H. Bronnimann, B. Chazelle, and J. Matousek, Product range spaces, sensitive sampling, and derandomization, SIAM J. Computing 28 (1999), 1552--1575.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... running time for computing d(L; L ) because we still need ) time for constructing the pairs (L i ; L i ) If we are interested in computing a pair of lines with the minimum vertical distance, the running time can be improved to O(n ) 232] An earlier attempt by Bronnimann et al. [50] to derandomize Clarkson Shor algorithm had an error. 8.3 Distance between polytopes We wish to compute the Euclidean distance d(P 1 ; P 2 ) between two given convex polytopes P 1 and P 2 in R . If the polytopes intersect, then this distance is 0. If they do not intersect, then this distance ....

H. Bronnimann, B. Chazelle, and J. Matousek, Product range spaces, sensitive sampling, and derandomization, Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci., 1993, pp. 400--409.

....to compute a closest pair in L i Theta L 0 i , for each i k, which can be done using parametric searching. The decision procedure is: For a given real number r, determine whether d(L i ; L 0 i ) r, for each i k. Since lines in 3 space have 3 An earlier attempt by Bronnimann et al. [44] to derandomize Clarkson Shor algorithm had an error. Geometric Optimization January 24, 1997 Proximity Problems 28 four degrees of freedom, each of these subproblems can be transformed to the following point location problem in R 4 : Given a set S of n points in R 4 (representing the lines ....

H. Bronnimann, B. Chazelle, and J. Matousek, Product range spaces, sensitive sampling, and derandomization, Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci., 1993, pp. 400--409.

....we use the following additional notation: Given a set of segments X and a set of trapezoids T , we write T [X] to denote the set of conflict lists X oe for oe 2 T , and write jT [X]j to denote P oe2T jX oe j. 3.1. Gradation. The algorithm is a variant of the RIC approach [10] and follows [6, 5]. Given parameters and 0 , it chooses a sequence of subsets of S, called a gradation: S 0 S 1 : S l Gamma1 S l = S where S i Gamma1 is a (1= sample from S i for i l and S l Gamma1 is a (1= 0 ) sample from S l = S (the need for two parameters will become clear when ....

H. Bronnimann, B. Chazelle, and J. Matousek. Product range spaces, sensitive sampling, and derandomization. In Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci., 1993, 400--409.

....3 , A (S) can be computed by a randomized algorithm whose running time is O(n 3 Gamma1=19 ) with high probability. Remark 5. 2 The algorithm can be derandomized without increasing the asymptotic bound on the running time, using the deterministic construction of nets by Bronnimann et al. [12]. 5.2 Computing A (E4 ; T4 ) Let A be a set of a segments in R 3 , each of which is an edge of VorN (S) Let B be a set of b triangles in R 3 , each of which is a part of a 2 face of Vor F (S) We want to compute A (A; B) a minimum width shell centered at an intersection point of a ....

H. Bronnimann, B. Chazelle, and J. Matousek, Product range spaces, sensitive sampling, and derandomization, Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci., 1993, pp. 400--409.

....S, that is, the maximum distance between two points of S. A very simple O(n log n) expected time randomized algorithm (which is worst case optimal) was given by Clarkson and Shor [33] but no optimal deterministic algorithm is known. The best known deterministic solution is due to Bronniman et al. [23], and runs in O(n log 3 n) time. It is based on parametric searching, and uses some interesting derandomization techniques. See also [25, 85, 104] for earlier close to linear time algorithms based on parametric searching. Closest line pair. Given a set L of n lines in the R 3 , we wish to ....

H. Bronnimann, B. Chazelle, and J. Matousek, Product range spaces, sensitive sampling, and derandomization, Proc. 34th IEEE Symp. Found. of Comp. Sci., 1993, pp. 400--409.

