J. S. Salowe, "Shallow interdistance selection and interdistance enumeration", Proc. WADS '91, Lecture Notes in Computer Science, 519 (Springer-Verlag, Berlin, 1991) 117--128.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of points may lie within a given distance of each other. The problem was originally solved by Bentley, Stanat, and Williams [5] in worst case time O(3 d dn log n 3 d k) where d is the dimension and k the number of pairs reported. Algorithms for problem 1 have also been used by Salowe [21, 22] and Lenhof and Smid [17] as subroutines in parametric search methods for solving Problems 2 and 4. Problem 3 is a generalization of the well known nearest neighbors problem. For classification problems, it is more robust than a simple nearest neighbors search. The graph of k nearest neighbors to ....

....# is any arbitrarily small constant. Salowe has also solved the interdistance selection problem for the L # metric in d dimensions in O(n log d n) time [20] for k # n, and has since extended these results to get an O(n log n k) time algorithm for Problem 2 that works for any L p metric [21, 22]; however the value of k must be known in advance, and the distances are not enumerated in order. As a sub step, Salowe also presents an algorithm to solve Problem 1, the fixed radius nearneighbor search problem, in time O(n log n k) for L p metrics in d dimensions. This algorithm was inspired by ....

J. S. Salowe, "Shallow interdistance selection and interdistance enumeration", Proc. WADS '91, Lecture Notes in Computer Science, 519 (Springer-Verlag, Berlin, 1991) 117--128.

....k = 1) The algorithm is based on the fair split tree. The constant factor in the update time is exponential in the dimension. We modify the fair split tree to reduce it. 1 Introduction The dynamic closest pair problem is one of the very well studied proximity problem in computational geometry [6, 17 20, 22, 24 26, 28 31]. We are given a set S of n points in k dimensional space, k 1, and a distance metric L t , for 1 t 1. The point set is modified by insertions and deletions of points. Each point p is given as a k tuple of real numbers (p 1 ; p k ) The closest pair of S is a pair (p; q) of distinct ....

....of size O(n) that supports insertions in O(log k Gamma1 n) amortized time. Schwarz, Smid and Snoeyink [26] presented a data structure of size O(n) that maintains the closest pair in O(log n) amortized time per insertion. Several algorithms are obtained for the dynamic closest pair problem [19, 20, 22, 24, 29 31]. In [20, 22, 29] the problem is solved with O( p n log n) update time using O(n) space. In [19] Kapoor and Smid gave data structures of size S(n) that maintain the closest pair in U(n) amortized time per update, where for k 3, size S(n) O(n) and time U(n) O(log k Gamma1 n log log n) ....

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J. S. Salowe. Shallow interdistance selection and interdistance enumeration. International Journal of Computational Geometry & Applications, 2, 1992, pp. 49--59.

....techniques for maintaining the minimal L r distance of a point set in d dimensional space. The data structure of [153] uses O(n) space and supports updates in O(n 2=3 log n) time, by giving a method to compute the O(n 2=3 ) smallest distances defined by a set of n points in O(n log n) time. By [49,141], which show how to compute the n smallest distances in O(n log n) time, the update time is improved to O(n 1=2 log n) for arbitrary dimension. In [156] the update time is reduced to O(log d n log log n) amortized, while the space is slightly increased to (n log d n) We briefly describe ....

J.S. Salowe, "Shallow Interdistance Selection and Interdistance Enumeration," Lecture Notes in Computer Science 519 (1991), 117--128.

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