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# Local Spanners Revisited

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LIPIcs.SWAT.2024.2.pdf
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## Acknowledgements

The authors thanks Mohammad Abam for providing us with an early version of [Mohammad Ali Abam and Mohammad Sadegh Borouny, 2021]. The authors also thank the reviewers for the numerous detailed comments.

## Cite As

Stav Ashur and Sariel Har-Peled. Local Spanners Revisited. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SWAT.2024.2

## Abstract

For a set P ⊆ ℝ² of points and a family ℱ of regions, a local t-spanner of P is a sparse graph G over P, such that for any region r ∈ ℱ the subgraph restricted to r, denoted by G ∩ r, is a t-spanner for all the points of r ∩ P. We present algorithms for the construction of local spanners with respect to several families of regions such as homothets of a convex region. Unfortunately, the number of edges in the resulting graph depends logarithmically on the spread of the input point set. We prove that this dependency cannot be removed, thus settling an open problem raised by Abam and Borouny. We also show improved constructions (with no dependency on the spread) of local spanners for fat triangles, and regular k-gons. In particular, this improves over the known construction for axis-parallel squares. We also study notions of weaker local spanners where one is allowed to shrink the region a "bit". Surprisingly, we show a near linear-size construction of a weak spanner for axis-parallel rectangles, where the shrinkage is multiplicative. Any spanner is a weak local spanner if the shrinking is proportional to the diameter of the region.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• Geometric graphs
• Fault-tolerant spanners

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## References

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