....covering, we can quickly find O(c) sized covers. 1 Introduction A set system (X; R) is a set X along with a collection R of subsets of X , which are sometimes called ranges [25] Such entities have also been called hypergraphs and range spaces in the computational geometry literature (e.g. see [5, 10, 11, 12, 13, 14, 15, 16, 20, 24, 25, 34, 36, 35, 41, 37, 39, 40]) and they can be used to model a number of interesting computational geometry problems. There are a host of NP hard problems defined on set systems, with one of the chief such problems being that of finding a set cover of minimum size (e.g. see [21, 23] where a set cover is a subcollection C ....

....time O(jY j D 1 ) It is a witness oracle if, for any set R of R jY , it can provide a set R 0 of R such that R = R 0 Y in O(jX j) time. If (X; R) has a subsystem oracle of degree D, and VC exponent d, it is clear that d D. Under this assumption, it has also been shown by Matousek et al. [36, 10] that one can find a (1=r) net for (X; R) of size O(dr log(dr) in O(d) 3D r D log D (rd)jX j time, for both the uniform and weighted cases. The scrupulous reader should verify that their algorithm works also in the log RAM model with identical running time. 3 The Main Algorithm The goal ....

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H. Bronnimann, B. Chazelle, and J. Matousek. Product range spaces, sensitive sampling, and derandomization. In Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS 93), pages 400--409, 1993.

....(y 1 ; y 10 ) j y 9 0 (p) 8 X i=1 y i i (p) y 10 ) Set (oe) h 1 (oe) 10 (oe)i 2 R 10 . Let P = T p2S H p be the convex polyhedron defined by the intersection of the 2n corresponding halfspaces. P has O(n 5 ) faces and can be computed in O(n 5 ) time [8]. A cylindrical shell (with nonhorizontal axis) oe contains S if and only (oe) 2 P . Let Psi R 4 Theta (R ) 2 denote the 6 dimensional set of all cylindrical shells (with nonhorizontal axis) that contain S. Then ( Psi) is the intersection of P with the 6 dimensional surface Phi = ....

H. Bronnimann, B. Chazelle, and J. Matousek, Product range spaces, sensitive sampling, and derandomization, Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci., 1993, pp. 400--409.

....and Friedman [CF90] Mat91a] Chazelle and Matousek [CM96] present a cleaner exposition of the algorithm and give an explicit dependence of the constants on the dimension d. Srivastav [Sri95] gives a somewhat more efficient implementation of the polynomial sampling subroutine. Bronnimann et al. BCM93] modify the algorithm to work with so called sensitive approximations instead of approximations. A subset A X is a sensitive approximation for (X; R) if fi fi fi fi jRj jX j Gamma jR Aj jAj fi fi fi fi 2 i s jRj jX j j for every set R 2 R (this definition may ....

....approach fails: the reason is (roughly) that the maximum complexity of the intersection of n halfspaces in dimensions d (d odd) and d Gamma 1 has the same order of magnitude, and hence boundary effects can be too large. Ramos [Ram97a] observed that by combining the methods of Bronniman et al. BCM93] and of Matousek and Schwarzkopf [MS93] one can get an O(nr bd=2c Gamma1 ) deterministic algorithm for computing 0 shallow (1=r) cuttings in the full range of r s. Another direction of generalization is to consider other surfaces than hyperplanes. In this case also the simplices of the ....

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H. Bronnimann, B. Chazelle, and J. Matousek. Product range spaces, sensitive sampling, and derandomization. In Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci., pages 400--409, 1993. Revised version is to appear in SIAM J. Computing .

....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 ....

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H. Bronnimann, B. Chazelle, and J. Matousek. Product range spaces, sensitive sampling, and derandomization. In Proc. 34th IEEE Sympos. Found. Comput. Sci., pages 400--409, 1993.

....is given by a subsystem oracle. This means that given a subset A X, we can compute the system Sj A (in the form of an incidence matrix) in time O(jAj d 1 ) where d is a constant called the dimension of the subsystem oracle. The current most efficient version of the algorithm was given in [BCM93] For a fixed oracle dimension d, it can compute a (1=r) net for S of size O(r log r) in time O(jXjr d log d r) in particular, if r is a constant, a (1=r) net is found in time linear in the cardinality of X. This result allows one to make most of divide and conquer algorithms in ....

