J. S. Salowe. Enumerating interdistances in space. Internat. J. Comput. Geom. Appl., 2:49--59, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....S; x 6= yg. More generally, a sequence (p i ; q i ) 1 i k, of pairs, where p i ; q i 2 S, p i 6= q i , is called a sequence of k closest pairs of S, if the distances jp i q i j, 1 i k, are the k smallest elements in the multiset fjxyj : x; y 2 S; x 6= yg. The following result is due to Salowe [12], and Lenhof and Smid [9] Theorem 3 ( 12, 9] Given a set S of n points in R d and a positive integer k, a sequence of k closest pairs in S can be computed in O(n log n k) time. Our construction of fault tolerant spanners is based on the notion of wellseparated pairs, which is due to ....

....; q i ) 1 i k, of pairs, where p i ; q i 2 S, p i 6= q i , is called a sequence of k closest pairs of S, if the distances jp i q i j, 1 i k, are the k smallest elements in the multiset fjxyj : x; y 2 S; x 6= yg. The following result is due to Salowe [12] and Lenhof and Smid [9] Theorem 3 ([12, 9]) Given a set S of n points in R d and a positive integer k, a sequence of k closest pairs in S can be computed in O(n log n k) time. Our construction of fault tolerant spanners is based on the notion of wellseparated pairs, which is due to Callahan and Kosaraju [4] Before we can define ....

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J. S. Salowe. Enumerating interdistances in space. Internat. J. Comput. Geom. Appl., 2:49--59, 1992.

....O(n log D 1 n) space. When both insertions and deletions are allowed, Smid [22] described a data structure which uses O(n log D n) space and runs in O(log D n log log n) amortized time per update. Another data structure due to Smid [21] with improvements stemming from results of Salowe [17] and Dickerson, Drysdale, and Sack [7] uses O(n) space and requires O( # n log n) time for updates. Very recently, after a preliminary version of this paper was presented, Kapoor and Smid [13] devised a deterministic data structure of linear size which achieves polylogarithmic amortized update ....

J. S. Salowe, Enumerating interdistances in space, Internat. J. Comput. Geom. Appl., 2 (1992), pp. 49--59.

....D Gamma1 n) space. When both insertions and deletions are allowed, Smid [Smi92] described a data structure that uses O(n log D n) space and runs in O(log D n log log n) amortized time per update. Another data structure due to Smid [Smi91] with improvements stemming from results of Salowe [Sal92] and Dickerson and Drysdale [DD91] uses O(n) space and requires O( p n log n) time for updates. Very recently, after a preliminary version of this paper was presented, Kapoor and Smid [KS94] devised a deterministic data structure of linear size that achieves polylogarithmic amortized update ....

J. S. Salowe. Enumerating interdistances in space. Internat. J. Comput. Geom. Appl., 2:49--59, 1992.

....depend on the dimension D and, in the last two lines, on D and k. The update times are either worst case (w) or amortized (a) mode dimension update time space reference insertions D 2 log n (w) n [17, 18] deletions D 2 (log n) D (a) n(log n) D Gamma1 [23] dynamic D 2 p n log n (w) n [16, 20] dynamic D 2 (log n) D log log n (a) n(log n) D [22] dynamic D 3 (log n) D Gamma1 log log n (a) n this paper dynamic 2 log n log log n (a) n log n= log log n) k this paper dynamic 2 (log n) 2 = loglog n) k (a) n this paper It seems that in higher dimensions it is impossible to ....

J. S. Salowe, Enumerating interdistances in space, International Journal of Computational Geometry & Applications, 2 (1992), pp. 49--59.

....has also received much attention. One of the earliest reported algorithms is by Dickerson et al. 12] who used the Delaunay triangulation to enumerate the k closest pairs in O( n k) log n) time in two dimensions. Their algorithm actually enumerates the distances in sorted order. Salowe [33] was the first to give an O(n log n k) algorithm for any fixed dimension. His algorithm employs the A preliminary version of this work appeared in Proc. 14th ACM Sympos. Comput. Geom. pages 279 286, 1998. y New address: Department of Computer Science, University of Waterloo, Waterloo, ....

....the Euclidean metric are natural under this definition. The following combinatorial property of Salowe will be important. The original proof was for the L1 metric only, but readjusting constants immediately imply the statement for any natural distance function. Proposition 3. 1 (Salowe [33]) If the distance function is natural, then K(P; ar) O(jP j K(P; r) for any constant a. The enumeration algorithms of Salowe [33] and Lenhof and Smid [24] are based on a lemma similar to the below; a proof is included for completeness. We will use this in our solutions to both the ....

