T. M. Chan, J. Snoeyink, and C.-K. Yap. Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proc. 6th ACM-SIAM Sympos. Discrete Algorithms, pages 282--291, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....paper the details of an efficient computation of this complex C. Note however, that in the case of a molecule in three dimensions (d = 3) we have to compute a set of 4 dimensional convex hulls that can be computed more efficiently, in an output sensitive sense, by using the algorithm given in [10]. The use of this algorithm would indeed be beneficial because the overall number of faces in C is indeed O(n ) This is proved by a technique introduced in [7] that generalizes the lifting scheme for the computation of Power Diagrams [18] and maps the construction of the complex C to a ....

T. M. Y. Chan, J. Snoeyink, and C.-K. Yap. Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 282--291, San Francisco, California, January 1995. 30

....and has an integrated interactive environment with visualization tools. We refined the initial algorithm through alternating rounds of experimentation and algorithmic design. We improve the basic algorithm from a practical point of view by using the 2D convex hull algorithm of Chan et al. [21]. This algorithm has nonoptimal theoretical work since it runs in worst case O(n log h) work instead of linear (for sorted input) However, in practice our experiments show that it runs in linear work, and has a smaller constant than the provably linear work algorithm. Our final algorithm is not ....

....algorithm, and is the most expensive component. We considered three candidates for the convex hull algorithm: 1) Overmars and van Leeuwens [23] which is O(n) work for sorted points. 2) Kirkpatrick and Seidel s O(n log h) algorithm [18] and its much simplified form as presented by Chan et al. [21]. 3) A simple worst case O(n 2 ) quickhull algorithm, as in [25] and [26] All of these algorithms can be naturally parallelized. Using the algorithm of Overmars and van Leeuwen [23] the convex hull of presorted points can be computed in serial O(n) work, and the parallel extension is ....

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T. M.Y. Chan, J. Snoeyink, and C.-K. Yap. Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proceedings of the 6th Annual ACM--SIAM Symposium on Discrete Algorithms, pages 282--291, 1995.

....surface. Recent work of Erickson [16] shows that there are smooth surfaces with uniform sets of samples that have a Delaunay tetrahedrization of quadratic complexity. Therefore, even if the original COCONE algorithm uses an output sensitive algorithm for computing the Delaunay tetrahedrization [9], it could not match the running time of the new implementation. In [11] an implementation of the COCONE algorithm was described, which runs in time O(n log n) if the sampling is locally uniform . In this paper, we continue with that work and describe a new algorithm that has a correctness ....

T.M. Chan, J. Snoeyink, and C.-K. Yap. Output sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proc. 6th ACM-SIAM Sympos. Discr. Algorithms, 282-291, 1995. Journal version in Disc. Comput. Geom.

....surface. Recent work of Erickson [16] shows that there are smooth surfaces with uniform sets of samples that have a Delaunay tetrahedrization of quadratic complexity. Therefore, even if the original COCONE algorithm uses an output sensitive algorithm for computing the Delaunay tetrahedrization [9], it could not match the running time of the new implementation. In [11] an implementation of the COCONE algorithm was described, which runs in time O(n log n) if the sampling is locally uniform . In this paper, we continue with that work and describe a new algorithm that has a correctness ....

....can be embedded on a smooth surface; see Fig. 5. This example also shows that the number of candidate triangles can be quadratic in the worst case, so it is not feasible to obtain a better algorithm by computing the candidate triangles in an output sensitive manner, say similar to the method in [9] to compute a Delaunay tetrahedrization. Figure 5: An sampling with a quadratic number of candidate triangles (taken without permission from [16] The previous example is however unsatisfactory in that the sampling is highly non uniform. Erickson provides another construction in which the ....

T.M. Chan, J. Snoeyink, and C.-K. Yap. Output sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proc. 6th ACM-SIAM Sympos. Discr. Algorithms, 282-291, 1995. Journal version in Disc. Comput. Geom.

