Approximating the bottleneck plane perfect matching of a point set

A bottleneck plane perfect matching of a set of n   points in R2R2 is defined to be a perfect non-crossing matching that minimizes the length of the longest edge; the length of this longest edge is known as bottleneck  . The problem of computing a bottleneck plane perfect matching has been proved to be NP-hard. We present an algorithm that computes a bottleneck plane matching of size at least n5 in O(nlog2⁡n)O(nlog2⁡n)-time. Then we extend our idea toward an O(nlog⁡n)O(nlog⁡n)-time approximation algorithm which computes a plane matching of size at least 2n5 whose edges have length at most 2+3 times the bottleneck.