Let @math and @math be sets of @math red and @math blue points in the plane, respectively, with @math . Let @math be a perfect matching between points from @math and @math , using @math straight line segments to match the points, that is, each point is an endpoint of exactly one line segment, and each line segment has one red and one blue endpoint. We forbid line segments to cross. Denote the length of a longest line segment in @math with @math , which we also call the [ ] value, bn of @math . We aim to find a matching under given constraints that minimizes @math . Any such matching is called a [ ] bottleneck matching of @math .
In @cite_0 , Chang, Tang and Lee gave an @math -time algorithm for computing a bottleneck matching of a point set, but allowing crossings. This result was extended by Efrat and Katz in @cite_9 to higher-dimensional Euclidean spaces.
@cite_0 @cite_9
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Abstract We extend the planar results of (1992) to higher dimensions, and show that given a set A of 2n points in d -space it is possible to compute a Euclidean bottleneck matching of A in roughly O (n 1.5 ) time, for d6 , and in subquadratic time, for any constant d>6 . If the underlying norm is L  , then it is possible to compute a bottleneck matching of A in O (n 1.5 log 0.5 n) time, for any constant d2 . A metric space X is said to be absolutely Lipschitz extendable if every Lipschitz function f from X into any Banach space Z can be extended to any containing space YX, where the loss in the Lipschitz constant in the extension is independent of Y,Z, and f. We show that various classes of natural metric spaces are absolutely Lipschitz extendable. To cite this article: J.R. Lee, A. Naor, C. R. Acad. Sci. Paris, Ser. I 338 (2004).