On Two Problems Regarding Farthest Distances in Continuous Networks

Consider the continuum of points along the edges of a network, i.e., a connected, undirected graph with positive edge weights. We measure the distance between these points in terms of the network distance, i.e., the weighted shortest path distance. The continuous diameter of a network is the largest network distance between any two points on the network. We study two intertwined problems within this metric space: The first problem is to minimize the continuous diameter of a geometric network by introducing one or more shortcuts that may connect any two points along the network. The second problem is to develop efficient data structures that support queries for the farthest points from a query point along a network.