Dynamic Euclidean minimum spanning trees and extrema of binary functions

D. Eppstein
1995 Discrete & Computational Geometry  
We maintain the minimum spanning tree of a point set in the plane subject to point insertions and deletions, in amortized time O(n ~/2 log 2 n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in time O(n9 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
more » ... diameter of a point set and the bichromatic farthest pair. I. Introduction A dynamic geometric data structure is one that maintains the solution to some problem, defined on a geometric input such as a point set, as the input undergoes update operations such as insertions or deletions of single points. Dynamic algorithms have been studied for many geometric optimization problems, including closest pairs [7], [23], [25], [26], diameter [7], [26], width [4], convex hulls [15], [22], linear programming [2], [9], [18], smallest k-gons [6], [11], and minimum spanning trees (MSTs) [8]. Many of these algorithms suffer under a restriction that if deletions are allowed at all, they may only occur at certain prespecified times--the algorithms are not fully dynamic. A number of other papers have considered dynamic computational geometry problems under an average-case model that assumes that among a given set of points each point is equally likely to be inserted or deleted next [i0], [19]-[21], [24]. However, we are interested here in worst-case bounds.
doi:10.1007/bf02574030 fatcat:4spdvj3vjrhobjgqrqtjwqss4q