P. M. Vaidya, "An O(n log n) algorithm for the all-nearest-neighbors problem", Discrete and Computational Geometry 4 (1989) 101--115.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 Vaidya s optimal all nearest neighbors algorithm [26]. Lenhof and Smid [17] have also presented an algorithm to compute the k closest pairs (k known in advance and the pairs not generated in order) in O(n log n k) time, using an approach similar to that of Salowe. Our paper presents algorithms with similar asymptotic running times, using ....

....case; as with their solution to Problems 2 and 1, it searches the standard Delaunay triangulation. Eppstein and Erickson [13] solved the planar problem for the simpler L # metric in time O(n log n kn) Once again, however, these approaches were not efficient in higher dimensions. Vaidya [26] gives an alternate approach based on a modified form of quadtrees; his algorithm works in any dimension and requires O(kn log n) time. In this paper, we extend the result of [11] We show how to make use of the results of [3] on linear sized higher dimensional Delaunay triangulation to solve ....

P. M. Vaidya, "An O(n log n) algorithm for the all-nearest-neighbors problem", Discrete and Computational Geometry 4 (1989) 101--115.

....in IR d . An O(k 1=d n 1 Gamma1=d ) separator of the k nearest neighborhood graph of P can be computed in O(kn) time if the k nearest neighborhood graph is given. Otherwise, it can be computed in O(kn log k n log n) time. The second part of the corollary follows from results of Vaidya [35] and Drysdale [3] that the k nearest neighborhood graph of a set of points can be computed in O(kn log k n log n) time. Notice that in many applications such as finite element and finite difference methods, only the graph and its embedding are given. Unlike the random algorithm of [23] we need ....

Vaidya, P. M. "An O(n log n) algorithm for the All-nearest-neighbors problem". Discrete & Computational Geometry,4:101--115, 1989.

.... of [20] More recently, the Voronoi digram [21] has provided a useful tool in low dimensional Euclidian settings and Figure 1: vp tree decomposition Figure 2: kd tree decomposition the overall field and outlook of Computational Geometry has yielded many interesting results such as those of [22, 23, 24, 25] and earlier [26] Unfortunately neither the kd tree, or the constructions of computational geometry, seem to provide a practical solution for high dimensions. As dimension increases, the kd tree soon visits nearly every database element while other constructions rapidly consume storage space. ....

P. M. Vaidya, "An O(n log n) algorithm for the allnearest -neighbor problem," Discrete & Computational Geometry, vol. 4, no. 2, pp. 101--115, 1989.

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