B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Manuscript, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... marriage before conquest principle: it computes the merge of two subproblems before it recursively solves the subproblems. Edelsbrunner and Shi [17] applied marriage before conquest in 3 d to obtain an O(n log f) time convex hull algorithm. Using a different approach, Chazelle and Matousek [9] have reported that derandomizing an algorithm of Clarkson and Shor [10] gives an O(n log f) time algorithm in 3 d. Recently, Chan [4] has obtained a simple O(n log f) time method for both 2 d and 3 d convex hulls. In dimensions 3, the fastest output sensitive algorithms currently known ....

....4. We first consider the case in which Delta is a halfspace. For d = 4, the projected point set Delta (P ) is 3 dimensional. We can compute the facets of the upper hull F ( Delta (P ) and thus, the boundary B Delta , in O(jP j log jV (P )j) time by either Chazelle and Matousek s algorithm [9] or Chan s algorithm [4] This permits computations involving the region S Delta to be done efficiently, such as deciding if a point lies in the interior of S Delta . Lemma 4.4 Suppose that d = 4. Then the restricted point set P j int S Delta O(jP j log jV (P )j) time using O(jP j) space. ....

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B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Manuscript, 1993.

....here the remark that these estimates assume that the polytope P is given together with its convex hull , by which we mean that all the adjacency relations between vertices, edges and facets (2 dimensional faces ) of P are given. Computation of the convex hull of P requires O(n log n) time (see [11, 3, 2], for example) The algorithms operate by repeatedly removing a carefully chosen vertex from the current polytope. Thus, the basic step selects a vertex v from the current polytope P i , and replaces P i by conv(vert(P i )nfvg) the convex hull of all vertices of P i except v. When repeated n k ....

B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Computational Geometry: Theory and Applications 5 (1995), 27{ 32.

....[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 derandomizing a randomized incremental ....

B. Chazelle and J. Matou sek, Derandomizing an output-sensitive convex hull algorithm in three dimensions, Comput. Geom. Theory Appl., 5 (1995), pp. 27--32.

....[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 are known when the output size ....

B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Technical report, Dept. Comput. Sci., Princeton Univ., 1992.

....algorithm for computing the convex hull of a set of points in dimension 3. The expected complexity of their algorithm is optimal. Their algorithm uses as a basic primitive the deterministic algorithm of D.G. Kirkpatrick and R. Seidel and was derandomized later on by B. Chazelle and J. Matou#ek [CM92] In higher dimensions (d 4) for a long time the best known solution was the algorithm of R. Seidel [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 ....

B. Chazelle and J. Matou#ek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Technical report, Dept. Comput. Sci., Princeton Univ., 1992.

.... and this gives also a fast parallel algorithm for approximations (Goodrich [Goo93] The observation that partition trees can produce approximations in geometric situations quickly is also in [Mat92a] together with some applications; more applications can be found in Chazelle and Matousek [CM94] Matousek and Schwarzkopf [MS96] and Amato et al. AGR94] AGR95] The linearization produced by assigning a new coordinate to each monomial is well known in algebraic geometry (the so called Veronese map, see e.g. Har92] It has been used by Yao and Yao [YY85] to show that various ....

.... formulate their results in a dynamic form, as a time bound for a randomized incremental algorithm (see also De Berg et al. dBDS95] part (i) appears in Agarwal et al. AMS94] The idea of sampling from an approximation is implicit in [Mat91a] and it has been used in several other papers ( CM94] MS96] AGR94] AGR95] The notion of a (1=r) seminet was introduced by Amato et al. AGR94] inspired by a paper of Chazelle [Cha93b] As was observed by Ramos [Ram96] the problem with computing a (1=r) seminet in polynomial time under Axiom 2 (without locality ) can be often ....

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B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Comput. Geom.: Theor. Appl., 5:27--32, 1994.

....[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 derandomizing a randomized incremental ....

<F3.746e+05> B. Chazelle and J. Matou<F3.852e+05> sek,<F3.712e+05> Derandomizing an output-sensitive convex hull algorithm in three<F3.852e+05> dimensions, Comput. Geom., 5 (1995), pp. 27--32.

....for many geometric problems, additional parameters like the size of the output capture the complexity of the problem more accurately enabling us to design superior algorithms. Algorithms whose running times are sensitive to the output size have been actively pursued for problems like convex hulls [CS89, CM92, Cha95, Sei86, Mat92]. 1.1 Parallel output size sensitive algorithms The primary objective of designing parallel algorithms is to obtain very fast running time while keeping the total work (the processor time product) close to the best sequential algorithms. Problems for which output size sensitive algorithms are ....

