R Seidel. A convex hull algorithm optimal for point sets in even dimensions. Technical Report 81 14, Department of Computer Science, University of British Columbia, Vancouver, British Columbia, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....using the above mentioned next v algorithm or it can be found in form of a table in [27] There are plenty of algorithms for point set problems which are based on computing the orientation of a sequence of points. Prime examples are algorithms for the construction of convex hulls; see for example [66, 72, 73] or [21,54, 67] References to other applications can be found in [27] which also extends the concept of orientation to points with homogeneous coordinates in arbitrary dimensions. Sphere test. The crucial primitive operation for constructing Delaunay triangulations is to check if a triangle is ....

R Seidel. A convex hull algorithm optimal for point sets in even dimensions. Technical Report 81 14, Department of Computer Science, University of British Columbia, Vancouver, British Columbia, 1981.

.... 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 ....

....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 spaceefficient gift wrapping algorithms of Avis and Fukuda 1 [2] and ....

R. Seidel. A convex hull algorithm optimal for pointsets in even dimensions. M.Sc. thesis, Dept. Comput. Sci., Univ. British Columbia, Vancouver, BC, 1981. Report 81/14.

....now widely known as the double description method , appeared in the pioneering 1953 paper of Motzkin et al. 17] this paper seems to have been overlooked by the Computational Geometry community. Many of the same ideas were rediscovered and refined in the beneath and beyond method of Seidel [18], the randomized algorithm of Clarkson and Shor [8] and the derandomized algorithm of Chazelle [6] In some sense the algorithms [18] and [6] can be considered optimal. The upper bound theorem of McMullen states that for any polytope P defined by m halfspaces, size(P ) O(m bd=2c ) and this ....

....to have been overlooked by the Computational Geometry community. Many of the same ideas were rediscovered and refined in the beneath and beyond method of Seidel [18] the randomized algorithm of Clarkson and Shor [8] and the derandomized algorithm of Chazelle [6] In some sense the algorithms [18] and [6] can be considered optimal. The upper bound theorem of McMullen states that for any polytope P defined by m halfspaces, size(P ) O(m bd=2c ) and this bound is achieved (see, e.g. 3] The algorithms [18] and [6] solve the vertex enumeration problem in this time bound. However, it is ....

[Article contains additional citation context not shown here]

R. Seidel. A convex hull algorithm optimal for point sets in even dimensions. Technical report, University of British Columbia, Dept. of Computer Science, 1981.

....algorithms for computing the planar convex hull, the asymptotically optimal time of O(n log h) has been achieved by Kirkpatrick and Seidel [Kirkpatrick Seidel 86] where h is the number of edges in the convex hull. For higher dimensions, these can be computed in time O(n dd=2e ) for odd d [Seidel 81] and at logarithmic cost per face for even d 4 [Seidel 86] For good survey material on convex hulls see [Preparata Shamos 85, Dobkin Souvaine 87, Edelsbrunner 87, Graham Yao 90] 3.3.2 ff Hull An elegant generalization of the convex hulls for points in a plane is done by Edelsbrunner, ....

R. Seidel. A convex hull algorithm optimal for point sets in even dimensions. M.Sc. Thesis and Report 81/14, Dept. Comput. Sci., Univ. British Columbia, Vancouver, BC, 1981.

....is the opposite of the orientation of (p i 0 ; p i 1 ; p i d ) Simulation of Simplicity 8 There are plenty of algorithms for point set problems which are based on computing the orientation of a sequence of points. Prime examples are the construction of convex hulls (see [PH77] PS85] [Se81], Se86] or [Ed87] computing matrices as discussed in [GP83] and [Ed87] and finding convex subsets (see [CK80] EG89] and [Ed87] The remainder of this section considers the primitive operations required by the three dimensional convex hull algorithm of Preparata and Hong which is ....

R. Seidel. A convex hull algorithm optimal for point sets in even dimensions. Technical Report 81--14, Department of Computer Science, University of British Columbia, Vancouver, British Columbia, 1981.

....Kirkpatrick and Seidel obtained a deterministic algorithm for planar convex hulls with the same time bound [31] We also give a Las Vegas incremental algorithm requiring O(n log n) expected time for d = 3 and O(n bd=2c ) expected time for d 3. This improves known results for odd dimensions [36, 40, 41, 20]. For independently identically distributed points, the algorithm requires O(n) P 1rn f(r) r 2 expected time, where f(r) is the expected size of the convex hull of r such points. Here f(r) must be nondecreasing. The algorithm is not complicated. Spherical intersections and diametral ....

R. Seidel. A convex hull algorithm optimal for point sets in even dimensions. Technical Report 81/14, Univ. British Columbia, Dept. Computer Science, 1981.

....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 log n) worst case running time. The randomized incremental construction technique ....

