Computing Oriented Spanners and their Dilation

TL;DR

This work introduces the first sparse oriented constant-dilation spanner in Euclidean space, constructing an oriented (2+ε)-spanner with O(n) edges in O(n log n) time for constant-dimension Euclidean spaces by leveraging a WSPD and an approximate minimum-perimeter-triangle oracle. It proves that choosing an orientation to minimize oriented dilation on an undirected graph is NP-hard (even in the plane), and shows that computing the oriented dilation of a given graph in general metric spaces is APSP-hard, while offering a subcubic approximation algorithm based on WSPD and approximate shortest-path queries. The results highlight fundamental hardness barriers for orientation optimization and dilation computation, and provide practical algorithms for sparse oriented spanners and approximate dilation estimation. The work also outlines open problems, including improving the constant (below 2), achieving plane-oriented O(1) spanners for general 2D point sets, and refining shortest-path approximations to further enhance dilation estimates.

Abstract

Given a point set $P$ in a metric space and a real number $t \geq 1$, an \emph{oriented $t$-spanner} is an oriented graph $\overrightarrow{G}=(P,\overrightarrow{E})$, where for every pair of distinct points $p$ and $q$ in $P$, the shortest oriented closed walk in $\overrightarrow{G}$ 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 \emph{oriented dilation} of a graph $\overrightarrow{G}$ is the minimum $t$ for which $\overrightarrow{G}$ is an oriented $t$-spanner. 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 $\mathbb{R}^d$, where $d$ is a constant, we construct an oriented $(2+\varepsilon)$-spanner with $\mathcal{O}(n)$ edges in $\mathcal{O}(n \log n)$ time and $\mathcal{O}(n)$ space. Our construction uses the well-separated pair decomposition and an algorithm that computes a $(1+\varepsilon)$-approximation of the minimum-perimeter triangle in $P$ containing two given query points in $\mathcal{O}(\log n)$ 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 orientation is already given, computing the 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 $\mathbb{R}^d$, where $d$ is a constant.

Figures (1)

• Figure 13: Graph constructed for $\varphi= (x_1\lor \neg x_2 \lor \neg x_3)\land (x_2\lor \neg x_3 \lor x_4)$. For visibility, oriented points are placed diagonally instead of vertically. Only for one point, all $K_{2,2}$-gadgets are indicated by green parallelograms. If the oriented point at the end of the wire from $x_1$ (blue) is -- as indicated -- oriented the same way as the tree of knowledge (red), this corresponds to setting it to true, resulting in an oriented closed walk in the clause gadget (purple) that gives a dilation smaller than 1.043.

Theorems & Definitions (13)

• Definition 1: oriented dilation
• Lemma 2: Arya, Mount, Netanyahu, Silverman and Wu AryaMountDS1998
• Lemma 3
• Remark
• Theorem 4
• Lemma 5
• Lemma 6
• Theorem 7
• Theorem 8: $\textsc{APSP} \leq_3 \textsc{MinimumTriangle}$
• Theorem 9: $\textsc{APSP} \leq_3 \textsc{Odil}$