....the first O(n log n) time algorithm. Clarkson and Shor [CS89] presented the first (randomized) optimal output sensitive algorithm which ran in O(n log h) expected time, where h is the number of hull vertices. Their algorithm was subsequently derandomized optimally by Chazelle and Matousek [CM92]. Chan [Cha95] presented a very elegant approach for output sensitive construction of convex hulls using ray shooting that achieve optimal Theta(n log h) running times for dimensions two and three. In higher dimensions, the quest is still on to design optimal output sensitive algorithms. ....

B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex-hull algorithm in three dimensions. Tech. Rept., Princeton University, 1992.

....output sensitive convex hull algorithms are known. Kirkpatrick and Seidel [KS86] give an optimal O(n log h) planar convex hull algorithm. Edelsbrunner and Shi [ES91] established that O(n log 2 h) time is sufficient to find the convex hull in 3 . Subsequently, Chazelle and Matousek [CM92] have given an algorithm for the three dimensional case that achieves optimal O(n log h) time; their algorithm is obtained by derandomizing the optimal O(n log h) algorithm of Clarkson and Shor [CS89] The algorithms of Kirkpatrick and Seidel and Edelsbrunner and Shi can be implemented in ....

B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Technical report, Dept. Comput. Sci., Princeton Univ., 1992.

.... 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, with complexity measured as a function of both n and the output size h, are said to be output sensitive. In this note, we point out a simple output sensitive convex hull algorithm in E 2 and its extension in E 3 , ....

B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Comput. Geom. Theory Appl., 5:27--32, 1995.

....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 derandomizing a randomized incremental ....

B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Technical report, Dept. Comput. Sci., Princeton Univ., 1992.

....Afterwards, Edelsbrunner and Shi [ES91] Chapter 1. Introduction 10 proposed a deterministic algorithm with an O(n log 2 h) running time, by following the paradigm of Kirkpatrick and Seidel. An optimal O(n log h) time deterministic algorithm was finally obtained when Chazelle and Matousek [CM95] applied recently developed derandomization techniques to Clarkson and Shor s convex hull algorithm. A different randomized O(n log h) algorithm was recently reported by Clarkson [Cla94] and, according to his paper, can also be derandomized. The resulting algorithm is not very practical, since ....

....extension in three dimensions. In the worst case, this 2 d algorithm is faster than the method from the previous section since this algorithm does not perform median finding operations. The extension in 3 d is also simpler than the previous optimal derandomization method of Chazelle and Matousek [CM95]. Our idea here is to improve Jarvis s march and the gift wrapping method by applying a common grouping trick. This grouping idea can be applied to other problems besides the construction of convex hulls, and we will consider one example later in Section 2.3. Some variants of the method for convex ....

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B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Computational Geometry: Theory and Applications, 5:27--32, 1995. Bibliography 93

....and compute the convex hull. Via the duality transform, each facet of the convex hull defines a point of intersection of the halfspaces. Recent work on convex hulls and Delaunay triangulations has focused on variations of a randomized, incremental algorithm that has optimal expected performance [12] [15] 21] 28] 30] 37] Points are processed one at a time in a random order. In this paper, we propose and analyze a strategy for processing points in a more efficient order. The result is a faster algorithm for distributions with interior points. An incremental algorithm for the convex hull ....

.... R 2 , Kirkpatrick and Seidel found an optimal output sensitive algorithm for convex hull that runs in O(n log h) time, where h is the output size [32] Clarkson Shor give a 3 d convex 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 ....

B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Computational Geometry: Theory and Applications, 1991.

....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 algorithms are known when the output ....

B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Technical report, Dept. Comput. Sci., Princeton Univ., 1992.

.... marriage before conquest principle: it computes the merge of two subproblems before it recursively solves the subproblems. Edelsbrunner and Shi [18] applied marriage before conquest in 3 d to obtain an O(n log 2 f) time convex hull algorithm. Using a different approach, Chazelle and Matousek [10] have reported that derandomizing an algorithm of Clarkson and Shor [11] gives an O(n log f) time algorithm in 3 d. Recently, Chan [5] has obtained a simple O(n log f) time method for both 2 d and 3 d convex hulls. In dimensions higher than three, the fastest output sensitive algorithms currently ....

....4. We first consider the case in which Delta is a halfspace. For d = 4, the projected point set Delta (P ) is 3 dimensional. We can compute the facets of the upper hull F ( Delta (P ) and thus, the boundary B Delta , in O(jP j log jV (P )j) time by either Chazelle and Matousek s algorithm [10] or Chan s algorithm [5] This permits computations involving the region S Delta to be done efficiently, such as deciding if a point lies in the interior of S Delta . Lemma 4.4 Suppose that d = 4. Then the restricted point set P j int S Delta can be computed in O(jP j log jV (P )j) time using ....

