D. Dobkin, R. Lipton, "Multidimensional search problems," SIAM J. Computing, 5(1976), pp. 181--186.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 ....

D. Dobkin and R. Lipton. Multidimensional search problems. SIAM J. Comput., 5:181-- 186, 1976.

....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 ....

D. Dobkin and R. Lipton. Multidimensional search problems. SIAM J. Computing, 5:181--186, 1976.

....model is a well recognized and fundamental open problem (see below) Related work. There is an extensive body of research concerning nearest neighbor problems for small dimensional (e.g. 2 and 3) Euclidean space (see for example the text by de Berg et al. [10] Dobkin and Lipton s seminal paper [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] ....

D. Dobkin and R. Lipton. Multidimensional search problems. SIAM J. Comput., 5:181-186, 1976.

.... and structures based on space filling curves; more recent results are surveyed in [60] While some perform well in 2 3 dimensions, in high dimensional spaces 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. ....

D. Dobkin and R. Lipton. Multidimensional search problems. SIAM Journal on Computing, 5(1976):181--186.

....of hyperplanes. We dare not attempt to survey but the most relevant papers to our work. There are excellent solutions to nearest neighbor search in low (two or three) dimensions. For more information see, e.g. 29] In high dimensional space, the problem was first considered by Dobkin and Lipton [11]. They showed an exponential in d search 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 ....

D. Dobkin and R. Lipton. Multidimensional search problems. SIAM J. Comput., 5:181-186, 1976.

....are functions, to indicate that g(n) O(f(n) log n) Also note that in three dimensions, a unique transformation T (upto reflection) between two point sets P 1 and P 2 is determined by matching three points p; q, and r in P 1 with three points s; t, and u in P 2 . Furthermore, it is known that [20]: Proposition 1 Given a transformation T from P 1 to P 2 , in time O(n) we can determine for each point p 2 P 1 a corresponding point p 0 2 P 2 such that jT (p) Gamma p 0 j is minimized. The basic random sampling method is as follows. BASIC SAMPLE: For some constant c, perform (c log ....

D. Dobkin and R. Lipton. Multidimensional search problems. SIAM J. Computing 5:181--186, 1976.

....model is a well recognized and fundamental open problem (see below) Related work. There is an extensive body of research concerning nearest neighbor problems for small dimensional (e.g. 2 and 3) Euclidean space (see for example the text by de Berg et al. [9] Dobkin and Lipton s seminal paper [17] 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 [14] ....

D. Dobkin and R. Lipton. Multidimensional search problems. SIAM J. Comput., 5:181-186, 1976.

....case (and usually in typical cases as well) The field of computational geometry has developed a rich theory for the study of proximity problems. Establishing upper bounds on the time required to answer a nearestneighbor query in d appears to have been first undertaken by Dobkin and Lipton [9]; they provided an algorithm with query time O(2 d log n) and pre processing O(n 2 d ) This was improved by Clarkson [4] he gave an algorithm with query time O(exp(d) Deltalog n) and pre processing O(n dd=2e(1 ) here exp(d) denotes a function that grows at least as quickly as 2 d . ....

D. Dobkin and R. Lipton, "Multidimensional search problems," SIAM J. Computing, 5 (1976), pp. 181--186.

....There is a voluminous literature on algorithms for the Euclidean nearest neighbor problem, and we do not attempt a comprehensive survey of it here. Establishing upper bounds on the time required to answer a nearest neighbor query in R d appears to have been first undertaken by Dobkin and Lipton [17]; they provided an algorithm with query time O(2 d log n) and pre processing O(n 2 d 1 ) We use the term pre processing to refer to the sum of the pre processing time and storage required. This was improved by Clarkson [11] he gave an algorithm with query time O(exp(d) Delta log n) and ....

....described above; they do not require the use of complicated data structures, the manipulation of sets of hyperplanes in d dimensions, or the special handling of degenerate cases. As such, the algorithms appear to be simpler to implement even than the original method of Dobkin and Lipton [17]. Techniques and Basic Definitions We build our data structures from the projections of the set P onto random lines through the origin in R d . The use of random projections onto lines has appeared in a number of contexts in recent high dimensional geometric constructions and algorithms (e.g. ....

D. Dobkin, R. Lipton, "Multidimensional search problems," SIAM J. Computing, 5(1976), pp. 181--186.

....published an algorithm for computing the convex hull of a set of points in the plane based on sorting the points by angle and then scanning them. His work arose from the need of a statistics group within Bell Labs to be able to efficiently cluster data samples [Gr2] In 1974, Dobkin and Lipton [DL1, DL2] gave the 3 first algorithms for searching in spatial subdivisions. Their work arose from an open problem given in Knuth [Kn] asking how to preprocess a set of points to be able to find nearest neighbors efficiently. In 1975, Shamos and Hoey [S, SH1] proposed efficient algorithms for finding ....

Dobkin, D. and Lipton, R., "Multidimensional search problems", SIAM J. Computing, vol. 5, 1976, pp. 181-186.

No context found.

D. Dobkin, R. Lipton, "Multidimensional search problems," SIAM J. Computing, 5(1976), pp. 181--186.

No context found.

D. Dobkin and R. Lipton. Multidimensional search problems. SIAM J. Comput., 5:181-186, 1976.

No context found.

D. Dobkin and R. Lipton. Multidimensional search problems. SIAM J. Comput., 5:181-186, 1976.

No context found.

D. Dobkin, R. Lipton. Multidimensional Search Problems. SIAM Journal of Computing, 5 (1976), 181--186.

No context found.

D. Dobkin and R. Lipton. Multidimensional search problems. SIAM Journal on Computing, 5:181-186, 1976.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC