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# A Unified Framework for Hopsets

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LIPIcs.ESA.2022.81.pdf
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## Acknowledgements

We want to acknowledge the review of the anonymous reviewers from SODA 2022. This review drew our attention for some improvements that we applied in the paper.

## Cite As

Ofer Neiman and Idan Shabat. A Unified Framework for Hopsets. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 81:1-81:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.81

## Abstract

Given an undirected graph G = (V,E), an (α,β)-hopset is a graph H = (V,E'), so that adding its edges to G guarantees every pair has an α-approximate shortest path that has at most β edges (hops), that is, d_G(u,v) ≤ d_{G∪H}^(β)(u,v) ≤ α⋅ d_G(u,v). Given the usefulness of hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter α. In this work we devise a single algorithm that can attain all state-of-the-art hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm. In [Ben-Levy and Parter, 2020], given a parameter k, a (O(k^ε),O(k^{1-ε}))-hopset of size Õ(n^{1+1/k}) was shown for any n-vertex graph and parameter 0 < ε < 1, and they asked whether this result is best possible. We resolve this open problem, showing that any (α,β)-hopset of size O(n^{1+1/k}) must have α⋅β ≥ Ω(k).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Shortest paths
##### Keywords
• Graph Algorithms
• Shortest Paths
• Hopsets

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## References

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