Oriented spanners were presented at ESA'23 as an extension of the well-researched geometric spanners: Given a set P of points in a metric space and an oriented graph G, the oriented dilation of two points p,q ∈ P is the length of the shortest closed walk in G containing p and q divided by the minimum perimeter triangle of p and q. G is called a t-spanner, if the maximum dilation over all pairs of points in P is at most t. This paper presents the first constructions of sparse oriented spanners for metric spaces beyond the Euclidean space. Given an orientation of the complete graph (i.e. a tournament) with dilation t on n points that satisfies an additional short-cycle property, we show how to extract a (t+ε)-spanner with 𝒪(k) edges in 𝒪(kn²+T(n)) time, for any metric space admitting a well-separated pair decomposition with k pairs computable in T(n) time. We supplement this with an improved construction of tournaments for metric point sets, obtaining dilation 5/3. This improves the previous bound of 2 and approaches the lower bound of 1.5. Combined, for n points in a metric space with constant doubling dimension d, this yields a (5/3 + ε)-spanner with (1/ε)^{𝒪(d)}n edges computable in (1/ε)^𝒪(d) n³ time using 𝒪(n²) space. This improves the dilation over the (2+ε)-spanner for Euclidean point sets presented at SoCG’25 while applying to more general metric spaces. Moreover, we generalize the known (2+ε)-spanner to doubling spaces. In particular, an oriented (2+ε)-spanner with 𝒪(ε^{-d} n) edges can be constructed in (1/ε)^𝒪(d) n log n time using 𝒪(ε^{-d} n) space. Since the oriented dilation can be dominated by one pair of points, we also consider the oriented average dilation, which is the sum over the oriented dilation of all pairs of points divided by the number of pairs. While oriented (1+ε)-spanners do not exist for every point set, we present an algorithm that computes a spanner with average dilation 1+ε for point sets in a metric space of constant doubling dimension d: More concretely, our algorithm computes an oriented spanner with average dilation at most 1 + 𝒪(1/s) + s^𝒪(d)/n with s^𝒪(d) n edges in s^𝒪(d) n log n time using s^𝒪(d) n space, where s is any sufficiently large number that may depend on n.

Abstract Given a point set P in the Euclidean plane and a parameter t , we define an 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.

Given a point set P in a metric space and a real number t ≥ 1, an oriented t-spanner is an oriented graph G = (P, E), where for every pair of distinct points p and q in P, the shortest oriented closed walk in G that contains p and q is at most a factor t longer than the perimeter of the smallest triangle in P containing p and q. The oriented dilation of a graph G is the minimum t for which G is an oriented t-spanner. For arbitrary point sets of size n in ℝ^d, where d ≥ 2 is a constant, the only known oriented spanner construction is an oriented 2-spanner with binom(n,2) edges. Moreover, there exists a set P of four points in the plane, for which the oriented dilation is larger than 1.46, for any oriented graph on P. We present the first algorithm that computes, in Euclidean space, a sparse oriented spanner whose oriented dilation is bounded by a constant. More specifically, for any set of n points in ℝ^d, where d is a constant, we construct an oriented (2+ε)-spanner with 𝒪(n) edges in 𝒪(n log n) time and 𝒪(n) space. Our construction uses the well-separated pair decomposition and an algorithm that computes a (1+ε)-approximation of the minimum-perimeter triangle in P containing two given query points in 𝒪(log n) time. While our algorithm is based on first computing a suitable undirected graph and then orienting it, we show that, in general, computing the orientation of an undirected graph that minimises its oriented dilation is NP-hard, even for point sets in the Euclidean plane. We further prove that even if the oriented graph is already given, computing its oriented dilation is APSP-hard for points in a general metric space. We complement this result with an algorithm that approximates the oriented dilation of a given graph in subcubic time for point sets in ℝ^d, where d is a constant.

We study the problem of computing the diameter and the mean distance of a continuous graph, i.e., a connected graph where all points along the edges, instead of only the vertices, must be taken into account. It is known that for continuous graphs with m edges these values can be computed in roughly O(m²) time. In this paper, we use geometric techniques to obtain subquadratic time algorithms to compute the diameter and the mean distance of a continuous graph for two well-established classes of sparse graphs. We show that the diameter and the mean distance of a continuous graph of treewidth at most k can be computed in O(n log^O(k) n) time, where n is the number of vertices in the graph. We also show that computing the diameter and mean distance of a continuous planar graph with n vertices and F faces takes O(n F log n) time.

Given a point set P in the Euclidean plane and a parameter t, we define an oriented t-spanner as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest cycle in G through those points is at most a factor t longer than the shortest oriented cycle in the complete bi-directed graph. 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 𝒪(n⁸) time for n points, and a greedy algorithm that computes a 5-spanner in 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 an oriented 𝒪(1)-spanner.