H. Bronnimann, B. Chazelle, and J. Matousek. Product range spaces, sensitive sampling, and derandomization. In Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci., pages 400--409, 1993.

....points of Y . So the VC dimension of our set system cannot be more than 4 (or 2d for d dimensional points) Because of their properties, ffl nets have been used in many geometric algorithms and applications. Their use is also instrumental in the derandomization of divide and conquer algorithms ([BCM]) The derandomization of random algorithms is a general problem that allows us a better understanding of the importance of randomness as a computational resource. It also produces algorithms with guaranteed worst case performance. The only known way to deterministically and efficiently compute ....

H. Bronnimann, B. Chazelle and J. Matousek, Product range spaces, sensitive sampling, and derandomization. Proc. 34th Ann. IEEE Symp. Foundat. of Comp. Sci. (1993), 143-155.

....covering, we can quickly find O(c) sized covers. 1 Introduction A set system (X; R) is a set X along with a collection R of subsets of X , which are sometimes called ranges [25] Such entities have also been called hypergraphs and range spaces in the computational geometry literature (e.g. see [5, 10, 11, 12, 13, 14, 15, 16, 20, 24, 25, 34, 36, 35, 41, 37, 39, 40]) and they can be used to model a number of interesting computational geometry problems. There are a host of NP hard problems defined on set systems, with one of the chief such problems being that of finding a set cover of minimum size (e.g. see [21, 23] where a set cover is a subcollection C ....

....O(jY j D 1 ) It is a witness oracle if, for any set R of R jY , it can provide a set R 0 of R such that R = R 0 Y in O(jX j) time. If (X; R) has a subsystem oracle of degree D, and VC exponent d, it is clear that d D. Under this assumption, it has also been shown by Matousek et al. [36, 10] that one can find a (1=r) net for (X; R) of size O(dr log(dr) in O(d) 3D r D log D (rd)jX j time, for both the uniform and weighted cases. 3 The Main Algorithm The goal of this section is to prove our main theorem. Let s be a non decreasing function, let n = jX j, and let m = jRj. ....

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H. Bronnimann, B. Chazelle, and J. Matousek. Product range spaces, sensitive sampling, and derandomization. In Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci., pages 400--409, 1993.

....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 the randomized method of Clarkson and Shor [19] or the deterministic method of Bronnimann, et al. [9]. Like the Euclidean diameter problem, the discrete 1 center problem in IR 3 can then be solved by parametric search in O(n polylog n) time [9, 13, 43, 48] We now show how parametric search can be replaced by a randomized search, solving the three dimensional discrete 1 center problem in O(n ....

.... O(n) size and can be computed in O(n log n) time by the randomized method of Clarkson and Shor [19] or the deterministic method of Bronnimann, et al. 9] Like the Euclidean diameter problem, the discrete 1 center problem in IR 3 can then be solved by parametric search in O(n polylog n) time [9, 13, 43, 48]. We now show how parametric search can be replaced by a randomized search, solving the three dimensional discrete 1 center problem in O(n log n) expected time. The randomized reduction to the decision problem is more involved than in the diameter problem [19] Section 3) In particular, we need ....

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H. Bronnimann, B. Chazelle, and J. Matousek. Product range spaces, sensitive sampling, and derandomization. In Proc. 34th IEEE Sympos. Found. Comput. Sci., pages 400--409, 1993.

....sophisticated techniques. 3 Their algorithm yields an O(n log 3 n) time algorithm for computing the diameter. Recently, Ramos [237] and Bespamyatnikh [45] obtained O(n log 2 n) time algorithms for computing the diameter. Obtaining an optimal 3 An earlier attempt by Bronnimann et al. [51] to derandomize Clarkson Shor algorithm had an error. Geometric Optimization April 30, Proximity Problems 29 O(n log n) time deterministic algorithm for computing the diameter in R 3 still remains elusive. 8.2 Closest line pair Given a set L of n lines in R 3 , we wish to compute a closest ....

