Oriented Spanners

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 $P$ in the Euclidean plane and a parameter $t$, we define an \emph{oriented $t$-spanner} $G$ as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest closed walk in $G$ through those points is at most a factor $t$ longer than the shortest cycle in the complete graph on $P$. 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 $1$-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 a dynamic program to compute the graph of minimum oriented dilation that runs in $\mathcal{O}(n^7)$ time for $n$ points, and a greedy algorithm that computes a $5$-spanner in $\mathcal{O}(n\log n)$ time. Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in a plane oriented $t$-spanner with $t=7.2 \cdot t_g$, where $t_g$ is an upper bound on the dilation of the greedy triangulation.