🤖 AI Summary
This paper studies the construction of sparse directed graphs on point sets in the Euclidean plane, aiming to minimize the *directed dilation*—the supremum ratio between the length of the shortest directed closed walk between any two points and the corresponding shortest cycle length in the complete graph. We introduce the notion of *directed t-spanners*, prove that computing the minimum directed dilation is NP-hard for planar point sets, and provide three algorithmic results: (i) an optimal O(n⁷) dynamic programming algorithm and an O(n log n) greedy 5-spanner algorithm for 1D point sets; (ii) for convex point sets, we establish for the first time that the greedy triangulation yields a plane directed (7.2·t₉)-spanner; and (iii) we unify computational geometry, graph theory, and book embedding theory to reveal deep connections between directed stretchability and planar embeddings. Our work delivers the first theoretical guarantees and efficient algorithmic frameworks for constructing sparse directed metric graphs.
📝 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 $mathcal{O}(nlog n)$ 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.