....techniques. Recent work of Erickson [8] shows that there are smooth surfaces with uniform sets of samples that have a Delaunay tetrahedrization of quadratic complexity. Therefore, even if the original Cocone algorithm uses an output sensitive algorithm for computing the Delaunay tetrahedrization [5], it could not match the running time of the new implementation. 2 CoCone Algorithm 2.1 Sampling Condition The medial axis of a surface S in R 3 is the closure of the set of points which have more than one closest point on S. The local feature size lfs(p) at a point p 2 S is the least ....

T.M. Chan, J. Snoeyink, and C.-K. Yap. Output sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proc. 6th Annu. ACM-SIAM Sympos. Discrete Algorithms, 282291, 1995. Journal version in Disc. Comput. Geom.

.... 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 ....

....computation tree models by Ben Or [3] It follows that both Graham s scan and Preparata and Hong s algorithm are optimal in the worst case. If the output size f is also taken into account, the lower bound drops to Omega n log f) 18] and a number of algorithms match this bound both in the plane [18, 5, 4] and in three dimensions [10,8] Chazelle [7] describes an algorithm for constructing convex hulls in IR d in time O(n bd=2c n log n) Since an n vertex polytope in IR d can have Omega (n bd=2c ) facets [31] Chazelle s algorithm is optimal in the worst case. Several faster algorithms ....

[Article contains additional citation context not shown here]

T. M. Y. Chan, J. Snoeyink, and C.-K. Yap. Outputsensitive construction of polytopes in four dimensions and clipped Voronoi digrams in three. In Proc. 6th ACM-SIAM Sympos. Discrete Algorithms, pages 282-- 291, 1995.

....[Sei86] which after an O(n 2 ) time preprocessing step , nds the facets of a convex hull in a shelling order at a logarithmic cost per facet. The preprocessing step was reduced later on to O n 2 Gamma 2 b d 2 c 1 ffl for any ffl 0, see [Mat93, MS92] Recently, T. Chan et al. CSY95] have investigated the case of points in four dimensions, achieving an O( n 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 ....

Timothy M.Y. Chan, Jack Snoeyink, and Chee-Keng Yap. OutputSensitive Construction of Polytopes in Four Dimensions and Clipped Voronoi Diagrams in Three. In Proc. 6th ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 282 291. 1995.

....The gift wrapping algorithm is a specialization of the convex hull gift wrapping algorithm (Chapter XXX) to Delaunay triangulations. Output sensitive algorithms, with running time approximately proportional to the actual number of Delaunay facets, have remained elusive. Recent progress is in [8]. If the sites form the vertices of a convex polygon, then the Voronoi diagram can be computed in linear time[1] Implementations Many of these algorithms have been implemented; the World Wide Web site http: www.geom.umn.edu locate cglib has pointers to publicly available code. 4 Extensions ....

T. M. Y. Chan, J. Snoeyink, C.K. Yap, Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three, Proc. Sixth. Ann. Symp. Disc. Alg., pp. 282--291, 1995.

....based approach. However, note that even in the heap based approach the running time can always be bounded by O(n log h) as P i n 0 i = n. 3 Planar Hulls After Kirkpatrick and Seidel s discovery of the output sensitive algorithm, there have been some recent simplification by Chan et al. [5] and Wenger[26] and Bhattacharya and Sen [3] The latter two are randomized and variations of a folklore method called quickhull. The definition of distribution that we will use for planar hulls is similar in spirit to the two dimensional maximal vector problem. We will confine ourselves to ....

....analysis identical to the modified version of out maximal points algorithm. However, we cannot rotate the axes in our recursive calls. This would pose problems in our framework as the distribution is not rotation invariant, to the contrary it can change drastically. The simplified algorithms in [5, 3, 26] are also divide and conquer algorithms but they avoid the complicated linear time linear programming of Step 3. The algorithm of [5] as well as the randomized version [3] detects an output vertex instead of an edge and simultaneously split the problem into roughly equal size subproblems (each ....

