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# Relaxed Spanners for Directed Disk Graphs

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LIPIcs.STACS.2010.2489.pdf
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## Cite As

David Peleg and Liam Roditty. Relaxed Spanners for Directed Disk Graphs. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 609-620, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)
https://doi.org/10.4230/LIPIcs.STACS.2010.2489

## 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.
##### Keywords
• Spanners
• directed graphs

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