N. M. Amato, M. T. Goodrich and E. A. Ramos. Parallel algorithms for higher-dimensional convex hulls. In Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS 94), 683--694, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that the expected running time of the overall algorithm is O(n log n) It was a challenging open problem whether an O(n log n) time deterministic algorithm can be developed for computing the intersection of n congruent balls in 3 space. This has been answered in the affirmative by Amato et al. [34], following a series of near linear time but weaker deterministic algorithms [60, 206, 237] Amato et al. derandomized the Clarkson Shor algorithm, using several sophisticated techniques. Their algorithm yields an n) time algorithm for computing the diameter. Recently Ramos [236] obtained ....

N. M. Amato, M. T. Goodrich, and E. A. Ramos, Parallel algorithms for higher-dimensional convex hulls, Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci., 1994, pp. 683--694.

....example, in 2D we can use the divide and conquer parallel algorithm devel oped by Blelloch et al. 4] for Delaunay triangulation. Their algorithm uses O(nlog n) work and O(log 3 n) paallel time. We can alternatively use the random ized parallel algorithms of Reif and Sen [28] or by Amaro et al. [1], in both two and three dimensions. Both of these randomized parallel Delaunay triangulation algorithms have expected parallel running time O(log n) Using one of these adds a logarithmic factor to our worst case total parallel time complexity analysis. To the best of our knowledge, these are the ....

N.M. Amaro, M. T. Goodrich, and E. A. Ramos. Parallel algorithms for higher-dimensional convex hulls. Proc. 35th IEEE Syrup. Found. of Comp. Si., 683-694, 1994.

....In fact, the threedimensional version of our algorithm simplifies Edelsbrunner and Shi s algorithm in the same way as our two dimensional algorithm simplifies Kirkpatrick and Seidel s. The contour based approach used in a recent parallel 3 d convex hull algorithm by Amato, Goodrich, and Ramos [1] can also be interpreted as a form of primal dividing. There, contours play a role similar to the lower dimensional upper hulls of Delta (P ) in Section 4.3 and are used to ensure that the total problem size at any level of the recursion remains O(n) but since they describe their method in the ....

N. M. Amato, M. T. Goodrich, and E. A. Ramos. Parallel algorithms for higher-dimensional convex hulls. In Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science, pages 683--694, 1994.

....less ecient in practice than the brute force algorithm for almost any data set. Moreover this algorithm is not ecient in higher dimensions since the intersection of n balls of the same radius has size (n ) Recent attempts to solve the 3 dimensional diameter problem led to O(n log n) [AGR94,Ram97b] and O(n log n) deterministic algorithms [Ram97a,Bes98] Finally Ramos found an optimal O(n log n) deterministic algorithm [Ram00] All these algorithms use complex data structures and algorithmic techniques such as 3 dimensional convex hulls, intersection of balls, furthest point Voronoi ....

N. M. Amato, M. T. Goodrich, and E. A. Ramos. Parallel algorithms for higher-dimensional convex hulls. In Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci., pages 683-694, 1994.

....all other processors they can stop their search. Signi cant speedups were reported. Although this is the only work we are aware of speci cally relating to motion planning, 29 there have been some parallel methods proposed for collision detection [54, 53] and related geometric problems (see, e.g. [1, 6]) These results illustrate that signi cant reductions in planning time can be obtained with parallel methods. We note, however, that none of the above mentioned motion planning methods is truly parallel. The former [52] does not parallelize the search itself, while the latter [22] cannot be ....

N. M. Amato, M. T. Goodrich, and E. A. Ramos, \Parallel algorithms for higher{dimensional convex hulls," in Proc. IEEE Symp. Foundations of Computer Science (FOCS), pp. 683-694, 1994.

