Fully dynamic maintenance of Euclidean minimum spanning trees and maxima of decomposable functions

We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in time O(n^1/2 log^2 n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in time O(n^E) per update. Our algorithm uses a novel construction, the ordered nearest neighbors of a sequence of points. Any point set or bichromatic point set can be ordered so that this graph is a simple path. Our results generalize to higher dimensions, and to fully dynamic algorithms for maintaining maxima of decomposable functions, including the diameter of a point set and the bichromatic farthest pair