T. M. Chan, J. Snoeyink, and C. K. Yap. Primal dividing and dual pruning: Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams. Discrete & Computational Geometry, 18:433--454, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....n) lower bound, every convex hull algorithm must require (n log n) time for some inputs. Despite these matching upper and lower bounds, and probably because of the many applications of convex hulls, a number of other planar convex hull algorithms have been published since Graham s algorithm [1, 2, 4, 6, 11, 17, 21, 28, 29, 36]. Of particular note is the Ultimate( algorithm of Kirkpatrick and Seidel [21] that computes the convex hull of a set of n points in the plane in O(n log h) time, where h is the number of vertices of the convex hull. Later, the same result was obtained by Chan using a much simpler algorithm ....

....We describe four space ecient planar convex hull algorithms. The rst is in place, uses Graham s scan in combination with an in place sorting algorithm, and runs in O(n log n) time. The second and third algorithms run in O(n log h) time, are in situ and are based on algorithms of Chan et al. [4] and Kirkpatrick and Seidel [21] respectively. The fourth ( More Ultimate ) algorithm is based on an algorithm of Chan [3] runs in O(n log h) time and is in place. The rst two algorithms are simple, implementable, and ecient in practice. To justify this claim, we have implemented both ....

[Article contains additional citation context not shown here]

T. Chan, J. Snoeyink, and C. K. Yap. Primal dividing and dual pruning: Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams. Discrete & Computational Geometry, 18:433-454, 1997.

....log n) lower bound, every convex hull algorithm must require (n log n) time for some inputs. Despite these matching upper and lower bounds, and probably because of the many applications of convex hulls, a number of other planar convex hull algorithms have been published since Graham s algorithm [1,2,4,6,11,17,27,28,21,35]. Of particular note is the Ultimate( algorithm of Kirkpatrick and Seidel [21] that computes the convex hull of a set of n points in the plane in O(n log h) time, where h is the number of vertices of the convex hull. The same authors show that, on algebraic decision trees of any xed order, ....

....be available at run time. We describe three planar convex hull algorithms. The rst is in place, uses Graham s scan in combination with an in place sorting algorithm, and runs in O(n log n) time. The second algorithm runs in O(n log h) time, is in situ and is based on an algorithm of Chan et al. [4]. The third ( More Ultimate ) algorithm is based on an algorithm of Chan [3] runs in O(n log h) time and is in place. The rst two algorithms are simple, implementable, and ecient in practice. To justify this claim, we have implemented both algorithms and made the source code freely available ....

[Article contains additional citation context not shown here]

T. Chan, J. Snoeyink, and C. K. Yap. Primal dividing and dual pruning: Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams. Discrete & Computational Geometry, 18:433-454, 1997.

....are referred to as output sensitive algorithms. Readers familiar with the literature on output sensitive convex hull algorithms may recognize the expression O(n log k) as the running time of optimal algorithms for computing convex hulls of n point sets with k extreme points, in 2 or 3 dimensions [2, 4, 5, 12, 18]. This is no coincidence. Given a set of n points in , we can color them all red and add three blue points at infinity (see Figure 2) In this set, the only points that contribute to the nearest neighbour decision boundary are the three blue points and the red points on the convex hull of the ....

.... where l is the number of points that contribute to the decision boundary in S 1 and where T (1, k) O(1) and T (n, 0) O(n) An easy inductive argument that uses the concavity of the logarithm shows that this recurrence is maximized when l = k 2, in which case the recurrence solves to O(n log k) [5]. Theorem 1. The nearest neighbour decision boundary of a set of n real numbers can be computed in O(n log k) time, where k is the number of elements that contribute to the decision boundary. 3 A 2 Dimensional Algorithm In the 2 dimensional nearest neighbour decision boundary problem the Vorono ....

T. M. Chan, J. Snoeyink, and C. K. Yap. Primal dividing and dual pruning: Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams. Discrete & Computational Geometry, 18:433--454, 1997.

....are referred to as output sensitive algorithms. Readers familiar with the literature on output sensitive convex hull algorithms may recognize the expression O(n log k) as the running time of optimal algorithms for computing convex hulls of n point sets with k extreme points, in 2 or 3 dimensions [2, 4, 5, 12, 18]. This is no coincidence. Given a set of n points in R , we can color them all red and add three blue points at infinity (see Figure 2) In this set, the only points that contribute to the nearest neighbour decision boundary are the three blue points and the red points on the convex hull of the ....

.... where l is the number of points that contribute to the decision boundary in S 1 and where T (1; k) O(1) and T (n; 0) O(n) An easy inductive argument that uses the concavity of the logarithm shows that this recurrence is maximized when l = k=2, in which case the recurrence solves to O(n log k) [5]. Theorem 1. The nearest neighbour decision boundary of a set of n real numbers can be computed in O(n log k) time, where k is the number of elements that contribute to the decision boundary. 3 A 2 Dimensional Algorithm In the 2 dimensional nearest neighbour decision boundary problem the Vorono ....