....that the expected running time of the overall algorithm is O(n log n) It was a challenging open problem whether an O(n log n) time deterministic algorithm can be developed for computing the intersection of n congruent balls in 3 space. This has been answered in the affirmative by Amato et al. [31], following a series of near linear time but weaker deterministic algorithms [51, 177, 202] Amato et al. derandomized the Clarkson Shor algorithm, using several sophisticated techniques. 3 Their algorithm yields an O(n log 3 n) time algorithm for computing the diameter. Obtaining an optimal ....

N. M. Amato, M. T. Goodrich, and E. A. Ramos, Parallel algorithms for higher-dimensional convex hulls, Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci., 1994, pp. 683--694.

....illustrate that the algorithm effectively scales as n log n. It is once again worth asking whether the algorithms in these papers could be implemented efficiently in parallel. The difficult part is the parallel computation of the power diagram. Using algorithms due to Amato, Goodrich, and Ramos [AGR94], one can construct a power diagram for n 3 dimensional balls in parallel in O(log n) time in the EREW PRAM model. Once K has been derived, evaluating the inclusion exclusion formula takes constant time for each simplex in K and can be done in parallel. 4 An O(n) Algorithm for Volume and Surface ....

Nancy Amato, Michael Goodrich and Edgar Ramos. Parallel algorithms for higherdimensional convex hulls. In Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pages 683--693, 1994.

....solution of (P1) can be found in times (respectively) O(n# 2 log # 1 ) O(n 3 # 1 log # 1 ) O(n 4 log # 1 ) III) We highlight a linearization technique for geometric optimization problems. The above result uses Megiddo s parametric search and a new parallel convex hull algorithm in [AGR]. But it also requires an application of the linearization technique, which we believe has wider applicability. The heart of both approximation and parametric search algorithms is a decision scheme for a fixed optimization parameter. To obtain e#cient decision algorithms, it is often possible to ....

....) the time (resp. number of processors) of a parallel decision algorithm, then the optimal value (here, r # ) can be computed in sequential time O(PT p T s T p log P ) It remains to give a parallel version of the decision algorithm. Here we exploit the new parallel algorithm for convex hulls of [AGR]. For dimension d # 4, there is an algorithm with time O(log n) and work O(n #d 2# log c(#d 2# #d 2#) n) for some constant c 0. Further, with O(n #d 2# ) processors, the test for intersection of H with M can be done in constant time in an algebraic model (resp. a real RAM, see [Re] ....

N. Amato, M. Goodrich, E. Ramos, "Parallel algorithms for higher-dimensional convex hulls", IEEE FOCS, 1994, pp. 683--694.

....gave the first optimal deterministic convex hull algorithm. This algorithm is not output sensitive and thus is optimal only in the worst case sense, i.e. it runs in O(n log n n bd=2c ) which is the worst case number of faces possible in d 3 dimensions. Recently, Amato, Goodrich and Ramos [1] gave an optimal randomized parallel algorithm for higher dimensions. They provide a 3 d output sensitive algorithm, but their algorithm is not output sensitive for d 3. There are no output sensitive parallel algorithms for d 3, all the sequential methods known seem to be hard to parallelize, ....

N.M. Amato, M.T. Goodrich, and E.A. Ramos. Parallel algorithms for higher-dimensional convex hull. In 35th Annual Symposium on Foundations of Computer Science, 683--694, 1994. IEEE.

....(x, y) ## (x, y, x 2 y 2 ) maps these regions to halfspaces in R 3 ; # is feasible if the intersection of all these halfspaces meets the paraboloid z = x 2 y 2 . The result follows by applying parametric search [29] to a parallel algorithm that constructs the intersection [2, 24] and tests whether any of its features crosses the paraboloid. # We can of course combine the maximum angle with the many other criteria, including circumradius, for which the feasible regions are bounded by lines and circles. An alternate approach suggests itself, which may have a better chance ....

