K. Clarkson, "A randomized algorithm for closest-point queries," SIAM J. Computing, 17(1988), pp. 830--847.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to it can be found rapidly [6, 11, 17, 20, 22] Some deterioration in performance occurs because the imposition of structure results in a non optimal codebook. The second approach is to preprocess the unstructured codebook so that the complexity of nearest neighbor searching is reduced [8, 9, 14, 21, 28]. Although methods based on preprocessing an unstructured codebook appear to work well in low dimensions, their complexity increases so rapidly with dimension that their running time is little better than brute force linear search in moderately large dimensions [23] In this paper we show that if ....

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM Journal on Computing, 17(4):830--847, 1988.

....extension to polyhedral surfaces. These can be found in [26] More related work. Beyond the related results already mentioned above, we point out the recent breakthrough due to Koltun in the analysis of vertical decompositions in higher dimensions [20] For alternative decompositions see [5] 6] [9]. We are not aware of experimental results on vertical decompositions of three or higher dimensional arrangements. 2. IMPROVED OUTPUT SENSITIVE CONSTRUCTION Our algorithm is an improvement of the sweep based algorithm by de Berg et al. 10] for brevity we call the algorithm of [10] the NQO ....

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17:830-847, 1988.

....that the hyperplanes are in general position. This assumption can be removed by a more careful (and technically a little more complicated) treatment, or by standard perturbation arguments [Ede87] Canonical triangulations. Let C be a subcomplex of A(H) The canonical triangulation of C ( Cla88] which we denote by C , is defined as follows. Let C be a j dimensional cell of C (thus, a convex polytope) and let v be the bottom vertex of C, that is, the lexicographically smallest vertex of C. If j = 1 then C is a segment and it is already a (1 dimensional) simplex. If j 1, then we ....

....vertex of C. If j = 1 then C is a segment and it is already a (1 dimensional) simplex. If j 1, then we recursively triangulate the (j Gamma 1) dimensional faces of C and extend each (j Gamma 1) simplex to a j simplex using the vertex v. Unbounded cells require some care in this definition [Cla88] The canonical triangulation of a cell with m vertices has O(m) simplices. The canonical triangulation C of the subcomplex C is the simplicial complex obtained by triangulating each cell of C in this manner. Let R be a subset of H . For a (relatively open) simplex Delta 2 A(R) let K ....

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput. , 17:830--847, 1988.

....planes. Chazelle and Friedmaxx [8] proved that any family H of hyperplanes has a ( cutting of size O(ra) for any r larger than some constant ra depending on d. This ( cutting is constructed by triangulating the arrangement of O(r) specific hyperplanes of H according to some particular ciiterion [11]. 1Throughout this paper, we use the asymptotic notations based on sets (see [6] By using such a concept, the following upper bound on the number of incidences between families of points and restricted families of hyperplanes in can be de rived. Theorem 2.2 Let a(x, y) be the maximum number ....

K.L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comp., 17:830-847, 1988.

....systems. Graph based algorithms pre calculate the nearest neighbors of spatial objects and create new index structures for the pre calculated nearest neighbor information for e#cient search [5] Examples of such algorithms include the RNG algorithm [2] and the algorithms using Voronoi diagrams [7, 11]. For example, one can first derive the Voronoi diagram for a given set of spatial points followed by indexing the cells in the Voronoi diagram. Given a query point q, finding the nearest neighbor can be simplified to finding the Voronoi cell that contains q. Although graph based algorithms are ....

K. Clarkson. A randomized algorithm for closestpoint queries. SIAM Journal of Computing, 17:830--847, 1988.

....systems. Graph based algorithms pre calculate the nearest neighbors of spatial objects and create new index structures for the pre calculated nearest neighbor information for e#cient search [5] Examples of such algorithms include the RNG algorithm [2] and the algorithms using Voronoi diagrams [7, 11]. For example, one can first derive the Voronoi diagram for a given set of spatial points followed by indexing the cells in the Voronoi diagram. Given a query point q, finding the nearest neighbor can be simplified to finding the Voronoi cell that contains q. Although graph based algorithms are ....

