Minimizing the Diameter of a Spanning Tree for Imprecise Points

Abstract

We study the diameter of a spanning tree, i.e., the length of its longest simple path, under the imprecise points model, in which each point is assigned an own occurrence disk instead of an exact location. We prove that the minimum diameter of a spanning tree for n points each of which is selected from its occurrence disk can be computed in \(O(n^9)\) time for arbitrary disks and in \(O(n^6)\) time for unit ones. If the disks are disjoint, we improve the run-time respectively to \(O(n^8\log ^2n )\) and \(O(n^5)\). These results contrast with the fact that minimizing the sum of the edge lengths of a spanning tree for imprecise points is \(\mathsf {NP}\)-hard.