Abstract
We introduce the concept of local spanners for planar point sets with respect to a family of regions, and prove the existence of local spanners of small size for some families. For a geometric graph G on a point set \(P\) and a region R belonging to a family \({\mathcal {R}}\), we define \(G \cap R\) to be the part of the graph G that is inside R (or is induced by R). A local t-spanner w.r.t \({\mathcal {R}}\) is a geometric graph G on \(P\) such that for any region \(R \in {\mathcal {R}}\), the graph \(G\cap R\) is a t-spanner for \(K(P) \cap R\), where \(K(P)\) is the complete geometric graph on P. For any set \(P\) of n points and any constant \(\varepsilon > 0\), we prove that \(P\) admits local \((1 + \varepsilon )\)-spanners of sizes \(O(n \log ^{6} n)\) and \(O(n \log n)\) w.r.t axis-parallel squares and vertical slabs, respectively. If adding Steiner points is allowed, then local \((1 + \varepsilon )\)-spanners with O(n) edges and \(O(n \log ^2 n)\) edges can be obtained for axis-parallel squares and disks using O(n) Steiner points, respectively.
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Abam, M.A., Borouny, M.S. Local Geometric Spanners. Algorithmica 83, 3629–3648 (2021). https://doi.org/10.1007/s00453-021-00860-5
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DOI: https://doi.org/10.1007/s00453-021-00860-5