H.-P. Lenhof and M. Smid. Enumerating the k closest pairs optimally.InProc. 33rd Annu. IEEE Sympos. Found. Comput. Sci., pages 380--386, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for Vaidya s. The new algorithm is faster than the one described by Aggarwal et al. 2] for all values of k. Very recently, Datta et al. 12]developed new algorithms for each of the problems we discuss in Section 5, except variance, using a searchtechnique developed by Lenhof and Smid [27] that does not require the computation of nearest neighbors and follows the algebraic decision tree model. In the plane, their algorithms use the same time and less space than the solutions we present here# in higher dimensions, they improve both both time and space bounds. They also present ....

H.-P. Lenhof and M. Smid. Enumerating the k closest pairs optimally.InProc. 33rd Annu. IEEE Sympos. Found. Comput. Sci., pages 380--386, 1992.

....supported by the National Science Foundation under grant IRI 9610240. d 2. Salowe [21] shows how to compute the k th closest pair in O(n log d n) time under L1 . The first result for enumerating more than one pair was by Smid [25] who solved CP n 2=3 in O(n log n) time. Lenhof and Smid [19] followed with a solution to the general CP k problem that runs in O(n log n k) time. Unfortunately, when d is not viewed as a constant, all of the algorithms reported above suffer from the so called dimensionality curse the constant factors hidden in the O notation grow exponentially with ....

....force algorithm can solve the problem in O(dn 2 ) time. Although we may often assume that d is much smaller than n, in practice however, c d can quickly overcome a reasonably large n, even for very small d, thus making the brute force solution the preferred choice. For example the algorithm in [19] consists of two phases: an approximation phase in which 20 d (k 2n) n 2 ) must be satisfied, followed by an enumeration phase in which the processing time is at least Omega Gammaa d k 4 d n log n) in the worst case. For the best scenario of k = 1 in the approximation phase, d ....

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H.-P. Lenhof and M. Smid. Enumerating the k closest pairs optimally. In Proc. 33rd Annu. IEEE Sympos. Found. Comput. Sci., pages 380--386, 1992.

....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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H.-P. Lenhof and M. Smid. Enumerating the k closest pair optimally. Proc. 33rd Ann. IEEE Symp. Found. Comput. Sci., 1992, pp. 380-386.

.... k point subset of a set of n points in the plane, in time O(n log n k 3=2 n) and space O(kn k 5=2 ) 2 Very recently, Datta et al. 12] developed new algorithms for each of the problems we discuss in Section 5, except variance, using a search technique developed by Lenhof and Smid [27] that does not require the computation of nearest neighbors and follows the algebraic decision tree model. In the plane, their algorithms use the same time and less space than the solutions we present here; in higher dimensions, they improve both both time and space bounds. They also present ....

H.-P. Lenhof and M. Smid. Enumerating the k closest pairs optimally. In Proc. 33rd Annu. IEEE Sympos. Found. Comput. Sci., pages 380--386, 1992.

....in the set S in sorted order, in time O(n k log k) The value of k need not be known at the start of the enumeration. We show similar results for enumerating approximate distances. For the problem of enumerating the k smallest distances, the following was known. Salowe [13] and Lenhof and Smid [9] achieve O(n log n k) time for any dimension, but in both algorithms, the value of k must be known in advance and the distances are not enumerated in sorted order. In the plane, Dickerson et al. 8] show that given the Delaunay triangulation, the k smallest distances can be enumerated in O(n k ....

....index i is extracted, w k is the weight of the k th closest pair seen so far. Hence, the algorithm terminates at this moment, proving the claim. Now we estimate the running time. The number of pairs extracted from the queue is at most equal to the number of pairs having weight at most tw k . In [9, 13], it is shown that the latter is bounded by O(n k) Hence, after initializing the queue, which takes O(n) time, the algorithm performs O(n k) queue operations. This follows from the fact that the spanner G has bounded degree. Since each queue operation takes O(log n) time, the entire running ....

H.-P. Lenhof and M. Smid. Enumerating the k closest pairs optimally. Proc. 33rd Annu. IEEE Sympos. Found. Comput. Sci., 1992, pp. 380--386.

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