Improvements on bottleneck matching and related problems using geometry

Let A and B be two sets of n objects in R^d. We propose to use bottleneck matching as a convenient way for measuring the resemblance between them, and present several algorithms for computing, as well as approximating, this resemblance. The running time of all these algorithms is close to O(n^1.5). For instance, if the objects are points in the plane, the running time is O(n^1.5 log n). We also consider the problem of finding a translation of B that maximizes the resemblance to A under the bottleneck matching criterion. When A and B are point-sets in the plane, we present an O(n⁵ log n) time algorithm for determining whether for some translated copy the resemblance gets below a given ρ, improving the previous result of Alt, Mehlhorn, Wagener and Welzl by a factor of almost n. We use this result to compute the smallest such ρ in time O(n⁵ log² n), and give an efficient approximation scheme for this problem.