D. R. Chand and S. S. Kapur, "An algorithm for convex polytopes," J. Association for Computing Machinery, vol. 17, no. 1, pp. 78--86, Jan. 1970.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the descriptions of P and Q (after removing redundant points) 5. Face Lattice of Geometric Polytopes See comments on Problem 1. Many algorithms for the Vertex Enumeration Problem in fact compute the whole face lattice of the polytope. Swart [60] analyzing an algorithm of Chand and Kapur [10], proved that there exists a polynomial total time algorithm for this problem. For a faster algorithm see Seidel [56] Fukuda, Liebling, and Margot [22] gave an algorithm which uses working space (without space for the output) bounded polynomially in the input size, but it has to solve many linear ....

D. R. Chand and S. S. Kapur, An algorithm for convex polytopes, J. Assoc. Comput. Mach., 17 (1970).

....an open question if we can do better than that if we start with a simple polyhedron rather than an arbitrary set of points. One method which is fairly easy to understand, and not too di#cult to implement, even if not the most theoretically e#cient, is the Gift Wrapping method of Chand and Kapur [3]. It can be used in both two and three dimensions, and has the advantage of being readily adapted to cope with curved objects if necessary. 7.2 Algebraic Basis of the Algorithm To see if a polyhedron will fit in a box, we must consider rotations about two di#erent axes, which obviously makes the ....

D. R. Chand, S. S. Kapur. An Algorithm For Convex Polytopes. J. ACM, 17, (1), 78-86, 1970.

....of partial convex hulls converges e ectively both in the Hausdor metric and the Lebesgue measure to the convex hull of the N points. 1 Introduction Despite a huge number of algorithms and articles published on robustness issues related to the convex hull of a nite number of points in R d [4, 2, 3, 7, 25, 26, 28], the question of computability of the convex hull for general exact real number inputs has never been addressed in the literature. Robustness problems arise from the discrepancy between the unrealistic real RAM machine model [24] used to prove the correctness of algorithms, and real computers ....

D. R. Chand and S. S. Kapur. An algorithm for convex polytopes. ACM, 17:78-86, 1970.

....Overmars. Key words. computational geometry, convex polytopes, lower bounds, decision trees, adversary arguments AMS subject classifications. 68Q25, 68U05, 52B55, 52B05 1. Introduction. The construction of convex hulls is one of the most basic and well studied problems in computational geometry [2,3,5,10,11,12,13,15,17,18, 29,34,35,38,39,47,41,45,43,44,47,48]. Over twentyyears ago, Graham described an algorithm that constructs the convex hull of n points in the plane in O(n log n) time [29] The same running time was first achieved in three dimensions by Preparata and Hong [38] Yao [48] proved a lower bound of Omega n log n) on the complexity of ....

....facets [27] Seidel s algorithm is optimal in even dimensions, and Chazelle s algorithm is optimal in all dimensions, in the worst case. Several faster algorithms are known when the output size f is also considered, at least when the input points are in general position. In 1970, Chand and Kapur [13] described a gift wrapping algorithm that constructs convex hulls in arbitrary dimensions in time O(nf )# see also [47] Seidel s shelling algorithm runs in time O(n 2 f log n) 43] A divide and conquer algorithm of Chan, Snoeyink, and Yap [12] constructs four dimensional hulls in time ....

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D. R. Chand and S. S. Kapur, An algorithm for convex polytopes,J.ACM, 17 (1970), pp. 78-- 86.

....size of input. Under the nondegeneracy assumption (see 2.12) there is a polynomial algorithm for the convex hull problem. Few of the earlier polynomial algorithms are pivot based algorithms [CCH53, Dye83] solving the problem in dual form (the vertex enumeration problem) and a wrapping algorithm [CK70]. A more recent algorithm [AF92] based on reverse search technique [AF96] is not only polynomial but compact at the same time. Here, we say an algorithm is compact if its space complexity is polynomial in the input size only. In the general case, there is no known polynomial algorithm. The paper ....

D.R. Chand and S.S. Kapur. An algorithms for convex polytopes. J. Assoc. Comput. Mach., 17:78-86, 1970.

