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# Relaxed spanners for directed disk graphs

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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.
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https://hal.inria.fr/inria-00455800
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Submitted on : Thursday, February 11, 2010 - 11:34:01 AM
Last modification on : Thursday, October 15, 2020 - 2:42:03 PM
Long-term archiving on: : Friday, June 18, 2010 - 8:13:20 PM

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• HAL Id : inria-00455800, version 1

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David Peleg, Liam Roditty. Relaxed spanners for directed disk graphs. 27th International Symposium on Theoretical Aspects of Computer Science - STACS 2010, Inria Nancy Grand Est & Loria, Mar 2010, Nancy, France. pp.609-620. ⟨inria-00455800⟩

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