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Efficient Farthest-Point Queries in Two-Terminal Series-Parallel 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 weighted shortest path distance, called the network distance. Within this metric space, we study farthest points and farthest distances. We introduce a data structure supporting queries for the farthest distance and the farthest points on two-terminal series-parallel networks. This data structure supports farthest-point queries in $O(k + \log n)$ time after $O(n \log p)$ construction time, where $k$ is the number of farthest points, $n$ is the size of the network, and $p$ parallel operations are required to generate the network.