Dynamic Euclidean Minimum Spanning Trees and Extrema of Binary Functions

We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in amortized time O(n1/2 log n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in time O(n) per update. Our algorithm uses a novel construction, the ordered nearest neighbor path of a set of points. Our results generalize to higher dimensions, and to fully dynamic algorithms for maintaining minima of binary functions, including the diameter of a point set and the bichromatic farthest pair.