On Farthest-Point Information in Networks

Consider the continuum of points along the edges of a network, an embedded undirected graph with positive edge weights. Distance between these points can be measured as shortest path distance along the edges of the network. We introduce two new concepts to capture farthest-point information in this metric space. The first, eccentricity diagrams, are used to encode the distance towards farthest points for any point on the network compactly. With this, we can solve the minimum eccentricity feed-link problem, i.e., the problem to extend a network by one new point minimizing the largest network distance towards the new point. The second, network farthest-point diagrams, provide an implicit description of the sets of farthest points. A network farthest-point diagram is, in principle, a compressed farthest-point network Voronoi link diagram generated by the entire continuum of uncountably many points on the network at hand. We provide construction algorithms for data structures that allow for queries for the distance to farthest points as well as their location from any point on a network in optimal time. Thus, we establish first bounds on construction times and storage requirements of such data structures.