Efficient Farthest-Point Queries in Two-Terminal Series-Parallel Networks

TL;DR

This paper presents a data structure for efficiently answering farthest-point queries in two-terminal series-parallel networks, enabling quick retrieval of the farthest points and distances after a specialized construction process.

Contribution

It introduces a novel data structure that supports fast farthest-point and distance queries specifically tailored for two-terminal series-parallel networks.

Findings

Query time is $O(k + ext{log} n)$ after $O(n ext{log} p)$ construction.

Supports efficient computation of farthest points and distances in series-parallel networks.

Applicable to networks with multiple parallel operations during construction.

Abstract

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.