....seems to bean open question if wecan do betterthan that if we start with a simple polyhedron ratherthan an arbitrary set of poin ts. On method which is fairly easy to un5] I)55 an n too di#cult toimplemen t, even if nI the most theoretically e#cien t, is the Gift WrappinI method ofChan an Kapur [3]. Itcan be usedin both twoan threedimen1D ]I an has the advan tage of bein readily adapted to cope with curved objects ifnI BS I . 7.2 Algebraic Basis of the Algorithm To see if a polyhedron will fitin a box, we mustcon1B rotation about twodi#eren t axes, which obviously makes the algebra ....

D. R.Chan] S. S. Kapur. An Algorithm For Convex Polytopes. J. ACM, 17, (1), 78-86, 1970.

....=1 = 1; p ] 2 P; 0; m 2 IN 9 = 11) As in IR n , it can be shown that the open convex hull of a finite number of points is the intersection of a finite number of halfspaces H 1 ; H nH . If only one of the points is infinite as in our case then the algorithm of (Chand and Kapur, 1970) can be adopted to calculate the halfspaces, just by changing the notation from Eq. 7) to Eq. 8) It works for the general case as well, but then the proof must be modified if a face or an edge happens to be entirely contained in the infinite part. This is what the introduction of IB n is ....

Chand, D. R. and Kapur, S. S. (1970). An algorithm for convex polytopes. Journal Assoc. Computing Machinery, 17(1):78--86.

....as a parametric problem of one degree of freedom, i.e. the normal vector of the new facet is obtained by tilting the given facet in the free direction determined by the ridge and the normal vector of the given facet. We call this the wrapping procedure, a term introduced by Chand and Kapur [9]. The solution to the parametric problem is found by solving a linear program over the union of polytopes projected onto the two dimensional space, see Section 3. The adjacency relation can be used to design a simple algorithm moving from one facet to all adjacent ones remembering the visited ....

D. R. Chand and S. S. Kapur. An algorithm for convex polytopes. J. ACM, 17(1):78--86, 1970.

.... [Clarkson and Shor 1989] it was derandomized by Chazelle and Matousek [Chazelle and Matousek 1992] In higher dimensions, 4 Delta the best output sensitive algorithm is Seidel s shelling algorithm at O(n 2 h log n) when h = Omega Gamma n) Seidel 1986] and gift wrapping at O(nh) otherwise [Chand and Kapur 1970]. The Double Description Method is the dual of the Beneath Beyond Algorithm [Motzkin et al. 1953] It is the earliest incremental method for computing the convex hull. It is an excellent choice in high dimensions when the number of facets is much smaller than the maximum number of facets for r ....

Chand, D. and Kapur, S. 1970. An algorithm for convex polytopes. Journal of the ACM 7, 78--86.

.... that d dimensional convex hulls can have Omega (n bd=2c ) facets, the previously best lower bound for either of the problems we consider is only Omega n log n) 1 Introduction The construction of convex hulls is one of the most basic and well studied problems in computational geometry [4,5, 6, 7, 9, 10, 15, 18,20,21,22, 29, 24, 26, 27]. Over twentyyears ago, Graham described an algorithm that constructs the convex hull of n points in the plane in O(n log n) time [15] The same running time was first achieved in three dimensions by Preparata and Hong [21] Yao [30]proved a lower bound of Omega (n log n)onthe complexity of ....

....while maintaining the order of the volumes of the other collapsible simplices. 6 Our Models vs. Real Convex Hull Algorithms A large number of convex hull algorithms rely (or can be made to rely) exclusively on sidedness queries. These include the gift wrapping algorithms of Chand and Kapur [6] and Swart [29] the beneath beyond method of Seidel [24] Clarkson and Shor s [10] and Seidel s [27] randomized incremental algorithms, Chazelle s worstcase optimal algorithm [7] and the recursive partialorder algorithm of Clarkson [9] Seidel s shelling algorithm [26] and the ....

D. R. Chand and S. S. Kapur. An algorithm for convex polytopes. J. ACM, 17:78--86, 1970.

.... h) log 2 h) time algorithm for computing the convex hull of a set of n points where h denotes the output size. In higher dimensions, T. Chan [Cha95] realized many improvements on the convex hull computations and related problems, combining the gift wrapping method of D.R. Chand and S.S. Kapur [CK70] and G.F. Swart [Swa85] with recent results on data structures for ray shooting queries in polytopes (developed by P.K. Agarwal et J. Matou#ek [AM93] and rened by J. Matou#ek and O. Schwarzkopf [MS93] Computing the convex hull of a set of curved objects has been much less investigated. ....

