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 its own occurrence region instead of an exact location. We prove that the minimum diameter of a spanning tree of n points each of which is selected from its respective disk (or axis-parallel square) is polynomial-time computable, contrasting with the fact that minimizing the cost of a spanning tree, i.e. the sum of its edge lengths, for imprecise points is known to be NP-hard. This difference is very interesting since for exact points minimizing the cost seems faster than minimizing the diameter [\(O(n\log n)\) versus almost cubic running time]. As the first work regarding the minimum diameter of a spanning tree for imprecise points, our main objective is to investigate essential properties that distinguish the problem from NP-hard instead of minimizing the running time [\(O(n^9)\) for general disks or \(O(n^6)\) for unit ones]. Our algorithm mainly utilizes farthest-site Voronoi diagrams for disks and axis-parallel squares, and these diagrams would have their own merits.