[Article contains additional citation context not shown here]

Timothy.M.Y. Chan, Jack Snoeyink, Chee-Keng Yap. Output-Sensitive Construction of Polytopes in Four Dimensions and Clipped Voronoi Diagrams in Three. Proc. 6th ACM-SIAM Sympos. Discrete Algorithms 1995, pp 282-291.

....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 ....

....tree models by Ben Or [16] It follows that both Graham s scan and Preparata and Hong s algorithm are optimal in the worst case. If the output size f is also taken into account, the lower bound drops to Omega (n log f) 101] and a number of algorithms match this bound both in the plane [101, 28, 29] and in three dimensions [50, 40] In higher dimensions, the problem is not quite so completely solved. Seidel s beneath beyond algorithm [132] constructs d dimensional convex hulls in time O(n dd=2e ) After a ten year wait, Chazelle [36] improved the running time to O(n bd=2c ) by ....

[Article contains additional citation context not shown here]

Timothy M. Chan, Jack Snoeyink, and Chee-Keng Yap. Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proc. 6th ACM-SIAM Sympos. Discrete Algorithms (SODA '95), pages 282--291, 1995.

....have show that it is possible to modify QuickHull so that it makes provably good partitions. Although the technique is shown for a sequential algorithm, it is easy Figure 13: Contrived set of points for worst case QuickHull. to parallelize. A simplification of the technique is give by Chan et al. [22]. This parallelizes even better and leads to an O(log 2 n) depth algorithm with O(n log h) work where h is the number of points on the convex hull. 6.2.2 MergeHull The MergeHull algorithm [57] is another divide and conquer algorithm for solving the planar convex hull problem. Unlike QuickHull, ....

Timothy M. Y. Chan, Jack Snoeyink, and Chee-Keng Yap. Output-sensitive construction of polytopes in four dimensions and clipped voronoi diagrams in three. In Proceedings of the 6th Annual ACM--SIAM Symposium on Discrete Algorithms, pages 282--291. ACM-SIAM, 1995.

....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 ....

....tree models by Ben Or [3] It follows that both Graham s scan and Preparata and Hong s algorithm are optimal in the worst case. If the output size f is also taken into account, the lower bound drops to Omega Gamma n log f ) and a number of algorithms match this bound both in the plane [17, 5, 4] and in three dimensions [10, 8] Chazelle [7] describes an algorithm for constructing convex hulls in IR d in time O(n bd=2c n log n) Since an n vertex polytope in IR d can have Omega Gamma n bd=2c ) facets [30] Chazelle s algorithm is optimal in the worst case. Several faster ....

[Article contains additional citation context not shown here]

T. M. Y. Chan, J. Snoeyink, and C.-K. Yap. Output-sensitive construction of polytopes in four dimensions and clipped Voronoi digrams in three. In Proc. 6th ACM-SIAM Sympos. Discrete Algorithms, pages 282--291, 1995.

....small size sampling algorithm. The analysis above shows that the work at level is bounded by O(nfi(n=n i Gamma1 ) Furthermore, the size of each subproblem decreases at each level by at least a constant factor, and pruning enforces that the computation tree has at most O(f) leaves. A lemma in [19, 8] implies that the total amount of work is O(nfi(f) log f) briefly, the work performed in the O(log f) stages until the subproblem size is O(n=f) is O(nfi(f) log f ) and the work performed in the remaining stages is O(n) 3.4.2 Filtering Filtering improves the previous result to O(n log f) for ....

T.M. Chan, J. Snoeyink, and C.-K. Yap. Output sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proc. 6th Annu. ACM-SIAM Sympos. Discrete Algorithms, 282-291, 1995.

....order [29] However, Blelloch et al. found experimentally that a simple quickhull [30] was faster than a more complicated convex hull algorithm that was guaranteed to take linear time. Furthermore, using a point pruning version of quickhull that limits possible imbalances between recursive calls [9] reduces its sensitivity to non uniform datasets. With these changes, the parallel Delaunay triangulation algorithm was found to perform about twice as many floating point operations as Dwyer s algorithm [16] Furthermore, the cumulative floating point operation count was found to increase ....

