Oriented Spanners

TL;DR

The paper introduces oriented geometric spanners by defining oriented dilation and investigates how to build sparse oriented graphs with small dilation. It establishes NP-hardness for minimum dilation under edge-count constraints, analyzes one-dimensional cases with one-page and two-page book embeddings, and derives a linear-time greedy construction achieving a 5-spanner in 1D. In two dimensions, it proves hardness results, shows that orienting the greedy triangulation yields a plane constant-dilation spanner for convex point sets with a bound of $7.2\,t_g$, and demonstrates that common triangulations like Delaunay or minimum weight do not guarantee constant oriented dilation. The results highlight the nuanced interplay between planarity, orientation, and dilation, and outline several open avenues for efficient computation of minimum or near-minimum oriented spanners in higher dimensions.

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.

Figures (12)

• Figure 1: If $p$ and $p'$ are very close to each other and $p"$ is far away from both, any oriented graph will have arbitrarily high (directed) dilation.
• Figure 2: One-dimensional oriented $1$-spanner and its plane embedding in the Euclidean space
• Figure 3: Part of $G=(P,E)$ with $E=\{(p_i,p_{i+1}),(p_{j+2},p_j)\mid 1\leq i\leq n-1, 1\leq j\leq n-2\}$.
• Figure 4: $1$-PPB almost $1$-spanner
• Figure 5: All maximal $1$-PPB spanners for a one-dimensional set of $5$ points. The worst-case pair is highlighted.

Theorems & Definitions (14)

• Definition 1: oriented dilation
• Lemma 2
• Corollary 3: oriented $1$-spanner
• Corollary 5
• Proposition 6: two-page plane $2$-spanner
• Lemma 7
• Lemma 8
• Definition 10: blocker
• Theorem 11: one-page plane $5$-spanner
• Theorem 12: Optimal $1$-PPB