S. Meiser, "Point location in arrangements of hyperplanes," Information and Computation, 106(1993), pp. 286--303.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of nearest neighbor in the Hamming cube can be solved trivially by a single probe into a table of 2 cells, each containing d bits. The best algorithms for the exact nearest neighbor problem in Euclidean space take poly(d; log n) query time and need n Theta(d) space (see, for instance, [13]) For the approximate version of the problem, the best known algorithms are randomized and make O(log log d) probes to a table of size poly(n; d) 9, 12] The current best known cell probe randomized lower bound for the exact nearest neighbor problem (in both the Hamming and the Euclidean cases) ....

S. Meiser. Point location in arrangement of hyperplanes. Information and Computation, 106(2):286--303, 1993.

....error of . 9 5 Applications 5.1 The Approximated Nearest Neighbor to Ane Spaces problem Let F 1 ; F 2 ; F n be n k dimensional ats in R d , and let x 2 R d be a query point. To answer a Nearest Neighbor to Ane Spaces is to nd the at closest (in the Euclidean sense) to x. In [Mei93] Meiser presents a solution to the point location problem in arrangements of n hyperplanes in R d with running time O(d 5 log n) and space O(n d 1 ) That paper was a breakthrough in that it was the rst time (and to the best of our knowledge also the last) where an algorithm to the problem ....

S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106:286-303, 1993.

....approach should be to find say 5 good features for the problem. Indeed, in a new approach that uses a genetic algorithm, editing the training data and reducing the number of features used is carried out simultaneously [38] One of the fastest nearest neighbor search algorithms is due to Meiser [42] and runs in time O(d 3 log n) where d is the dimension. More recent algorithms which achieve efficiency at the expense of knowing the distribution of query points in advance have been found by Clarkson [12] One approach to practical applications is of course to sacrifice finding the exact ....

S. Meiser. Point location in arrangements of hyperplanes. Inform. Comput., 106:286--303, 1993.

....For higher dimensions, Clarkson [43] in one of the earliest applications of random sampling in computational geometry, presented a data structure of size O(n dd=2e ) that can answer a query in 2 O(d) log n time. The query time can be improved to O(d 5 log n) using a technique of Meiser [147]. Note that the query time of the above approach is exponential in d, so it is impractical even for moderate values of d (say d 10) Several heuristics have been developed, especially in higher dimensions, which use practical data structures such as kd trees, R trees, R trees, and Hilbert ....

S. Meiser, Point location in arrangements of hyperplanes, Inform. Comput., 106 (1993), 286{ 303.

....been studied extensively, especially in low dimension, where good solutions are known (see, for example [9] However, the combinatorial complexity of arrangements grows exponentially with the dimension, rendering the problem seemingly intractable. Indeed, following a long list of contributions [18, 12, 36, 28, 1, 29], currently the best algorithms can find a nearest neighbor in time poly(d; log n) but they need exponential (n Theta(d) storage. On the other hand, there is little evidence in the form of concrete lower bounds to support the curse of dimensionality conjecture [13] i.e. the belief that in ....

S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106(2):286--303, 1993.

....space. For example, the closest pair problem, furthest pair (or diameter) problem, many variants of clustering (including MST) and nearest neighbor search all belong to this class. If the dimension d is low, these problems enjoy very efficient (exact or approximate) solutions (e.g. see [29, 4, 26, 1] or the textbooks [28, 27] However, the running time and or space requirements of these algorithms grow exponentially with the dimension. This is unfortunate, since the high dimensional versions of the above problems are of major and growing importance to a variety of applications, usually ....

S. Meiser. "Point location in arrangements of hyperplanes", Information and Computation, 106(1993):286--303.

....and preprocess it for point location queries [219] For higher 42 Pankaj Agarwal and Jeff Erickson dimensions, Clarkson [86] presented a data structure of size O(n dd=2e ) that can answer a query in 2 O(d) log n time. The query time can be improved to O(d 3 log n) using Meiser s technique [193]. A nearest neighbor query for a set of points under the Euclidean metric can be formulated as an instance of the ray shooting problem in a convex polyhedron in R d 1 , as follows. We map each point p = p 1 ; p d ) 2 S to a hyperplane p in R d 1 , which is the graph of the function ....

