Relaxed spanners for directed disk graphs

TL;DR

This paper investigates the optimality of spanners in directed disk graphs within metrics of constant doubling dimension and shows that slight radius perturbations enable the construction of significantly smaller spanners.

Contribution

It proves the size bounds of existing spanners are essentially optimal and introduces a method to create smaller spanners by allowing minor radius perturbations.

Findings

Existing spanner size bounds are tight for constant doubling dimension metrics.

Allowing a small increase in radii yields much smaller spanners.

The proposed algorithm is simple and efficiently implementable.

Abstract

Let $(V,\delta)$ be a finite metric space, where $V$ is a set of $n$ points and $\delta$ is a distance function defined for these points. Assume that $(V,\delta)$ has a constant doubling dimension $d$ and assume that each point $p\in V$ has a disk of radius $r(p)$ around it. The disk graph that corresponds to $V$ and $r(\cdot)$ is a \emph{directed} graph $I(V,E,r)$, whose vertices are the points of $V$ and whose edge set includes a directed edge from $p$ to $q$ if $\delta(p,q)\leq r(p)$. In \cite{PeRo08} we presented an algorithm for constructing a $(1+\eps)$-spanner of size $O(n/\eps^d \log M)$, where $M$ is the maximal radius $r(p)$. The current paper presents two results. The first shows that the spanner of \cite{PeRo08} is essentially optimal, i.e., for metrics of constant doubling dimension it is not possible to guarantee a spanner whose size is independent of $M$. The second result shows that by slightly relaxing the requirements and allowing a small perturbation of the radius assignment, considerably better spanners can be constructed. In particular, we show that if it is allowed to use edges of the disk graph $I(V,E,r_{1+\eps})$, where $r_{1+\eps}(p) = (1+\eps)\cdot r(p)$ for every $p\in V$, then it is possible to get a $(1+\eps)$-spanner of size $O(n/\eps^d)$ for $I(V,E,r)$. Our algorithm is simple and can be implemented efficiently.