T. M. Chan, J. Snoeyink, and C. K. Yap. Primal dividing and dual pruning: Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams. Discrete & Computational Geometry, 18:433--454, 1997.

....algorithmic implications. For example, we can construct the Delaunay triangulation of a three dimensional point set in O( 3 log n) expected time using the standard randomized incremental algorithm [43] or in O( 3 log 2 n) time using the deterministic algorithm of Chan, Snoeyink, and Yap [16]. Using the history graph of the randomized incremental algorithm, which has expected size O( 3 log n) we can answer nearest neighbor queries in O(log 2 n) expected time. Since the Euclidean minimum spanning tree of a set of points is a subcomplex of the Delaunay triangulation, we can ....

T. M. Chan, J. Snoeyink, and C. K. Yap. Primal dividing and dual pruning: Output-sensitive construction of 4-d polytopes and 3-d Voronoi diagrams. Discrete Comput. Geom. 18:433-454, 1997.

....time sequentially and involves computing a g k th and h k th intersection point of G k and H k respectively on the line t. There are various cases, and we refer the reader to [16, pp 339 341] for details. We use the following optimal selection algorithm from Chaudhuri, Hagerup and Raman [6]. 11 Lemma 8 For all integers n 4, selection problems of size n can be solved in O(log n= log log n) CRCW time using n log log n= log n processors. Using this as a subroutine, we can compute the sign of a line t in O(log n= log log n) time. Consider the set of lines L = H [ G. Suppose l 1 ; l ....

....a simple algorithm for this problem using random sampling followed by verification. We follow the lines of our previous algorithms, with the modification that testing with respect to a line is done using approximate selection. We use the following algorithm for parallel approximate selection from [6]. Lemma 9 For all integers n 4 and t log log 4 n, approximate selection with relative accuracy 2 t= log log 4 n can be achieved in O(t) time, using an optimally in O(n) operations. Furthermore, for q 1, a relative accuracy of 2 q can be achieved in O(q log log 4 n) time, using ....

T. Chan, J. Snoeyink and C. Yap. Primal dividing and dual pruning: output-sensitive construction of 4-d polytopes and 3-d Voronoi diagrams. Discrete and Computational Geometry, 18, 1997, 433 -- 454.

....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 ....

....computation tree models by Ben Or [7] 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) 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 ....

[Article contains additional citation context not shown here]

T. M. Chan, J. Snoeyink, and C.-K. Yap, Primal dividing and dual pruning: Output-sensitive construction of 4-d polytopes and 3-d Voronoi diagrams, Disc. Comput. Geom., 18 (1997), pp. 433--454.

....to the gift wrapping method, runs in time O(n 2 f log(n) for facet enumeration and O(n 2 L log(n) for computing the facial lattice. There are a couple of divide and conquer algorithms that construct the convex hull in: ffl 4 dimensions in O( n f) log 2 f) time and O(n f) space ([9]) ffl 5 dimensions in O( n f) log 3 f) time ( 1] Erickson showed that, in the worst case, Omega Gamma n dd=2e Gamma1 n log n) time is needed to determine the number of convex hull facets ( 5] For odd dimensions, this matches Chazelle s upper bound. 12 7 Two dimensions revisited ....

....were developed that run in O(n log f) time. We mention here three of them and refer to the original papers for implementation and analysis details. Chan, Snoeyink and Yap proposed a deterministic variation of Quickhull which runs in time O(n log f) and can be generalized to higher dimensions ([9]) This algorithm also finds the median slope, and is faster than Kirckpatrick Seidel s algorithm by a constant factor. The median finding is the most costly operation in both algorithms. Let cn be the time to find the median of n numbers. Then Kirckpatrick and Seidel s algorithm spends, in the ....

M.C. Timothy, J. Snoeyink, and C. Yap. Primal dividing and dual pruning: Output-sensitive construction of 4-d polytopes and 3-d Voronoi diagrams. preliminary version of this paper in Proceedings 6th ACMSIAM SODA, pages 282-291, 1995.

....geometry, convex polytopes, degeneracy, lower bounds, decision trees, adversary arguments AMS subject classifications. 68Q25, 68U05, 52B55, 52B05 PII. S0097539797315410 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, 48]. Over 20 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 ## n log n) on the complexity of ....

....computation tree models by Ben Or [7] 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 ## 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 O(n #d 2# ) time. After a 10 year wait, Chazelle [15] improved the running time to O(n #d 2# ) by ....

[Article contains additional citation context not shown here]

<F3.746e+05> T. M. Chan, J. Snoeyink, and C.-K.<F3.852e+05> Yap,<F3.712e+05> Primal dividing and dual pruning: Output-sensitive construction of<F3.852e+05><F3.712e+05> 4-d polytopes and<F3.852e+05><F3.712e+05> 3-d Voronoi<F3.852e+05> diagrams, Discrete Comput. Geom., 18 (1997), pp. 433--454.