....time. The algorithm is therefore quite efficient for the whole range of output sizes f from Theta(1) to Theta(n 2 ) For example, when f = Theta(n) the algorithm runs in O(n log 2 n) time, which is a significant improvement over the O(n 2 ) running time of a worst case optimal algorithm [Sei81]. The previous output sensitive method by Seidel [Sei86] combined with Matousek s improvement [Mat93] achieves O(n 4=3 log O(1) n) time in this case. The basic strategy behind our algorithm is divide and conquer. In order to obtain an output sensitive method, the subproblems we solve ....

R. Seidel. A convex hull algorithm optimal for point sets in even dimensions. Technical Report 81-14, Department of Computer Science, University of British Columbia, Vancouver, B.C., 1981.

....we consider is only Omega Gamma n log n) This research was partially supported by a GAANN Fellowship. New Lower Bounds for Convex Hull Problems in Odd Dimensions 1 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, 14, 17, 19, 20, 21, 28, 23, 25, 26]. 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. The same running time was achieved in three dimensions by Preparata and Hong [20] Yao [29] proved a lower bound of Omega Gamma n log n) on the complexity of ....

....simplices. 6 Our Computational 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 [28] the beneath beyond method of Seidel [23], Clarkson and Shor s [10] and Seidel s [26] randomized incremental algorithms, Chazelle s worst case optimal algorithm [7] and the recursive algorithm of Clarkson [9] Seidel s shelling algorithm [25] and the space efficient gift wrapping algorithms of Avis and Fukuda 2 [2] and Rote [21] ....

R. Seidel. A convex hull algorithm optimal for point sets in even dimensions. M.Sc. thesis, Dept. Comput. Sci., Univ. British Columbia, Vancouver, BC, 1981. Report 81/14.

.... d dimensional convex hulls can have Omega Gamma n bd=2c ) facets, the previously best lower bound for either of the problems we consider is only Omega Gamma 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 twenty years 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 Gamma n log n) on the complexity ....

....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 spaceefficient gift wrapping algorithms of Avis and Fukuda 1 [2] and ....

R. Seidel. A convex hull algorithm optimal for point sets in even dimensions. M.Sc. thesis, Dept. Comput. Sci., Univ. British Columbia, Vancouver, BC, 1981. Report 81/14.

....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 twenty years 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 Gamma n log n) on the ....

....is also taken into account, the lower bound drops to Omega Gamma n log f) 34] and a number of algorithms match this bound both in the plane [34, 12, 10] and in three dimensions [18, 16, 10] In higher dimensions, the problem is not quite so completely solved. Seidel s beneath beyond algorithm [41] constructs d dimensional convex hulls in time O(n dd=2e ) After a ten year wait, Chazelle [15] improved the running time to O(n bd=2c ) by derandomizing a randomized incremental algorithm of Clarkson and Shor [18] see also [44] Since an n vertex polytope in IR d can have Omega Gamma n ....

[Article contains additional citation context not shown here]

R. Seidel, A convex hull algorithm optimal for point sets in even dimensions, M.Sc. thesis, Dept. Comput. Sci., Univ. British Columbia, Vancouver, BC, 1981. Report 81/14.

....at Urbana Champaign. z This research supported in part by NSF Grant CCR 9118874. 1.1 Related work Optimal deterministic sequential algorithms have long been known for the cases d = 2; 3 [28, 47] In higher dimensions, d 4, Seidel proposed two deterministic algorithms. His first algorithm [54] ran in O(n logn n dd=2e ) time 1 , which is optimal for even d, and later he gave an O(n bd=2c log n) solution [55] For some time, the only solutions optimal in higher dimensions were the randomized incremental algorithm of Clarkson and Shor [15] and the subsequent randomizedmethod of ....

R. Seidel. A convex hull algorithm optimal for point sets in even dimensions. M.Sc. thesis, Dept. Comput. Sci., Univ. British Columbia, Vancouver, BC, 1981. Report 81/14.

....now widely known as the double description method , appeared in the pioneering 1953 paper of Motzkin et al. 31] this paper seems to have been overlooked by the Computational Geometry community. Many of the same ideas were rediscovered and refined in the beneath and beyond method of Seidel [35] (in the facet enumeration setting) the randomized algorithm of Clarkson and Shor [14] and the derandomized algorithm of Chazelle [12] Finally we should mention that so called Fourier Motzkin elimination can be viewed just as a dual formulation of the double description method and thus falls in ....

....of the current polytope) As we shall see, the same algorithm applied to the same input can have vastly different running times when different insertion orders are used. Thus choosing a good insertion order is crucial. As a manifestation of this consider the incremental algorithms of Seidel [35] and of Chazelle [12] If performance is measured only in terms of input size and the dimension d is kept fixed, then these algorithms are asymptotically worst case optimal (for even d in case of [35] and for general d in case of [12] Seidel s algorithm relies crucially on the use of a ....

[Article contains additional citation context not shown here]

R. Seidel. A convex hull algorithm optimal for point sets in even dimensions. Technical report, University of British Columbia, Dept. of Computer Science, 1981.

No context found.

R. Seidel. A convex hull algorithm optimal for point sets in even dimensions. Tech. Rept. 81/14, Dept. Computer Science, Univ. of British Columbia, Vancouver, 1981. 26

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC