B. Chazelle, "New techniques for computing order statistics in Euclidean space", Proc. 1st Annual ACM Symposium on Computational Geometry, June 1985, 125--134. based Symposium  Home/Search   Document Not in Database   Summary   Related Articles   Check

....McElfresh, and Welzl [12] For this application, k is not known in advance (the enumeration is terminated when the triangulation is complete) and the pairs are required in nondecreasing order of distance. Closely related to Problem 2 is the following problem recently investigated by Chazelle [8], and by Agarwal, Aronov, Sharir, and Suri [1] Problem 4 (Selecting Distances) Given a finite set S of n points, let d 1 # # d ( n 2 ) be the distances determined by the pairs of points in S. For a given positive integer k # # n 2 # , determine the value of d k and find a pair of ....

....k points with minimum diameter, circumradius, or variance, could all be solved efficiently using algorithms for Problem 3 as a subroutine, improving previous techniques based on kth order Voronoi diagrams. Problem 3 has also been used for contouring in geographic information systems. Chazelle [8] presented a subquadratic solution to Problem 4 based on the batching technique of Yao [27] The algorithm works in any dimension in time O(n 2 #(d) log 1 #(d) n) where#(d) 1 (2 d 1) Thus for dimension 2 2 the running time is O(n 9 5 log 4 5 n) More recently, Agarwal, Aronov, ....

B. Chazelle, "New techniques for computing order statistics in Euclidean space", Proc. 1st Annual ACM Symposium on Computational Geometry, June 1985, 125--134. based Symposium

....algorithm is O(T log n m log(2n=m) when m n. If the matrix M is sorted by columns only then the overall runtime of the optimization algorithm is O(T maxflog m; log ng n log m) There have been some attempts to apply these techniques to geometric selection and optimization problems (e.g. [3, 4, 16]) In this paper we show some more such applications. In general our approach is the following: We examine geometric properties of the optimal solution for the particular problem. This leads to the determination of constraints on the set S of potential solutions for the problem. The constraints ....

....to perform an efficient search in the set of strips that are candidates for the optimal solution. The k th Triangle Area Problem. Given a set S of n points in the plane, determine which three points define the triangle with the k th largest area. This problem was stated and solved by Chazelle [3]. He presented an O(n 2 log 2 n) time algorithm based on Cole s scheme [5] for searching among similar lists, combined with the Frederickson and Johnson selection algorithm. Chazelle has conjectured that this bound can be improved to O(n 2 log n) In Section 3 we show that the geometric ....

B. Chazelle, "New techniques for computing order statistics in Euclidean space", Proc. 1st ACM Symp. on Computational Geometry, 1985, pp. 125--134.

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