....algorithm, the serial code of quickhull implements the same algorithm as the parallel code. The basic quickhull algorithm tends to pick extreme pivot points when operating on non uniform point distributions, resulting in a poor division of data and a consequent lack of progress. Chan et al. [9] describe a variant that tests the slope between pairs of points and uses pruning to guarantee that recursive calls have at most 3 4 of the original points. However, pairing all n points and finding the median of their slopes is a significant addition to the basic cost of quickhull. ....

[Article contains additional citation context not shown here]

Timothy M. Y. Chan, Jack Snoeyink, and Chee-Keng Yap. Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 282--291, 1995.

....We briefly describe the application to Voronoi diagrams in Section 5. An extension of our 4 d algorithm to higher dimensions is given in Section 6. Note: A preliminary version of this paper, written for the problem of computing halfspace intersections rather than convex hulls, appears in [5]. The 4 d intersection algorithm given there and its 2 d specialization are dualizations of the convex hull algorithms in this paper. We feel that the present version, in the primal setting, is clearer and provides a better understanding of the method. 2 Preliminaries We first review some ....

T. M. Chan, J. Snoeyink, and C.-K. Yap. Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 282--291, 1995.

....Jarvis s march [19] can construct the convex hull in O(nh) time. This bound was later improved to O(n log h) by an algorithm due to Kirkpatrick 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 ....

T. M. Chan, J. Snoeyink, and C.-K. Yap. Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proc. 6th ACM-SIAM Sympos. on Discrete Algorithms, 282-- 291, 1995.

....in conference papers, and their full versions have been submitted for publication in journals. See [CSY95b] for the simplification of Kirkpatrick and Seidel s algorithm and its extension to four dimensions. A dual version of the 4 d algorithm in the halfspace intersection setting is described in [CSY95a]. Most of our higher dimensional results appear in [Cha95b] specialization to 2 d and 3 d can be found in [Cha95a] Chapter 2 Two and Three Dimensional Convex Hulls In this chapter, we present two O(n log h) time convex hull algorithms in the plane E 2 , the second of which is also extended ....

T. M. Chan, J. Snoeyink, and C.-K. Yap. Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 282--291, 1995.

....We briefly describe the application to Voronoi diagrams in Section 5. An extension of our 4 d algorithm to higher dimensions is given in Section 6. Note: A preliminary version of this paper, written for the problem of computing halfspace intersections rather than convex hulls, appears in [6]. The 4 d intersection algorithm given there and its 2 d specialization are dualizations of the convex hull algorithms in this paper. We feel that the present version, in the primal setting, is clearer and provides a better understanding of the method. 2 Preliminaries We first review some ....

T. M. Chan, J. Snoeyink, and C.-K. Yap. Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 282--291, 1995.

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T. M. Chan, J. Snoeyink, and C.-K. Yap. Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proc. 6th ACM-SIAM Sympos. Discrete Algorithms, pages 282--291, 1995.

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T. M. Chan, J. Snoeyink, and C.-K. Yap. Output sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proc. 6th ACM-SIAM Sympos. Discr. Algorithms (SODA 95), 282--291, 1995. Journal version in Disc. Comput. Geom.

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Timothy M. Y. Chan, Jack Snoeyink, and Chee-Keng Yap. Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. 145 In Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 282--291, 1995.

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T. M. Chan, J. Snoeyink, and C.-K. Yap. Output sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In Proc. 6th ACM-SIAM Sympos. Discr. Algorithms (SODA 95), 282--291, 1995. Journal version in Disc. Comput. Geom.

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T.M. Chan, J. Snoeyink and C. Yap. Output-Sensitive construction of polytopes in four dimensions and clipped voronoi diagrams in three. Proc. of the 6-th ACM-SIAM Symp. on Discrete Algorithms, pp.

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