N. Amato, M. T. Goodrich, and E. A. Ramos. Parallel algorithms for higher dimensional convex hulls. 35th IEEE Symp. Foundations of Comp. Sci., 1994, pp. 683--694; http://www.cs.tamu.edu/research/robotics/ Amato/Papers/focs94.300.ps.gz. 10

....of the halfspaces (b) in R d 1 ) for each ball b 2 B(L 0 ) Under the standard primal dual transformation, this intersection is transformed into the convex hull (in R d 1 ) of the points dual to hyperplanes bounding these halfspaces. It is computed using the algorithm of Amato et al. [2] in O(log n) parallel steps, using O(n dd=2e log c n) processors, for some constant c 0 (these bounds refer to d space for d 4) Applying this procedure in parallel to each L 0 L, we can generate S in O(log n) parallel steps using O(n d3d=2e log c n) processors. Turning to ....

N. M. Amato, M. T. Goodrich and E. R. Ramos, Parallel Algorithms for HigherDimensional Convex Hulls, Proceedings 6th Annual ACM Symposium on Theory of Computing, 1994, 683--694.

.... (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 geometrically defined set systems can be embedded into the set ....

....has been used by Yao and Yao [YY85] to show that various geometrically defined set systems can be embedded into the set systems defined by simplices on point sets in some IR k ; see also [AM94] for more information. Linearization has been used for derandomization in an essential way in [MS96] AGR94] and [AGR95] 4.4 Higher moment bounds and seminets First we give three examples for a future reference. Example A. Let X be a set of lines in the plane. For any subset S X, define T (S) as the set of all trapezoids in the vertical decomposition of the arrangement 3 of S. Example B. Let X ....

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N. M. Amato, M. T. Goodrich, and E. A. Ramos. Parallel algorithms for higherdimensional convex hulls. In Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci., pages 683--694, 1994.

....our randomized technique will guide us to the right subproblem quickly, with the aid of the decision algorithm. 1. 2 Previous Approaches One of the most general approaches for reducing geometric optimization problems to their decision problems is parametric search, invented by Megiddo [54] see [1, 4, 7, 12, 15, 17, 26, 34, 53, 59, 61, 64] for just a partial list of examples) The basic idea is to simulate the decision algorithm compare the optimum with t with the parameter t being the unknown optimum itself. In most instances, the branching points of the simulation require testing the signs of low degree polynomials in t, ....

....A better approach in IR 3 is to first solve the decision problem, which involves the construction of an intersection of congruent balls. This intersection has O(n) size and can be computed in O(n log n) time by a randomized incremental method of Clarkson and Shor [24] or its derandomization [7, 12]. Like the Euclidean diameter problem, the discrete 1 center problem in IR 3 can then be solved by parametric search in O(npolylog n) time [7, 12, 17, 53, 59] We now show how parametric search can be replaced by a randomized search, solving the threedimensional discrete 1 center problem in O(n ....

[Article contains additional citation context not shown here]

N. M. Amato, M. T. Goodrich, and E. Ramos. Parallel algorithms for higher-dimensional convex hulls. In Proc. 35th IEEE Sympos. Found. Comput. Sci., pages 683--694, 1994.

....n) time algorithm based on Chazelle s [19] optimal sequential algorithm for triangulation and efficient construction of planar separators. The optimal algorithm for three dimensional convex hulls appears in [76] the version presented here uses a simpler Filtering scheme used in Amato et al. [7]. The algorithm for two dimensional arrangements is based on Hagerup et al. s [49] optimal speed up algorithms for constructing arrangements of hyperplanes in d dimensions. Ramaswamy and Rajasekaran [73] describe optimal speed up algorithms for constructing Voronoi diagrams of line segments using ....

....bits suffice for the algorithms described in this chapter. Bounding the number of random bits has important ramifications in derandomization. Recently there have been some improved results obtained by efficient derandomization of randomized parallel algorithms Goodrich [41] and Amato et al. [7]. Among other impressive results they also obtain a deterministic work optimal three dimensional convex hull algorithm based on the algorithm of Reif and Sen. An important problem that was not discussed in this chapter is the fixed dimensional linear programming. Randomization has been very ....