S. Meiser, Point location in arrangements of hyperplanes, Inform. Comput., 106 (1993), 286-- 303.

....a linear complex, relying on theorem 1 from the previous section. The complexity bound is nontrivial for small dimensions n. There are quite sophisticated methods for recognizing linear complexes by means of (sequential) linear decision trees with complexity O(n O(1) log m) see [M88] [M93]) This bound is sharp (due to [SY82] B83] ignoring n (so, for small n relative to m) Unfortunately, it is unclear how to adjust these methods for MAC s. Therefore, we make use of a much more general method of cylindrical algebraic decomposition [C75] which provides a worse dependency on the ....

S. Meiser, Point location in arrangements of hyperplanes, Information and Computation, v. 106, 1993, p. 286--303.

....problem with a topological complexity at most O(log N ) Obviously, the bound is sharp. Let us also mention that for linear polynomials deg(f i ) 1; 1 i m the range searching problem can be solved even with a small computational complexity log O(1) N by linear decision trees [M88] [M93]. 2 Divide and conquer of the signs vectors The desired in the theorem TDT will be designed (notice that the proof is nonconstructive) in two stages. At the first one we design a TDT T 0 which solves the range searching problem with respect to the equality to zero, i.e. if for two input points ....

S. Meiser, Point Location in Arrangements of Hyperplanes, Information and Computation, 1993, 2, pp. 286--303.

.... 0, that can answer a query in 2 O(d) log n time. The data structure can be constructed in O(n dd=2e ) expected time. This paper was one of the earliest applications of random sampling in computational geometry. The query time can be improved to O(d 5 log n) using a technique of Meiser [153]. Note that the query time of the above approach is exponential in d, so it is impractical even for moderate values of d (say d 10) This exponential dependence on dimension is called the curse of dimension. Several heuristics have been developed, especially in higher dimensions, which use ....

S. Meiser, Point location in arrangements of hyperplanes, Inform. Comput., 106 (1993), 286-- 303.

....Heide [32] asked whether any union of m hyperplanes of R n can be recognized by R ovs branching trees of depth (n log m) O(1) His construction does not quite yield that result because it uses certain bounds on the size of the hyperplanes equations. In [14] we used a construction of Meiser [30] to give a second proof of (i) this construction also yields a positive answer to Meyer auf der Heide s question. In fact Meiser s construction almost answers that question, except that he has a non degeneracy assumption on the arrangement, and that he allows multiplications in his computation ....

S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106(2):286-303, 1993.

....in this LSA have integer coefficients of size (n log q) O(1) Here we say that a LSA solves the range searching problem if two points of R n arriving to the same leaf always belong to the same face. Part (i) answers a question of [15] This question was almost answered in a paper by Meiser [13], the main caveat being that multiplications are used in his algorithm (thus the implicit computation model is the algebraic decision tree instead of the LSA) Range searching has been studied by many other authors, see e.g. 6] and the references there. Part (ii) follows from a fairly ....

S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106(2):286--303, 1993.

....for point location in space subdivisions. The algorithm requires O(n 2 ) space, and has query time O(log 2 n) The best solutions for Nearest Neighbor Search in three dimensions come from the application of algorithms for arbitrary dimensions d to the case d = 3. Clarkson [26] and Meiser [66] give algorithms that have optimal query time O(log n) but require O(n 2 ffi ) space. Yao and Yao [89] give an algorithm that achieves linear space, but requires barely sublinear query time. We investigate these algorithms in detail in the following section. 9 5 Exact Nearest Neighbor Search ....

....q, at each node of the tree we identify the simplex that contains the point q by performing a simple brute force search over the O(s dd=2e ) simplices in R S . The query time of the algorithm is O(s dd=2e log n) and the required space is O (n dd=2e(1 ) The Meiser Algorithm Meiser [66] refines the idea of Clarkson by improving the query time of the algorithm. The set N is a set of m hyperplanes, and A(N) denotes the arrangement of the hyperplanes. The algorithm takes a random sample S of N , and builds the arrangement A(S) We construct the triangulation Delta(A(S) of the ....

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S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106(2):286--303, 1993.

....Remark. We note that what prevents achieving the conjectured lower bounds is the fact that the query time of the linear storage data structures is also away from its conjectured lower bound O(n 1bd=2c ) The dependency on d in the query time can be made O(d 4 log d log n) as shown by Meiser [25] (he states it as O(d 5 log n) but this slightly better bound follows from his construction) It seems an open problem to reduce this dependency. Can randomization help What if we are interested only in expectations (not worst case) It would appear that O(d 2 log n) is a lower bound. 5 ....