D. R. Chand and S. S. Kapur. An algorithm for convex polytopes. J. ACM, 17:7886, 1970.

....in [24, 36, 43] PL sets and mappings between such sets have been proposed by E.D. Sontag, see [48, 49] Algorithms on polyhedral sets are treated in several references. The algorithm by N.V. Chernikova is proposed in [18] and further discussed in [22, 58] Additional references on algorithms are [17, 52, 21, 37]. A.7 Software for polyhedral sets Software packages on polyhedral sets are discussed in [30, 60] Software packages for piecewiselinear hybrid systems are discussed in [27, 28, 31] 35 B Modeling a class of hybrid systems by piecewise linear systems In this appendix it is proven that the ....

D.R. Chand and S.S. Kapur. An algorithm for convex polytopes. J. ACM, 17:78--86, 1970.

....bases may exceed the number of facets by far. Therefore we propose a variation of the method that takes degeneracies into account explicitly: Instead of visiting all feasible bases, the algorithm visits all facets. The manner of visiting facets is analogous to the convex hull algorithm of Chand and Kapur #1970# as it is described and analyzed in Swart #1985#. Section 2 gives an overview of the method in a quite general way. Section 3 describes the method from a di#erent point of view: as a seach of the face lattice. Section 4 describes speci#c pivoting rules and section 5 gives some implementation ....

D. R. Chand and S. S. Kapur #1970# An algorithm for convex polytopes, J. Assoc. Comput.

....bases may exceed the number of facets by far. Therefore we propose a variation of the method that takes degeneracies into account explicitly: Instead of visiting all feasible bases, the algorithm visits all facets. The manner of visiting facets is analogous to the convex hull algorithm of Chand and Kapur [1970] as it is described and analyzed in Swart [1985] Section 2 gives an overview of the method in a quite general way. Section 3 describes the method from a different point of view: as a seach of the face lattice. Section 4 describes specific pivoting rules and section 5 gives some implementation ....

D. R. Chand and S. S. Kapur [1970] An algorithm for convex polytopes, J. Assoc. Comput.

....on the bases of P , where each edge corresponds to a pivot , the replacement of exactly one hyperplane in a basis. All vertices of this graph are generated, from which the extreme points of P are readily computed. Representatives of this class are the gift wrapping algorithm of Chand and Kapur [5], Seidel s algorithm [19] and the reverse search algorithm of Avis and Fukuda [1] For a simple polytope P , pivoting algorithms can solve all of the above problems in time polynomial in size(P ) For non simple polytopes the number of bases may be exponential in the size of P . For such ....

D.R. Chand and S.S. Kapur. An algorithm for convex polytopes. J. ACM, 17:78--86, 1970.

.... and Seidel [20] who also provided a matching lower bound; a simplification of their algorithm has been recently reported by Chan, Snoeyink, and Yap [2] In E 3 , one can obtain an O(nh) time algorithm using the gift wrapping method, an extension of Jarvis s march originated by Chand and Kapur [3]. A faster but more involved algorithm in E 3 was discovered by Edelsbrunner and Shi [13] having running time O(n log 2 h) Finally, by derandomizing an algorithm of Clarkson and Shor [8] Chazelle and Matousek [7] succeeded in attaining optimal O(n log h) time in E 3 . These algorithms, ....

....faces) of the convex hull; with the aid of a dictionary, we can easily produce the set of h vertices and 3h Gamma 6 edges together with their adjacency and order information in additional O(h log h) time. The higher dimensional analogue of Jarvis s march is Chand and Kapur s gift wrapping method [3, 25, 26], which computes the hull facets one at a time as follows: from a given facet f , we generate its three adjacent facets f j by performing a wrapping step about each of the three edges e j of f (j = 1; 2; 3) Here, a wrapping step about e j is to compute a point p j 2 P that maximizes the angle ....