H. Bronnimann, B. Chazelle, and J. Matousek, Product range spaces, sensitive sampling, and derandomization, Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci., 1993, pp. 400--409.

....algorithm may be possible; we plan to discuss some particular cases in the final version. 1. 2 Basic approach The basic approach uses divide and conquer based on geometric partitioning together with a pruning of the input to the subproblems that enforces a 1 Another algorithm presented in [5] is flawed and apparently has not been fixed. global invariant on the total size of the subproblems in each level of the recursion. It dates back to Clarkson and Shor s [12] randomized output sensitive algorithm for computing halfspace intersections in R 3 . That approach was refined by Reif ....

H. Bronnimann, B. Chazelle, and J. Matousek. Product range spaces, sensitive sampling, and derandomization. In Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS 93), 400--409, 1993.

....easily implemented to run in O(logn(log log n) d Gamma1 ) time using linear work on an EREW PRAM. 1 Introduction The study of randomized algorithms and methods for reducing the amount of perfect randomness needed for geometric algorithms has proven to be a very rich area of research (e.g. see [1, 2, 4, 5, 14, 15, 22, 42, 58, 57]) Indeed, randomized geometric algorithms are typically simpler and more efficient than their deterministic counterparts and studying the limitation of the randomness needed by such algorithms often yields insights into the specific properties of randomization that are needed to achieve this ....

.... ffl approximation if, for each range R 2 R, fi fi fi fi jY Rj jY j Gamma jRj jX j fi fi fi fi ffi jRj jX j ffl: This notion is a combined measure of the absolute and relative error between jY Rj=jY j and jRj=jX j, and it is somewhat similar to a notion Bronnimann et al. [14] refer to as a sensitive ffl approximation 1 . Note that this notion also subsumes that of an ffl net, for any ffi relative ffl approximation is automatically an (ffl= 1 Gamma ffi ) net. Our specific interest in this paper is in the design of fast and efficient deterministic methods for ....

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Herv'e Bronnimann, Bernard Chazelle, and Jir'i Matousek. Product range spaces, sensitive sampling, and derandomization. In Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS 93), 400--409, 1993.

....among all the ranges in C on S with coloring . The motivation for solving the maximum bichromatic discrepancy problem is in the construction of ffl nets, which are used extensively in many geometric algorithms and applications. In particular, it plays a crucial role in the Bronnimann et al. [BCM93] result on the derandomization of divide and conquer algorithms. However, the only known deterministic way to compute ffl nets efficiently is by constructing an ffl approximation. Matousek et al. MWW93] shows that a range space having maximum bichromatic discrepancy ffi has a 2ffi jSj ....

H. Bronnimann, B. Chazelle and J. Matousek, Product range spaces, sensitive sampling, and derandomization, Proc. 34th Ann IEEE Symp. Foundat. of Comp. Sci., (1993), pp 143-155.

....of segments f(X; r) O(r k(r=n) 2 ) so to have the right parameters in (iii) and (iv) of Theorem 2. 4, we need to verify that f(A; r) O(r k(r=n) 2 ) A (1=r) approximation A for (S; R(S; T(S) is also a (1=r) approximation for (S; R(S; S) Then, by a result of Bronnimann et al. [6] on product range spaces, A can also be used to estimate the number of intersections in A(S) inside any convex region. More precisely, for a set of segments X, let v(X; oe) denote the number of intersections between segments in X that lie in oe, then Lemma 2.5 Let A be a (1=r) approximation for ....