S. Meiser. Point location in arrangements of hyperplanes. Information and Computation 106 (1993), 286-303.

....O(2 d log n) search time. Clarkson [15] improves the storage requirement to O Gamma n (1 ffi)dd=2e Delta , paying d O(d) log n search time. Improvements by Yao and Yao [48] Matousek [36] and Agarwal and Matousek [1] still give exponential in d storage and search time. Finally, Meiser [37], gives the best result to date (in terms of search time) O(d 5 log n) search time using O Gamma n 2d ffi Delta storage. 3 2 For definiteness we say that a two sided error protocol returns the correct answer with probability at least 2=3. A one sided error protocol for a decision ....

S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106(2):286--303, 1993.

....problems which involve the notion of a distance between points in a d dimensional space. For example, the closest pair problem, furthest pair (or diameter) problem and nearest neighbor search all belong to this class. If the dimension d is low, these problems have very efficient solutions [13, 3, 12]. However, the running time and or space requirements of these algorithms grow exponentially with the dimension. This is unfortunate, since the high dimensional versions of the above problems are of major and growing importance to a variety of applications, usually involving similarity search or ....

S. Meiser. "Point location in arrangements of hyperplanes", Information and Computation, 106(1993):286--303.

....and video databases, protein databases, data mining and pattern recognition. The problem was formally posed in 1969 by Minsky and Papert [MP69] and was a subject of extensive study since then. Many efficient algorithms where discovered for the case where X is a low dimensional Euclidean space [Cl88, Me93]. However, many (if not most) of recent applications require X to be either high dimensional or non Euclidean, or both. In these cases the known algorithms become inefficient due to the so called curse of dimensionality their query times and or storage requirements grow exponentially with the ....

....query time, we conclude that there exists a provable trade off between the accuracy and efficiency for the c NNS problem. 1.1 Related work In this section we review the work on approximate nearest neighbor in metric spaces. We note that for the exact case the best known result is due to Meiser [Me93], who obtained an algorithm with storage O(n d ) and query time O(d 5 ) this can be improved to O(d 4 log d log n) query time and roughly O( n= log n) bd=2c ) preprocessing ( AE] p. 46) The storage requirements can be significantly reduced when the underlying norm is l 1 for ....

S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106(1993):286--303.

....diagram of S and preprocess it for point location queries in O(n log n) time [242] For higher dimensions, Clarkson [89] presented a data structure of size O(n dd=2e ) that can answer a query in 2 O(d) log n time. The query time can be improved to O(d 3 log n) using a technique of Meiser [214]. A nearest neighbor query for a set of points under the Euclidean metric can be formulated as an instance of the ray shooting problem in a convex polyhedron in R d 1 , as follows. We map each point p = p 1 ; p d ) 2 S to a hyperplane p in R d 1 , which is the graph of the function ....

S. Meiser, Point location in arrangements of hyperplanes, Inform. Comput., 106 (1993), 286-- 303.

....algorithm using (roughly) a doubleexponential in d (summing up time and space) data structure. This was improved and extended in subsequent work of Clarkson [5] Yao and Yao [34] Matousek [27] Agarwal and Matousek [1] and others, all requiring query time exponential in d. Recently, Meiser [28] obtained a polynomial in d search algorithm using an exponential in d size data structure. For approximate nearest neighbor search, Arya et al. 3] gave an exponential in d time search algorithm using a linear size data structure. Clarkson [6] gave a search algorithm with improved dependence on ....

S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106(2):286--303, 1993.

....O(2 d log n) search time. Clarkson [14] improves the storage requirement to O Gamma n (1 ffi)dd=2e Delta , paying d O(d) log n search time. Improvements by Yao and Yao [46] Matousek [34] and Agarwal and Matousek [1] still give exponential in d storage and search time. Finally, Meiser [35], gives the best result to date (in terms of search time) O(d 5 log n) search time using O Gamma n 2d ffi Delta storage. 3 In the approximate nearest neighbor search (approximate NNS) problem, a query is answered by finding a database point whose distance from the query is within a ....

S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106(2):286--303, 1993.

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S. Meiser, "Point location in arrangements of hyperplanes," Information and Computation, 106(1993), pp. 286--303.

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S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106(2):286--303, 1993.

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S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106:286-303, 1993.

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S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106:286-303, 1993.

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