geometric predicates. If these predicates are not computed correctly (for example due to round o errors) the algorithm may easily give incorrect results. There are di erent notions of robustness [DSB92, For89] For some algorithms, strategies exist to deal with inexact predicates. However, in general this is very di cult to achieve. One way to deal with robustness problems is to perform exact arithmetic, which implies exact geometric computation [Sch96] 2.2 Generality The applications of the Cgal ....

S. Fortune. Stable maintenance of point-set triangulations in two dimensions. In Proceedings of the 30th IEEE Symposium on Foundations of Computer Science, pages 494-499, 1989.

....Robustness, efficiency, generality, and ease of use are the primary design goals. 4.1 Robustness What is robustness Dey et al. 5] define robustness as the ability of a geometric algorithm to deal with degeneracies and the inaccuracies during various numerical computations. Fortune [8] calls an algorithm robust if it always computes an answer that is the correct answer for some perturbation of the input. He calls it stable if that perturbation is small. Sometimes an algorithm is already called robust if it does not fail. A safe way to robustness is exact computation. It simply ....

S. Fortune. Stable maintenance of point-set triangulations in two dimensions. In Proceedings of the 30th IEEE Symposium on Foundations of Computer Sience, pages 494--499, 1989.

....P rep can be thought of as a set of constraints on the topology and position of the implicit model polygon. Any real polygon P satisfying these constraints is considered a model for P rep . Under the representation and model approach, the definition of robustness is very close to that of Fortune [2]. Consider a geometric problem P as a function from an input space consisting of models to an output space, P : I O, and consider an algorithm A as function from a different input space consisting of representations to the same output space, A : R O. Given a representation x rep , the set of ....

S. Fortune. Stable maintenance of point set triangulations in two dimensions. In IEEE Annual Symposium on Foundations of Computer Science, pages 494--499, 1989.

....trees to the technique of [69] they show that the dynamic point inclusion problem on a collection of compound objects defined by CSG trees of total size n can be solved using an O(n) space data structure that supports query and update operations in O(log n) time. 9. 6 Approximation Fortune [59] considers geometric algorithms implemented using approximate arithmetic. An algorithm is robust if it always produces an output that is correct for some perturbation of its input; it is stable if the perturbation is small. An assertion of stability should be accompanied by a measure of the ....

....some perturbation of its input; it is stable if the perturbation is small. An assertion of stability should be accompanied by a measure of the relative perturbation bound. Perturbation can be measured as a function of the problem size n and the relative accuracy of the approximate arithmetic. In [59] a technique to maintain the triangulation of a planar point set is presented, in which the triangulation is stored as a combinatorial planar graph embedding. The following operations are supported: point location, adding and deleting points, and changing edges. It achieves the stability assertion ....

S. Fortune, "Stable Maintenance of Point-set Triangulations in Two Dimensions," Proc. 30th Symp. on Foundations of Computer Science (1989), 494--499.

....by a disk (or box) that contains it. But if the lines are close to parallel, the diamond is long and skinny, and the replacement disk is very large. Some attempt has been made to apply error analysis techniques to geometric algorithms as a whole, rather than just to the geometric primitives[41, 57, 65, 66]. It is not immediately clear what kind of statement can be made about a geometric algorithm implemented using floating point arithmetic. It is too much to require that the algorithm obtain the correct answer for all inputs, because it may be impossible to distinguish among inputs with distinct ....

....error analysis for some simple geometric algorithms. An example is the problem of computing convex hulls in two dimensions. The input is a set of n points in E 2 . The output is a simple polygon, the indices of the points on the boundary of 37 the convex hull in order around the hull. Fortune[41] gives a variant of Graham s algorithm that can be implemented in floating point arithmetic. The algorithm can be applied to any set of points, in general position or not. The correctness claim is that the computed polygon is the true boundary of the convex hull of a slightly perturbed set of ....

S. Fortune. Stable maintenance of point-set triangulation in two dimensions, manuscript, AT&T Bell Laboratories. An abbreviated version appeared in Proc. 30th Annual Symp. Found. Comp. Sci. 494--499, 1989.

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S. Fortune, Stable maintenance of point-set triangulation in two dimensions, 30th Ann. Symp. on the Found. of Comp. Sci., 494--499, 1989.

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S. Fortune. Stable maintenance of point-set triangulations in two dimensions. In Proceedings of the 30th IEEE Symposium on Foundations of Computer Sience, pages 494--499, 1989.

No context found.

Steven Fortune. Stable Maintenance of Point Set Triangulations in Two Dimensions. 30th Annual Symposium on Foundations of Computer Science, pages 494--499. IEEE Computer Society Press, 1989.

No context found.

S. Fortune. Stable Maintenance of Point Set Triangulations in Two Dimensions. In Proc. of 30th Annual Symposium on Foundations of Computer Science, pages 494--499. IEEE Computer Society Press, 1989.

No context found.

S. Fortune, "Stable maintenance of point-set triangulation in two dimensions," unpublished manuscript, AT&T Bell Laboratories. (An abbreviated version appeared in Proc. 30th Ann. Symp. on Foundations of Computer Science, 1989, pp. 494-499.)

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S. Fortune, "Stable Maintenance of point set triangulations in two dimensions", IEEE Annual Symposium on Foundations of Computer Science, pp. 494-499, 1989.

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