....(R) jS oe j Cnff(r) Randomized algorithms Efficient randomized algorithms are known for both problems on segments we are considering. Here, we point out that for both, algorithms with equal asymptotic performance can be obtained using the approach used by Chazelle [10] and Bronnimman et al. [6] for half space intersection. The sampling is global, that is, at a given stage a single global sample determines the sample in each subproblem, rather than an independent sample for each subproblem. As a result, one maintains T (R i ) for a single sample R i of S at each stage. This has the ....

[Article contains additional citation context not shown here]

Herv'e Bronnimann, Bernard Chazelle, and Jir'i Matousek. Product range spaces, sensitive sampling,and derandomization. In Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci., pages 400--409, 1993.

.... 1 Introduction There is a growing interest in algorithms working on sets of data that are too large to be fit in the internal memory of computers, and that consequently need to perform input output accesses to external storage devices, like disks and CD ROMs (see e.g. 4, 11, 19, 21, 29, 37] These devices are roughly 10 6 times slower than internal memory in terms of access time. In many applications, this disparity has given rise to an input output (or I O) bottleneck, in which the time spent on moving data between internal and external memory dominates the overall execution ....

....develop a general randomized approach suitable to solve I O efficiently not only the segment intersections problem, but also several others geometric problems like convex hulls, Voronoi diagrams, batched planar point location. We study these problems in the external memory model , introduced in [37] Here a computer consists of a processing unit, an internal memory of size M and an (unbounded) external memory partitioned into blocks of size B, B M . Each access to the external memory transfers from to the internal memory one block of B items, e.g. integers, pointers, characters. The goal ....

[Article contains additional citation context not shown here]

H. Bronnimann, B. Chazelle, and J. Matousek. Product range spaces, sensitive sampling, and derandomization. In Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci., 1993, 400--409.

.... 3) There is a data structure with O(n 1 ) preprocessing and O(n) space, such that an LP query on H can be answered in O(n 1 Gamma1=bd=2c 2 O(log n) time. Remarks: 1. We can make the query algorithm deterministic by applying derandomization methods on the computation of nets [CM93, BCM93]. The space and query time remain the same; however, the cost of preprocessing becomes quite high (but polynomial) since we need to construct (1=r) nets with r close to n= log n. For certain specific halfspace range reporting structures, it may be possible to reduce this preprocessing cost. 2. ....

H. Bronnimann, B. Chazelle, and J. Matousek. Product range spaces, sensitive sampling, and derandomization. In Proc. 34th IEEE Sympos. Found. Comput. Sci., pages 400--409, 1993.

.... optimal in higher dimensions were the randomized incremental algorithm of Clarkson and Shor [15] and the subsequent randomizedmethod of Seidel [56] Recently, Chazelle [10] gave the first deterministic algorithm that is optimal in higher dimensions, which was simplified by Bronnimann et al. [8]. The optimality of the above algorithms is measured with respect to the worst case size complexity of the resulting convex hull. However, when the size of the output is considered, it may be possible to beat the worst case lower bounds since the size of the convex hull may range from O(1) to ....

.... ball intersection together with parametric search [43] as in previous works [11, 40] we obtain a sequential algorithm for computing the diameter of a point set in IR 3 with running time O(n log 3 n) that is arguably simpler than the algorithm with the same running time by Bronnimann et al. [8]. We present some important constructions for hyperplane set systems in the next section. In Section 3 we give our convex hull methods for d 4, and we give some specialized methods for d = 3 in Section 4. 2 Hyperplane Set Systems We begin by describing a general framework for set systems, which ....

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Herve Bronnimann, Bernard Chazelle, and Jir Matousek. Product range spaces, sensitive sampling, and derandomization. In Proc. 34th Annu. IEEE Sympos. Found.Comput. Sci. (FOCS 93), pages400--409, 1993.

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H. Bronnimann, B. Chazelle, and J. Matousek, Product range spaces, sensitive sampling, and derandomization, SIAM J. Computing 28 (1999), 1552--1575.

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H. Bronnimann, B. Chazelle, and J. Matousek. Product range spaces, sensitive sampling, and derandomization. SIAM Journal of Computing, 1999.

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