Euclidean Chains and Their Shortcuts

Abstract

A Euclidean graph is a straight line embedding of a graph in the plane such that there is no crossing between any pair of edges and the length of an edge is the Euclidean distance between its two endpoints. A Euclidean chain C = (x ₁,x ₂,…, x _n ) is a planar straight line graph with vertex set {x ₁,x ₂,…, x _n } and edge set {x _i x _i + 1: 1 ≤ i ≤ n − 1}. Given a Euclidean chain C in the plane, we study the problem of finding a pair of points on C such that the new Euclidean graph obtained from C by adding a straight line segment (called shortcut) between this pair of points has the minimum diameter. We also study the ratio between the diameter of the new graph and the length of C. We give necessary and sufficient conditions for optimal shortcuts. We present three approximation algorithms for computing the optimal shortcuts of chains. One of them is a fully polynomial-time approximation scheme (FPTAS). We introduce two types of chains, strongly monotonic chain and simple chain. We provide properties for these two types of chains and their shortcuts.