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# An Improved Random Shift Algorithm for Spanners and Low Diameter Decompositions

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LIPIcs.OPODIS.2021.16.pdf
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## Acknowledgements

The first author would like to thank Merav Parter for mentioning that there was still some room for improvement in the spanner construction.

## Cite As

Sebastian Forster, Martin Grösbacher, and Tijn de Vos. An Improved Random Shift Algorithm for Spanners and Low Diameter Decompositions. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.OPODIS.2021.16

## Abstract

Spanners have been shown to be a powerful tool in graph algorithms. Many spanner constructions use a certain type of clustering at their core, where each cluster has small diameter and there are relatively few spanner edges between clusters. In this paper, we provide a clustering algorithm that, given k ≥ 2, can be used to compute a spanner of stretch 2k-1 and expected size O(n^{1+1/k}) in k rounds in the CONGEST model. This improves upon the state of the art (by Elkin, and Neiman [TALG'19]) by making the bounds on both running time and stretch independent of the random choices of the algorithm, whereas they only hold with high probability in previous results. Spanners are used in certain synchronizers, thus our improvement directly carries over to such synchronizers. Furthermore, for keeping the total number of inter-cluster edges small in low diameter decompositions, our clustering algorithm provides the following guarantees. Given β ∈ (0,1], we compute a low diameter decomposition with diameter bound O((log n)/β) such that each edge e ∈ E is an inter-cluster edge with probability at most β⋅ w(e) in O((log n)/β) rounds in the CONGEST model. Again, this improves upon the state of the art (by Miller, Peng, and Xu [SPAA'13]) by making the bounds on both running time and diameter independent of the random choices of the algorithm, whereas they only hold with high probability in previous results.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Sparsification and spanners
• Theory of computation → Distributed algorithms
##### Keywords
• Spanner
• low diameter decomposition
• synchronizer
• distributed graph algorithms

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