....to a subquadratic term which is n 4=3 (times a polylog factor) for d = 3; 4. On the other hand, the gift wrapping approach by Swart [31] has running time O(nf ) and has been improved by Chan [6] for d = 3; 4 the resulting time is O(n log f (nf) 4=3 log c n) Chan, Snoeyink and Yap [7, 8] made important progress by giving an algorithm that runs in time O( n f) log 2 f) for d = 3. Our contribution is to extend this to d = 4 with an additional log factor: O( n f) log 3 f ) Our new insight leads to an approach for arbitrary dimension, but unfortunately it runs into ....

....d = 3. Our contribution is to extend this to d = 4 with an additional log factor: O( n f) log 3 f ) Our new insight leads to an approach for arbitrary dimension, but unfortunately it runs into difficulties, so the results that are obtained in higher dimensions are not very interesting. As in [8], some marginal improvement is possible using trade offs between preprocessing and query times in closest point data structures [6] A negative feature of both Chan et al. s algorithm and ours is that nondegeneracy is needed for their analysis. To handle arbi trary input the algorithm is ....

[Article contains additional citation context not shown here]

T.M. Chan, J. Snoeyink, and C.-K. Yap. Primal dividing and dual pruning: Output-sensitive construction of 4-d polytopes and 3-d Voronoi diagrams. Submitted to Discrete Comput. Geom.

....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 ....

....tree models by Ben Or [7] 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) 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 ....

[Article contains additional citation context not shown here]

T. M. Chan, J. Snoeyink, and C.-K. Yap, Primal dividing and dual pruning: Output-sensitive construction of 4-d polytopes and 3-d Voronoi diagrams, Disc. Comput. Geom., 18 (1997), pp. 433--454.

....computational geometry. Chapter 5 summarizes our work and concludes with open problems and remarks on directions for further research. Remark : Most of the results of this thesis have been presented 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 ....

T. M. Chan, J. Snoeyink, and C.-K. Yap. Primal dividing and dual pruning: output-sensitive construction of 4-d polytopes and 3-d Voronoi diagrams. Submitted to Discrete & Computational Geometry.

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T. M. Chan, J. Snoeyink, and C. K. Yap. Primal dividing and dual pruning: Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams. Discrete & Computational Geometry, 18:433--454, 1997.

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T. Chan, J. Snoeyink, and C. K. Yap. Primal dividing and dual pruning: Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams. Discrete & Computational Geometry, 18:433-454, 1997.

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T. Chan, J. Snoeyink, and C. K. Yap. Primal dividing and dual pruning: Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams. Discrete & Computational Geometry, 18:433-454, 1997.

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Chan, T. M., Snoeyink, J., and Yap, C. K. Primal dividing and dual pruning: Output-sensitive construction of 4-d polytopes and 3-d Voronoi diagrams. Discrete Comput. Geom. 18 (1997), 433--454.

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T.M. Chan, J. Snoeyink and C.-K. Yap. Primal dividing and dual pruning: Output-sensitive construction of 4-d polytopes and 3-d Voronoi diagrams. Discrete Comput. Geom. 18 (1997), 433--454.

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T. M. Chan, J. Snoeyink, C. K. Yap, Primal dividing and dual pruning: Output-sensitive construction of 4-d polytopes and 3-d Voronoi diagrams, Discrete Comput. Geom. 18 (1997) 433--454. 10

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Chan, T. M., Snoeyink, J., and Yap, C. K. Primal dividing and dual pruning: Output-sensitive construction of 4-d polytopes and 3-d Voronoi diagrams. Discrete Comput. Geom. 18 (1997), 433--454.

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T. M. Chan, J. Snoeyink, and C. K. Yap. Primal dividing and dual pruning: Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams. Discrete & Computational Geometry, 18:433--454, 1997.

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T. M. Chan, J. Snoeyink, and C. K. Yap. Primal dividing and dual pruning: Output-sensitive construction of 4-d polytopes and 3-d Voronoi diagrams. Discrete Comput. Geom., 18:433-454, 1997.

No context found.

T. M. Chan, J. Snoeyink, and C. K. Yap. Primal dividing and dual pruning: Outputsensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams. Discrete & Computational Geometry, 18:433--454, 1997.

No context found.

T. Chan, J. Snoeyink, C. K. Yap, Primal dividing and dual pruning: Outputsensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams, Discrete & Computational Geometry 18 (1997) 433--454.

No context found.

T. M. Chan, J. Snoeyink, and C. K. Yap. Primal dividing and dual pruning: Outputsensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams. Discrete & Computational Geometry, 18:433--454, 1997.

No context found.

Chan, T. M., Snoeyink, J., and Yap, C. K. Primal dividing and dual pruning: Output-sensitive construction of 4-d polytopes and 3-d Voronoi diagrams. Discrete Comput. Geom. 18 (1997), 433--454.

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