K. Clarkson. A randomized algorithm for closest-point queries. SIAM Journal of Computing, 17:830--847, 1988.

....count higher dimensional Delaunay simplices: triangles in R 4 , tetrahedra in R 6 , and so on. Standard range searching techniques can be used to answer nearest neighbor queries in R 3 in O(log n) time using O(n 2 = polylog n) space, or in O( p n polylog n) time using O(n) space [2, 17, 22, 34, 50]. Using these data structures, we can compute the Euclidean spanning tree of a three dimensional point set in O(n 4=3 ) time [1] All these results ultimately rely on the simple observation that the Delaunay triangulation of a random sample of a point set is signi cantly less complex (in ....

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput. 17:830-847, 1988.

....O(1) n) search time using O(1= d n log O(1) n storage. For the large data sets we are interested in, even polylogarithmic factors and polynomial factors in (1= may exceed reasonable storage limits. If space is not an issue, then there are a number of alternatives. For example, Clarkson [Cla88] presents a data structure that has O(d O(d) search time and O(n (1 )dd=2e ) space. There is a wealth of literature on methods for dimension reduction for high dimensional data sets. A good survey is presented by Carreira Perpi n an [CP96] However, our interest is on how high ....

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17(4):830{ 847, 1988.

....of the boundary of P is obtained inductively by joining, for j d 1, each j dimensional simplex in the boundary of a (j 1) dimensional face to the lowest vertex in the face. Then, a triangulation of P is obtained by joining each (d 1) simplex in the triangulation of the boundary to the origin o [5] (the simplices are open so that they form a partition of the polytope, that is, a point is in a unique simplex) Linear Programming. Given the linear function w : R d R and the set of constraints H , the linear programming problem is to determine the minimum of w in P = T H . We ....

....with 0 1, as guaranteed by the partition theorem for (n v =r v ) shallow hyperplanes; and (iii) a child for each of the O(r v ) sets in v . 3 Basic Data Structures We use two basic data structures, which provide a trade o between storage space and query time. They are due to Clarkson [5] and Matou sek and Shwarzkopf [15] Afterwards, we re ne them to achieve better bounds. These data structures have a common generic tree structure that allows the use of a single general query procedure, presented in the next section. 4 3.1 Generic Tree Structure The data structures we employ ....

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K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput. 17 (1988), 830-847.

....the method required n 2 O (d) space and O(2 d log n) query time. For d = 2, one can construct the Voronoi diagram of S and preprocess it for point location queries in O(n log n) time using O(n) space so that an NN query can be answered in O(log n) time [154] 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 ....

K. L. Clarkson, A randomized algorithm for closest-point queries, SIAM J. Comput., 17 (1988), 830-847.

....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 ....

K. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17:830-- 847, 1988.

....bounds as we are trying to bound the maximum of the procedures spawned from any given node. 62 12.3 FURTHER READING Random sampling was introduced into parallel computational geometry by Reif and Sen [39] conference version) independently around the same time as the seminal papers of Clarkson [6, 8] and Haussler and Welzl [17] All these papers primarily exploited the net property, namely the (almost) even partitioning of the problem using a random subset. The subsequent papers of Clarkson [9] and Clarkson and Shor [7] re ned the techniques considerably and developed very general and ....

K. Clarkson. A randomized algorithm for closest point queries. SIAM Journal on Computing, 17:830-847, 1988.

....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 ....

K. Clarkson. "A randomized algorithm for closest-point queries", SIAM Journal on Computing, 17(1988), pp. 830-847.

....kd trees correspond to unit vector projection with the canonical basis. More recently, the Voronoi digram [2] has provided a useful tool in low dimensional Euclidean settings and the overall eld and outlook of Computational Geometry has yielded many interesting results such as those of [31, 9, 8, 17] and earlier [12] It appears that [12] may be the rst work focusing on worst case bounds. Very recently Kleinberg [22] gives two algorithms for an approximate form of the nearest neighbor problem. The space requirements of the rst are prohibitive but the second, which almost always nds ....

Clarkson, K. L. A randomized algorithm for closest-point queries. SIAM Journal on Computing 17, 4 (August 1988).

....randomly pretransformed data. Our criterion for pruning branches during search, and its analysis, are the primary contributions of this paper and distinguish our methods from kd tree search. More recently, the eld of computational geometry has yielded many interesting results such as those of [32, 11, 10, 19] and earlier [14] Very recently a number of papers have appeared striving for more ecient algorithms with worst case time bounds for the approximate nearest neighbor problem [1, 24, 25, 22] These exploit properties of random projections beyond the simple projection distance dominance fact ....

Clarkson, K. L. A randomized algorithm for closestpoint queries. SIAM Journal on Computing 17, 4 (August 1988).

....If m n 4 ffl we dualize the problem. The lines spanning edges of a triangle are mapped into three Plucker points and the line l ae , spanning 12 a query ray ae is mapped into a Plucker hyperplane. We select a random sample of r Plucker hyperplanes and we form its canonical triangulation [Cla88] We can make sure during the process that we produce simplices having at least one vertex. We obtain M = O(r 4 log r) simplices covering the zone of Pi. Each simplex is cut by O( n r log r) hyperplanes [Cla87] Now we consider every triple of simplices in turn. The number of triples is ....

K.L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. on Computing, 17:830--847, 1988.

....1997 1 Introduction About ten years ago, the field of range searching, especially simplex range searching, was wide open. At that time, neither efficient algorithms nor nontrivial lower bounds were known for most range searching problems. A series of papers by Haussler and Welzl [149] Clarkson [85, 86], and Clarkson and Shor [89] not only marked the beginning of a new chapter in geometric searching, but also revitalized computational geometry as a whole. Led by these and a number of subsequent papers, tremendous progress has been made in geometric range searching, both in terms of developing ....

....information systems [222, 236] Let us assume that the distance between points is measured in the Euclidean metric. For d = 2, one can construct the Voronoi diagram of S 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 ....

K. L. Clarkson, A randomized algorithm for closest-point queries, SIAM J. Comput., 17 (1988), 830--847.

....number of points on the convex hull [15] Their algorithm is based on linear time algorithms by Megiddo and Dyer for solving linear programming problems in the plane. Starting with seminal work by Clarkson, randomized algorithms have played an increasingly important role in computational geometry [8, 7, 9]. These algorithms always produce the correct solution but their running time depends upon random choices made during the execution of the algorithm. As such, they are called Las Vegas random algorithms. Clarkson and Shor in [9] gave randomized incremental algorithms for constructing convex hulls ....

Clarkson, K. L. A randomized algorithm for closest-point queries. SIAM J. Comput. 17 (1988), 830--847.

....16 ] quad trees [ 17 ] vp trees [ 39 ] and hB trees [ 22 ] none of which significantly improve performance for high dimensions. Clarkson describes a randomized algorithm which finds the closest point in d dimensional space in O(log 2 n) operations using a RPO (randomized post office) tree [ 10 ] However, the time taken to construct the RPO tree is O(n dd=2e(1 ffl) and the space required to store it is also O(n dd=2e(1 ffl) This makes it impractical when the number of points n is large or or if d 3. 3 The Algorithm 3.1 Searching by Slicing We illustrate the proposed high ....

....using equation (26) and the time taken for the few (1 ) additional 3 searches necessary when a point was not found within the hypercube. Although 3 When a point was not found within the hypercube, we incremented ffl by 0.1 and searched again. This 22 0 0.05 0.1 0.15 0.2 0.25 0.3 0. 35 5 10 15 20 25 Time (secs. Dimensions Proposed algorithm k d tree Exhaustive search 0 0.2 0.4 0.6 0.8 1 1.2 5 10 15 20 25 Time (secs. Dimensions Proposed algorithm k d tree Exhaustive search (a) b) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 5 10 15 20 25 Time (secs. ....

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K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Computing, 17(4):830--847, August 1988.

....[ f Gamma(e) j e is an edge corresponding to a hyperplane in L Delta g : Note that jU Delta j; jL Delta j st. We map the lines containing the edges of U Delta to their Plucker hyperplanes in R 5 and preprocess their lower envelope for point location queries, using an algorithm of Clarkson [19]. That is, we preprocess the hyperplanes into a data structure, so that we can quickly determine whether a query point lies below all hyperplanes. Similarly, we map the edges of L Delta to their Plucker hyperplanes and preprocess their upper envelope for similar point location queries. Each point ....

....Similarly, we map the edges of L Delta to their Plucker hyperplanes and preprocess their upper envelope for similar point location queries. Each point location structure requires O( st) 2 ffi ) space and preprocessing time, for any ffi 0, and answers a query in O(log st) time; see [19] for details. Our data structure needs one more ingredient: For each simplex Delta 2 Xi and for each polyhedron P i we consider the first (marked) edge, if any, of Gamma i that contributes a hyperplane to U Delta [L Delta . For each such edge e, let f 1 , f 2 be the two faces of P i incident ....

