Oriented Spanners
Kevin Buchin, Joachim Gudmundsson, Antonia Kalb, Aleksandr Popov, Carolin Rehs, Andr\'e van Renssen, Sampson Wong

TL;DR
This paper studies oriented spanners in Euclidean spaces, focusing on computing sparse graphs with small dilation, providing algorithms for one-dimensional sets and results for convex two-dimensional sets.
Contribution
It introduces algorithms for constructing oriented spanners with minimal dilation in one-dimensional and convex two-dimensional point sets, including a dynamic program and a greedy approach.
Findings
Dynamic programming computes minimum dilation graphs in O(n^7) time for 1D.
Greedy algorithm produces a 5-spanner in O(n log n) time.
For convex sets, greedy triangulation yields a plane oriented spanner with dilation at most 7.2 times the greedy triangulation's dilation.
Abstract
Given a point set in the Euclidean plane and a parameter , we define an \emph{oriented -spanner} as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest closed walk in through those points is at most a factor longer than the shortest cycle in the complete graph on . We investigate the problem of computing sparse graphs with small oriented dilation. As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a -spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present…
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