D. R. Chand and S. S. Kapur. An algorithm for convex polytopes. J. ACM, 17:78--86, 1970.

....are in harmony with me and sustain me. 22. Laugh away these facts and truths if you can. Carl Theodore Heisel, The Circle Squared Beyond Refutation, 31 Chapter 3 Convex Hull Problems The construction of convex hulls is perhaps the oldest and best studied problems in computational geometry [6, 10, 11, 12, 29, 28, 30, 36, 49, 50, 91, 101, 110, 123, 130, 132, 134, 136, 142]. Over twenty years ago, Graham described an algorithm that constructs the convex hull of n points in the plane in O(n log n) time [91] The same running time was first achieved in three dimensions by Preparata and Hong [123] Yao [154] proved a lower bound of Omega (n log n) on the complexity of ....

....an n vertex polytope in IR d can have Omega (n bd=2c ) facets [87] Seidel s algorithm is optimal in even dimensions, and Chazelle s algorithm is optimal in all dimensions, in the worst case. Several faster algorithms are known when the output size is also considered. In 1970, Chand and Kapur [30] described an algorithm that constructs convex hulls in time O(nf) where f is the number of facets in the output. An algorithm of Chan, Snoeyink, and Yap [28] constructs four dimensional hulls in time O( n f) log 2 f) and 32 a recent improvement by Amato and Ramos [6] constructs ....

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D. R. Chand and S. S. Kapur. An algorithm for convex polytopes. J. ACM, 17:78--86, 1970.

....it The convex hull problem in its three dimensional setting (d = 3) has also been studied intensively. Preparata and Hong [PH77] presented the first O(n log n) time algorithm in E 3 , based on divide and conquer. The first output sensitive algorithm is the giftwrapping method of Chand and Kapur [CK70], which works in arbitrary dimensions and is a generalization of Jarvis s march in two dimensions (although historically, Chand and Kapur s method appeared before Jarvis s) As analyzed by Swart [Swa85] the method runs in O(nh) time. For a long time the gift wrapping method was the only ....

....of conditional probabilities. The problem of finding a simple optimal output sensitive algorithm for computing convex hulls in E 3 thus remained. For dimensions d 4, much attention was directed to devising efficient worst case convex hull algorithms. The gift wrapping method by Chand and Kapur [CK70] was shown to run in O(n bd=2c 1 ) time [Swa85] in the worst case. Seidel [Sei81] improved this time bound to O(n dd=2e ) using a different approach called the beneath beyond method. In a later paper [Sei86] Seidel exploited a shelling order to obtain a second algorithm with an O(n bd=2c ....

[Article contains additional citation context not shown here]

D. R. Chand and S. S. Kapur. An algorithm for convex polytopes. Journal of the Association for Computing Machinery, 17:78--86, 1970.

....hull of each path, taking the union of the convex hulls where the paths merge. Since the union of a set of convex hulls is not necessary itself a convex hull, the convex hull of the resulting polyhedron must also be determined. The convex hull can be found using any of the following algorithms: [8, 7, 10]. The if then else construct can now be replaced by the constraints which define the minimal maximal changes of each variable. This is known as collapsing the branching construct. To determine the set of new vertices following a conditional statement, each vertex of the current polyhedron is ....

S. S. Kapur D. R. Chand. An algorithm for convex polytopes. Journal of the ACM, 17:78--86, January 1970.

.... hull algorithm with optimal outputsensitive expected time [16] it was derandomized by Chazelle and Matousek [12] In higher dimensions, the best output sensitive algorithm is Seidel s shelling algorithm at O(n 2 h log n) when h = Omega Gamma n) 40] and gift wrapping at O(nh) otherwise [11]. The Double Description Method is the dual of the Beneath Beyond Algorithm [36] It is the earliest incremental method for computing the convex hull. It is an excellent choice in high dimensions when the number of facets is much smaller than the maximum number of facets for r vertices (f r ) ....

D.R. Chand and S.S. Kapur. An algorithm for convex polytopes. Journal of the ACM, 7:78--86, 1970.

No context found.

D. R. Chand and S. S. Kapur, "An algorithm for convex polytopes," J. Association for Computing Machinery, vol. 17, no. 1, pp. 78--86, Jan. 1970.

No context found.

D.R.Chand, S.S. Kapur. An Algorithm for Convex Polytopes. JACM, 17:1:78-- 86, 1970.

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D. R. Chand, and S. S. Kapur. An algorithm for convex polytopes. J. ACM, 17(1):78--86, January 1970.

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