K.L. Clarkson, A randomized algorithm for closest-point queries, SIAM J. on Comput., 17 (1988), 830--847.

....requires at most O(n d) operations where n is the number of songs in the database and d is the number of feature dimensions stored per song. Clearly there are more efficient algorithms for computing nearest neighbors. Most deterministic algorithms have a query time of at least#st1 d) log(n) [4, 1], while Kleinberg s # approximate algorithm [12] has a query time of O( d log 2 d) d log n) and a preprocessing step which requires O( n log d) 2d ) storage. However, we feel that our use of a brute force technique is reasonable for two reasons. First, it performs very well given the size ....

K. Clarkson. A randomized algorithm for closestpoint queries. SIAM J. Computing, 17:830--847, 1988.

....the method required n 2 O(d) space and O(2 d log n) query time. For d = 2, one can construct the Voronoi diagram of S and preprocess it for point location queries in O(n log n) time using O(n) space so that an NN query can be answered in O(log n) time [162] For higher dimensions, Clarkson [47], presented a data structure of size O(n dd=2e ) for any constant 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 ....

K. L. Clarkson, A randomized algorithm for closest-point queries, SIAM J. Comput., 17 (1988), 830--847.

....several O(jRj log jRj) time algorithms are known [32, 54] among the simplest of which are based on the randomized incremental paradigm. Let CT 0 (R) denote the collection of (closed) full dimensional simplices in the canonical triangulation of the ( 0) level, as defined for instance by Clarkson [31] (also called the bottom vertex triangulation) The precise definition of this triangulation is not important to us except in the proof of the sampling lemma below. The only facts we need is that the triangulation is lineartime constructible and that the simplices in CT 0 (R) are all vertical ....

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17:830--847, 1988.

....max 1iD jp i Gamma q i j. The planar version of the nearest neighbor problem can be solved optimally, i.e. with O(log n) query time using O(n) space, by means of Voronoi diagrams. See [15] In higher dimensions, however, the situation is much worse. The best results known are due to Clarkson [7] and Arya et al. 1] In [7] a randomized data structure is given that finds a nearest neighbor of a query point in O(log n) expected time. This structure has size O(n dD=2e ffi ) where ffi is an arbitrarily small positive constant. In [1] the problem is solved with an expected query time of ....

....The planar version of the nearest neighbor problem can be solved optimally, i.e. with O(log n) query time using O(n) space, by means of Voronoi diagrams. See [15] In higher dimensions, however, the situation is much worse. The best results known are due to Clarkson [7] and Arya et al. 1] In [7], a randomized data structure is given that finds a nearest neighbor of a query point in O(log n) expected time. This structure has size O(n dD=2e ffi ) where ffi is an arbitrarily small positive constant. In [1] the problem is solved with an expected query time of O(n 1 Gamma1=d(D 1) 2e ....

K. L. Clarkson, A randomized algorithm for closest-point queries, SIAM J. Comput., 17 (1988), pp. 830--847.

....that can be used 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 ....

....be its bottom, that is, the vertex with the minimum x d coordinate. By induction hypothesis, all subfaces of f have been triangulated. We triangulate the face f by extending all simplices on its boundary to cones with apex v. The cones are cylinders if the vertex v lies at infinity. It is proven [26] that the triangulation of a convex polytope with n facets has O ( n bd=2c ) simplices 2 . The triangulation of a Voronoi diagram has O ( n dd=2e ) simplices, and the triangulation of an arrangement of hyperplanes has O (n d ) simplices. Finally, it is worth noting that the algorithms ....

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K. Clarkson. A randomized algorithm for closest-point queries. SIAM Journal of Computing, 17:830--847, 1988.

....[18] marks the beginning of work on the Euclidean case of arbitrary dimension. They achieve a discretization of the problem, so that (super ) exponential storage can be used to answer queries quickly. Dobkin and Lipton use O i n 2 d 1 j storage to allow 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 ....

K. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17:830-847, 1988.

....in time polynomial in jA Delta j by the method of conditional probabilities. Bibliography and remarks. The ideas leading to the construction of cuttings via nets appeared in the early works of Clarkson [Cla88a] and Haussler and Welzl [HW87] The bottom vertex triangulation is from Clarkson [Cla88b] The existence of asymptotically optimal cuttings was first established by Chazelle and Friedman [CF90] by the method with seminets; derandomization of their method was elaborated in [Mat91a] Mat95a] In the plane, a deterministic construction of a (1=r) cutting of size O(r 2 ) was ....

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17:830--847, 1988.

....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 ....

K. Clarkson. "A randomized algorithm for closest-point queries", SIAM Journal on Computing, 17(1988), pp. 830-847.

....can be constructed in O(log n) in an n node butterfly network with bounded buffer size. 7 Bibliographic Notes Random sampling was introduced into parallel computational geometry by Reif and Sen [75] conference version) independently around the same time as the seminal papers of Clarkson [25, 26] and Haussler and Welzl [51] All these papers primarily exploited the net property, namely the (almost) even partitioning of the problem using a random subset. The subsequent papers of Clarkson [27] and Clarkson and Shor [30] refined the techniques considerably and developed very general and ....

K.L. Clarkson. A randomized algorithm for closest point queries. SIAM Journal on Computing, 17(4), August 1988, pp. 830--847.

....d = 3, several O(jRj log jRj) time algorithms are known, among the simplest of which are based on the randomized incremental paradigm. Let CT 0 (R) denote the collection of (closed) full dimensional simplices in the canonical triangulation of the ( 0) level, as defined for instance by Clarkson [28] (also called the bottom vertex triangulation) The precise definition of this triangulation is not important to us except in the proof of the sampling lemma below. The only facts we need is that the triangulation is lineartime constructible and that the simplices in CT 0 (R) are all vertical ....

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17:830--847, 1988.

....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 ....

K. Clarkson. A randomized algorithm for closest-point queries. SIAM Journal on Computing, 17(1988):830--847.

.... they all exhibit poor behavior in the worst case and in typical cases as well (e.g. see Arya, Mount, and Narayan [4] Dobkin and Lipton [23] were the first to provide an algorithm for nearest neighbors in d , with query time O(2 d log n) and preprocessing 1 cost O(n 2 d 1 ) Clarkson [16] reduced the preprocessing to O(n dd=2e(1 ffi) while increasing the query time to O(2 O(d log d) log n) Later results, e.g. Agarwal and Matousek [1] Matousek [51] and Yao and Yao [65] all suffer from a query time that is exponential in d. Meiser [52] obtained query time O(d 5 log n) ....

K. Clarkson. A randomized algorithm for closest-point queries. SIAM Journal on Computing, 17(1988):830--847.

....solving the nearest neighbor problem, either in time or space, seems to grow extremely rapidly. Clarkson presented a randomized O(logn) expected time algorithm for finding nearest neighbors in fixed dimension based on computing Voronoi diagrams of randomly sampled subsets of points (the RPO tree) [6]. However in the worst case the space needed by his algorithm grows roughly as O(n dd=2e ffi ) and this is too high for our applications. Yao and Yao [24] observed that nearest neighbor searching can be solved in linear space and barely sublinear time O(n f(d) where f(d) log(2 d ....

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM Journal on Computing, 17(4):830--847, 1988.

....such as simplices, that are each homeomorphic to a ball and have constant description complexity. Ideally, the number of cells after the refinement should be proportional to the overall complexity of the arrangement. For arrangements of hyperplanes the well known bottom vertex triangulation [15] meets this criterion. For more general arrangements such refined decompositions are more difficult to find. For example, for algebraic hypersurfaces of constant maximum degree in d dimensional space (d 3) the best decomposition technique known so far results in O(n 2d Gamma3 fi(n) cells ....

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17:830--847, 1988.

....used in many existing systems. The well known standard (nonchromatic) nearest neighbor problem can be thought of as a special case of the chromatic problem, where every point has its own color. Algorithms and data structures for the standard nearest neighbor problem have been extensively studied [4, 8, 10, 15, 16, 17, 28, 31, 32, 35]. For existing approaches, as the dimension grows, the complexity of answering exact nearest neighbor queries increases rapidly, either in query time or in the space of the data structure used to answer queries. The growth rate is so rapid that these methods are not of real practical value in even ....

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM Journal on Computing, 17(4):830--847, 1988.

....1997 1 Introduction About ten years ago, the field of range searching, especially simplex range searching, was wide open. At that time, neither efficient algorithms nor nontrivial lower bounds were known for most range searching problems. A series of papers by Haussler and Welzl [161] Clarkson [88, 89], and Clarkson and Shor [92] not only marked the beginning of a new chapter in geometric searching, but also revitalized computational geometry as a whole. Led by these and a number of subsequent papers, tremendous progress has been made in geometric range searching, both in terms of developing ....

....the distance between points is measured in the Euclidean metric, though a more complicated metric can be used depending on the application. For d = 2, one can construct the Voronoi 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 ....

K. L. Clarkson, A randomized algorithm for closest-point queries, SIAM J. Comput., 17 (1988), 830--847.

....) For more details about the D dimensional post office problem, we refer the reader to the chapter on Voronoi diagrams in this handbook. We just mention here that the best results currently known either use a large amount of space, or their query time is almost linear. More precisely, Clarkson [42] gives a randomized data structure that finds a nearest neighbor of a query point in O(log n) expected time. This structure has size O(n dD=2e ffi ) where ffi is an arbitrarily small positive constant. In [11] Arya and Mount solve the problem with an expected query time of O(n ....

K.L. Clarkson. A randomized algorithm for closest-point queries. SIAM Journal on Computing 17 (1988), pp. 830--847.

....in k dimensions with n constraints can be solved in O(3 k 2 n) time. We note here some of the new recent developments for linear programming. There are several randomized algorithms for this problem, of which the best expected complexity, O(k 2 n k k=2 O(1) log n) is due to Clarkson[47], which is later improved by Matousek et al. 91] to run in O(k 2 n e O( p k ln k) log n) Clarkson s algorithm[47] is applicable to work in a general framework, which includes various other geometric optimization problems, such as smallest enclosing ellipsoid. The best known deterministic ....

....for linear programming. There are several randomized algorithms for this problem, of which the best expected complexity, O(k 2 n k k=2 O(1) log n) is due to Clarkson[47] which is later improved by Matousek et al. 91] to run in O(k 2 n e O( p k ln k) log n) Clarkson s algorithm[47] is applicable to work in a general framework, which includes various other geometric optimization problems, such as smallest enclosing ellipsoid. The best known deterministic algorithm for linear programming is due to Chazelle and Matousek[42] which runs in O(k 7k o(k) n) time. 2.7 ....

[Article contains additional citation context not shown here]

K. L. Clarkson, "A Randomized Algorithm for Closest-Point Queries," SIAM J. Comput., 17,4 (August 1988), 830-847.

....solving the nearest neighbor problem, either in time or space, seems to grow extremely rapidly. Clarkson presented a randomized O(logn) expected time algorithm for finding nearest neighbors in fixed dimension based on computing Voronoi diagrams of randomly sampled subsets of points (the RPO tree) [5]. However in the worst case the space needed by his algorithm grows roughly as O(n dd=2e ffi ) and this is too high for our applications. Yao and Yao [20] observed that nearest neighbor searching can be solved in linear space and barely sublinear time O(n f(d) where f(d) log(2 d ....

....from Voronoi diagrams, we can project the points onto a paraboloid in dimension d 1 and reduce the problem of reporting the points lying 10 Arya and Mount within a given sphere to the problem of reporting the points lying within a given halfspace. Using standard results from random sampling [5, 15] we can compute the largest empty sphere centered at the query point in O(logn) expected queries to a halfspace range reporter as follows. The idea is to decompose the point set into a chain of k = dlg ne random subsets: S 0 ae S 1 ae : ae S k = S where jS i j = 2 i . Given q, for i = 1; ....

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM Journal on Computing, 17(4):830--847, 1988.

No context found.

K. Clarkson, "A randomized algorithm for closest-point queries," SIAM J. Computing, 17(1988), pp. 830--847.

No context found.

K. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17:830-847, 1988.

No context found.

K. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17:830847, 1988.

No context found.

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17:830{ 847, 1988.

No context found.

K. L. Clarkson. A Randomized Algorithm for Closest-point Queries. SIAM J. Comput., 17:830---848, 1988.

No context found.

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17(4):830--847, 1988.

No context found.

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17:830--847, 1988.

No context found.

K. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Comput., 17:830-847, 1988.

No context found.

K. L. Clarkson. A randomized algorithm for closest-point queries. SIAM Journal on Computing, 17(4):830--847, 1988.

No context found.

Clarkson, K.L. A randomized algorithm for closestpoint queries, SIAM J. Comput., 17 (1988), 830--847.

No context found.

K. L. Clarkson. A randomized algorithm for closestpoint queries. SIAM J. Comput. 17 (1988) 830--847.

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