A.C. Yao and F.F. Yao. A general approach to d-dimension geometric queries. In Proc. of 17th STOC, pp. 163--168, 1985. 17  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 ....

A.C. Yao and F.F. Yao. A general approach to d-dimension geometric queries. In Proc. of 17th STOC, pp. 163--168, 1985. 29

....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 in d dimensional Spaces 5.1 Algorithms based on Voronoi diagrams In this section we present a set of ....

....algorithm will search the whole tree, resulting in O(dn) query time. The authors prove that under some assumptions on the distribution of the data and the query points the average query time is O(d log n) The space required by the tree is O(dn) The idea of space partition is used by Yao and Yao [89] to tackle a more general problem. Their algorithm is designed for generic geometric queries; a generic geometric query is defined as the semigroup sum P x2P Q(x; q) where P is the data base, and q is the query point. The function Q(x; q) is defined as one of k bilinear functions f i (x; q) ....

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A. C. Yao and F. F. Yao. A general approach to d-dimension geometric queries. In Proceedings of the 17th Symposium on Theory of Computing, pages 163--168, 1985.

....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 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 ....

A.C. Yao and F.F. Yao. A general approach to d-dimension geometric queries. In Proc. of 17th STOC, pp. 163--168, 1985.

....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 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 ....

A.C. Yao and F.F. Yao. A general approach to d-dimension geometric queries. In Proc. of 17th STOC, pp. 163--168, 1985.

....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] 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 ....

A.C. Yao and F.F. Yao. A general approach to d-dimension geometric queries. In Proc. of 17th STOC, pp. 163--168, 1985.

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A.C. Yao and F.F. Yao. A general approach to d-dimension geometric queries. In Proc. of 17th STOC, pp. 163--168, 1985. 17

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A.C. Yao and F.F. Yao. A general approach to d-dimension geometric queries. In Proc. of 17th STOC, pp. 163--168, 1985. 17

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