N.M. Amato, M.T. Goodrich and E.A. Ramos. Parallel Algorithms for Higher-Dimensional Convex Hulls. Proc. of the 35th Annual FOCS, pages 683-- 694, 1994.

....be constructed in O(log log n Delta log h) time with optimal work with high probability. In three dimensions, Goodrich and Ghouse [GG91] described an O(log 2 n) expected time, O(minfn log 2 h; n log ng) work method, which is output sensitive but not work optimal. More recently, Amato et.al([AGR94]) gave a deterministic O(log 3 n) time, O(n log h) work algorithm for convex hulls in R 3 on the EREW PRAM. In higher dimensions, Amato et al. AGR94] have shown that the convex hull of n points in R d can be constructed in O(log n) time with O(n log n n bd=2c ) work with high ....

....2 n) expected time, O(minfn log 2 h; n log ng) work method, which is output sensitive but not work optimal. More recently, Amato et.al( AGR94] gave a deterministic O(log 3 n) time, O(n log h) work algorithm for convex hulls in R 3 on the EREW PRAM. In higher dimensions, Amato et al. [AGR94] have shown that the convex hull of n points in R d can be constructed in O(log n) time with O(n log n n bd=2c ) work with high probability and in O(log n) time with O(n bd=2c log c(dd=2e Gammabd=2c) n) work deterministically, where c 0 is a constant. In this paper, we present a ....

N.M. Amato, M.T. Goodrich, and E.A. Ramos. Parallel algorithms for higher-dimensional convex hulls. Proc. of the 35th Annual FOCS, pages 683-- 694, 1994.

No context found.

N. M. Amato, M. T. Goodrich and E. A. Ramos. Parallel algorithms for higher-dimensional convex hulls. In Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS 94), 683--694, 1994.

.... in contrast to O(log 2 n) for the rst algorithm in this paper and the previous one [8] The approach of divide and conquer with partial clean up also simpli es other previous algorithms (3 d convex hulls, 2 d abstract Voronoi diagrams, 3 d diameter, single face in an arrangement of segments [7, 8, 10]) and also leads to the same time speed up for the corresponding parallel algorithms of some of them (3 d convex hulls, 2 d abstract Voronoi diagrams) We present here the algorithm for computing a single face in an arrangement of curve segments and leave the other applications for a ....

....often leads to nonoptimal algorithms, at least as far as the most basic analysis can tell. In fact, the literature is plagued with running times that are a factor n or log c n away from optimal. Some techniques have been used to correct this and obtain optimal algorithms: sparse cuttings [18, 7, 8], pruning [61, 7, 8] and biased sampling [51, 22, 59, 7] In particular, the algorithm in [8] achieves optimality through the use of a complicated pruning step which is what introduces the limitation of handling only line segments. New Results. In this paper, we present a simpli ed version of the ....

[Article contains additional citation context not shown here]

N. M. Amato, M. T. Goodrich, and E. A. Ramos. Parallel algorithms for higher-dimensional convex hulls. In Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS 94), 683-694, 1994.

....can be computed in polynomial time. Through the use of a (1=r) approximation, the time can be reduced to O(nr C ) and for linearizable configuration spaces, for r n ffl , to O(n log r) In parallel, k wise independence can only guarantee part 2 of the theorem for j = O(k) and not part 1) [3, 4]. Modeling the sampling with leveled DFAs, and fooling them with relative error, we can construct in parallel a sample as guaranteed by the sampling theorem, except for a constant multiplicative factor. Relative error is needed because of the exponential weighting that makes even small probability ....

N. M. Amato, M. T. Goodrich, and E. A. Ramos. Parallel algorithms for higher-dimensional convex hulls. In Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci., 1994, 683--694.

....often leads to nonoptimal algorithms, at least as far as the most basic analysis can tell. In fact, the literature is plagued with running times that are a factor n or log c n away from optimal. Some techniques have been used to correct this and obtain optimal algorithms: sparse cuttings [19, 7, 8], pruning [65, 7, 8] and biased sampling [54, 24, 63, 7] In particular, the algorithm in [8] achieves optimality through the use of a complicated pruning step which is what introduces the limitation of handling only line segments. New Results. In this paper, we present a simpli ed version of the ....

....algorithms, at least as far as the most basic analysis can tell. In fact, the literature is plagued with running times that are a factor n or log c n away from optimal. Some techniques have been used to correct this and obtain optimal algorithms: sparse cuttings [19, 7, 8] pruning [65, 7, 8] and biased sampling [54, 24, 63, 7] In particular, the algorithm in [8] achieves optimality through the use of a complicated pruning step which is what introduces the limitation of handling only line segments. New Results. In this paper, we present a simpli ed version of the algorithm in [8] ....

[Article contains additional citation context not shown here]

N. M. Amato, M. T. Goodrich, and E. A. Ramos. Parallel algorithms for higher-dimensional convex hulls. In Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS 94), 683-694, 1994.

....has size at most C=ffl. Although a semi splitter may fail to have all basic intervals of size at most ffln, still there is some control on these sizes in the form of the moment bounds. As a result, a semi splitter can be refined into a good splitter using a weaker splitter algorithm as follows [14, 15, 32, 7, 19]. 8 Assume that on input (X; ffl) algorithm A computes an (a; C; ffl) semi splitter for X , and that on input (X; ffl) algorithm B computes an ffl splitter of size D(1=ffl) c (i.e. too large) for X with c a. A better ffl splitter is computed as follows: First, compute an (a; C; ....

....t wise independent if for any indices j 1 ; j t , and 0 1 values c 1 ; c t : PrfI j1 = c 1 ; I j t = c t g = Q t i=1 PrfI j i = c i g. The following fact is essential for our algorithms [28, 29] 8 This construction has been extensively used in computational geometry [14, 15, 32, 7]. The idea of two levels of sampling also appears in the context of hashing [19] and possibly elsewhere) 9 In our algorithm, there is no advantage in using other more efficient constructions of probability spaces, like spaces with almost k wise independence [35, 5] Fact 3 For any s, 1 s ....

N.M. Amato, M.T. Goodrich, and E.A. Ramos. Parallel algorithms for higherdimensional convex hulls. FOCS'94, 683--694.

....can be reduced to O(nr c ) For a so called linearizable configuration space, for which the range space (X; fK(oe) oe 2 C(X)g) is linearizable in IR l for some l, there is an ffl = ffl(l) 0 ffl 1, so that for r n ffl , a good sample can be computed in O(n log r) time. In parallel [5, 6], k wise independence can guarantee (ii) of the theorem for j = O(k) but not (i) a weaker version, jK(oe)j C 0 dn r r ffi , follows from (ii) Modelling the sampling with levelled RFAs, and fooling them with relative error, we can construct in parallel a sample as guaranteed by the ....

N. M. Amato, M. T. Goodrich, and E. A. Ramos. Parallel algorithms for higher-dimensional convex hulls. In Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci., 1994, 683--694.

No context found.

N. M. Amato, M. T. Goodrich, and E. A. Ramos. Parallel algorithms for higherdimensional convex hulls. In Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci., pages 683694, 1994.

No context found.

N. M. Amato, M. T. Goodrich and E. R. Ramos, Parallel Algorithms for HigherDimensional Convex Hulls, Proceedings 6th Annual ACM Symposium on Theory of Computing, 1994, 683--694.

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N. Amato, M. Goodrich, and E. Ramos. Parallel Algorithms for HigherDimensional Convex Hulls. In Proc. of the 35th Annual IEEE Symposium on Foundations of Computer Science, pages 683#694, October 1994.

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N. M. Amato, M. T. Goodrich and E. R. Ramos, Parallel Algorithms for HigherDimensional Convex Hulls, Proceedings 6 Annual ACM Symposium on Theory of Computing 1994, 683--694.

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