[Article contains additional citation context not shown here]

B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Computational Geometry: Theory and Applications, 5:27--32, 1995.

....f ) Adding a filtering stage improves the performance to O(n log f) for f = O(n ffl ) for some ffl 0 and provides fast parallel algorithms. Pruning and filtering have been used widely before to achieve output sensitive algorithms, beginning with the work of Clarkson and Shor [14] for example [44, 11, 3, 28]. Also [31] is relevant. 3.4.1 Pruning The algorithm does not need to recurse on a cell oe of the cutting T i that does not contain a vertex of L Gamma F . This can be detected by computing LF (a) and LF (b) where a and b are the x coordinates of the left and right sides of oe. These are ....

B. Chazelle and J. Matousek. Derandomizing an output sensitive convex hull algorithm in three dimensions. Technical Report, Dept. of Computer Science, Princeton University, 1992.

....with n exponential probability (e.g. see [23] Of course, to apply the above lemma to our convex hull algorithm, we must also know the value of W , which seems to require that we know the value of h. We can get around this seeming circular argument, however, by a well known trick (e.g. see [6]) In apply this trick to our method we begin by setting h 0 = 2 and we then run our algorithm to construct the convex hull in O(log n) time and W = O(n log h 0 ) work, with n exponential probability, except that we stop processing should the method of Lemma 4.2 determine that the total work ....

B. Chazelle and J. Matousek, "Derandomizing an output-sensitive convex hull algorithm in three dimensions," Technical report, Dept. Comput. Sci., Princeton Univ., 1992.

....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) 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 Gamma n bd=2c ) facets [31] Chazelle s algorithm is optimal in the worst case. Several faster algorithms are known when the output ....

B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Technical report, Dept. Comput. Sci., Princeton Univ., 1992.

....depends only on the input size n and is insensitive to the output size f . An optimal O(n log f) time output sensitive algorithm in two dimensions was given by Kirkpatrick and Seidel [23] For dimension 3, Edelsbrunner and Shi [17] obtained an O(n log 2 f) time method, and Chazelle and Matousek [11] demonstrated that optimal O(n log f) time is possible by derandomizing an earlier algorithm due to Clarkson and Shor [13] In any fixed dimension, the gift wrapping algorithm of Swart [44] and the beneath beyond algorithm of Seidel [41] achieve O(nf) and O(n 2 f log n) time respectively. ....

B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Comput. Geom. Theory Appl., 5:27--32, 1995.

....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 derandomizing a randomized incremental ....

B. Chazelle and J. Matou sek, Derandomizing an output-sensitive convex hull algorithm in three dimensions, Comput. Geom. Theory Appl., 5 (1995), pp. 27--32.

....d 3, and frequently does for many natural distributions. The algorithm is somewhat simpler than one previously given for d = 3 with the same complexity. 4] Like the earlier one, this one also can be derandomized for d 3 to obtain a deterministic algorithm with the same asymptotic complexity. [3]) This result is perhaps best interpreted as a rough theoretical justification for algorithms like Quickhull[5, 10] which find extreme points on their way to computing the hull. As further evidence of the interest of this approach, x5 gives an output sensitive algorithm for the problem of ....

B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Technical report, Dept. Comput. Sci., Princeton Univ., 1992.

....The first output sensitive algorithm, due to Kirkpatrick and Seidel [33] computed the convex hull in IR 2 in O(n log h) time. Clarkson and Shor [15] gave an optimal randomized output sensitive solution for 3 dimensional convex hulls, which was optimally derandomized by Chazelle and Matousek [13]. In higher dimensions, the only deterministic output sensitive method known, due to Seidel [55] runs in time O(n 2 h log n) which can be slightly improved to O(n 2 Gamma(2= bd=2c 1) ffl h log n) for any fixed ffl 0, using a technique of Matousek [42] All of these methods for d 3 ....

.... parallel algorithm using optimal O(n logh) work, where h = jH j, but increased running time O(log 3 n) by applying the technique used in the sequential randomized output sensitive method of Clarkson and Shor [15] which was also used in its derandomized version by Chazelle and Matousek [13]. Suppose that we know the value of h = jH j, and that h n ffl , for some ffl 0 (otherwise the O(n log n) work method suffices) By Theorem 2.10(3) we obtain a 0shallow (1=h) cutting of size O(h) Everything proceeds as before except the contours are computed using a parallel version ....

B. Chazelle and J. Matousek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Technical report, Dept. Comput. Sci., Princeton Univ., 1992.

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