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
2065158139 2080059421
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 d≤6 , 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 d≥2 . 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 Y⊇X, 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). The problem of construction of planar Voronoi diagrams arises in many areas, one of the most important of which is in nearest neighbor problems. This includes clustering [ 141, contour maps [6] and (Euclidean) minimum spanning trees [23]. Shamos [22] gives several more applications. An JZ(N log N) time worst case lower bound can be shown for this problem by reducing it to sorting [2 11. The challenge is to construct an O(N log N) time algorithm. Shamos [213 and Shamos anti Hoey [23] describe an O(N log N) time divide-and-conquer algorithm for construction of the planar Euclidean Voronoi diagram. Lee and Wong [ 161 describe an O(N log N) time algorithm for the L1 and L, metrics in the plane, and Drysdale pnd Lee [8] present an O(N@g N)l *) t’ rme algorithm for the Voronoi diagram of N line segments (which they have since improved to O(N(log N)*) time). Shamos [2 11, Lee and Preparata [ 151, and Lipton and Tarjan [ 171 have produced fast algorithms for searching a Voronoi diagram (or any other straight-line planar graph). In this paper we describe an O(N log N) time algorithm for constructing a planar Euclidean Voronoi diagram which extends straightforwardly to higher dimensions. The fundamental result is that a K-dimensional Euclidean Voronoi diagram of N points can be constructed by transforming the points to K + I-space, Given a set of pointsV in the plane, the Euclidean bottleneck matching problem is to match each point with some other point such that the longest Euclidean distance between matched points, resulting from this matching, is minimized. To solve this problem, we definek-relative neighborhood graphs, (kRNG) which are derived from Toussaint's relative neighborhood graphs (RNG). Two points are calledk-relative neighbors if and only if there are less thank points ofV which are closer to both of the two points than the two points are to each other. AkRNG is an undirected graph (V,E r k) whereE r k is the set of pairs of points ofV which arek-relative neighbors. We prove that there exists an optimal solution of the Euclidean bottleneck matching problem which is a subset ofE r 17. We also prove that ¦E r k¦ < 18kn wheren is the number of points in setV. Our algorithm would construct a 17RNG first. This takesO(n2) time. We then use Gabow and Tarjan's bottleneck maximum cardinality matching algorithm for general graphs whose time-complexity isO((n logn)0.5m), wherem is the number of edges in the graph, to solve the bottleneck maximum cardinality matching problem in the 17RNG. This takesO(n1.5 log0.5n) time. The total time-complexity of our algorithm for the Euclidean bottleneck matching problem isO(n2 +n1.5 log0.5n).