Computing Oriented Spanners and their Dilation

TL;DR

This paper introduces an efficient algorithm for constructing sparse oriented spanners with bounded dilation in Euclidean space, and explores the computational complexity of optimizing graph orientation for minimal dilation.

Contribution

It presents the first polynomial-time algorithm for Euclidean oriented spanners with constant dilation and analyzes the NP-hardness of orientation optimization.

Findings

Constructed an ( n) edges oriented spanner with (n  log n) time.

Proved that finding an orientation minimizing dilation is NP-hard.

Provided an approximation algorithm for oriented dilation in general metric spaces.

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.