[Article contains additional citation context not shown here]

J. S. Salowe. Enumerating interdistances in space. Int. J. Comput. Geom. Appl., 2:49--59, 1992.

....6= yg. More generally, a sequence (p i ; q i ) 1 i k, of pairs, where p i ; q i 2 S, p i 6= q i , is called a sequence of k closest pairs of S, if the distances jp i q i j, 1 i k, are the k smallest elements in the multiset fjxyj : x; y 2 S; x 6= yg. The following result is due to Salowe [14], and Lenhof and Smid [10] Theorem 3 ( 14, 10] Given a set S of n points in R d and a positive integer k, a sequence of k closest pairs in S can be computed in O(n log n k) time. Our construction of fault tolerant spanners is based on the notion of wellseparated pairs, which is due to ....

....i ) 1 i k, of pairs, where p i ; q i 2 S, p i 6= q i , is called a sequence of k closest pairs of S, if the distances jp i q i j, 1 i k, are the k smallest elements in the multiset fjxyj : x; y 2 S; x 6= yg. The following result is due to Salowe [14] and Lenhof and Smid [10] Theorem 3 ([14, 10]) Given a set S of n points in R d and a positive integer k, a sequence of k closest pairs in S can be computed in O(n log n k) time. Our construction of fault tolerant spanners is based on the notion of wellseparated pairs, which is due to Callahan and Kosaraju [5] De nition 2 Let s 0 ....

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J. S. Salowe. Enumerating interdistances in space. Internat. J. Comput. Geom. Appl., 2:49-59, 1992.

....with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and or a fee. closest pairs in O( n k) log n) time in two dimensions. Their algorithm actually enumerates the distances in sorted order. Salowe [32] was the first to give an O(n log n k) algorithm for any fixed dimension. His algorithm employs techniques such as parametric searching [28] or Vaidya s all nearest neighbor method. Lenhof and Smid [23] subsequently pointed out a much simplified algorithm; the only geometric structure needed is ....

....the Euclidean metric are natural under this definition. The following combinatorial property of Salowe will be important. The original proof was for the L1 metric only, but readjusting constants immediately imply the statement for any natural distance function. Proposition 3. 1 (Salowe [32]) If the distance function is natural, then K(P; ar) O(jP j K(P; r) for any constant a. The enumeration algorithms of Salowe [32] and Lenhof and Smid [23] are based on a lemma similar to the below; a proof is included for completeness. We will use this in our solutions to both the ....

[Article contains additional citation context not shown here]

J. S. Salowe. Enumerating interdistances in space. Int. J. Comput. Geom. Appl., 2:49--59, 1992.

....in the algebraic computation tree model. We remark that Arya and Smid [15] have shown that the algorithm of [52] also works if we replace the Delaunay triangulation by any bounded degree spanner. See Section 6.1. The first optimal algorithm for the k closest pairs problem is due to Salowe [105]. He first combines a variant of Vaidya s algorithm [125] with the parametric search technique to compute the k th smallest L1 distance. Then, again using a variant of Vaidya s algorithm, he enumerates all pairs of points that have distance at most equal to Dffi. The number of these pairs is ....

J.S. Salowe. Enumerating interdistances in space. International Journal of Computational Geometry & Applications 2 (1992), pp. 49--59.

....Their algorithm is based on the k nearest neighbor Voronoi diagrams. Dickerson et al. 7] present an algorithm for the problem: enumerate all the k smallest distances in S in increasing order. Their algorithm works in time O(n log n k log k) and uses O(n k) space. Lenhof et al. 18] Salowe [19], Dickerson and Eppstein [9] also solved this problem but they just report the k closest pairs of points without sorting the distances, spending O(n log n k) time and O(n k) space. An algorithm for solving the second problem (for the smallest k distances) is also presented in [9] spending ....

J. Salowe, "Enumerating interdistances in space", Internat. J. Comput. Geom. Appl. 2, 49--59, 1992.

....the history of a fully dynamic point set in O(log n) amortized time per insertion or deletion. Note that recently there has been much interest in the dynamic closest pair problem. For the case where only deletions are allowed, see Supowit [14] For the fully dynamic case, see Smid [11, 13] Salowe [8] and Dickerson and Drysdale [3] The algorithm in this paper is based on the algorithm of Smid [12] To update the closest pair when a point is inserted, that algorithm makes some queries into a data structure for the k dimensional rectangular point location problem. In this data structure, one ....

J.S. Salowe. Enumerating interdistances in space. Internat. J. Comput. Geom. Appl. 2 (1992), pp. 49-59.

....the k smallest distances 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 ....

....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 ....

J.S. Salowe. Enumerating interdistances in space. Internat. J. Comput. Geom. Appl. 2 (1992), pp. 49--59.

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