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Lower Bounds on Sparse Spanners, Emulators, and Diameter-reducing shortcuts

Authors Shang-En Huang, Seth Pettie

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Shang-En Huang
  • University of Michigan, USA
Seth Pettie
  • University of Michigan, USA

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Shang-En Huang and Seth Pettie. Lower Bounds on Sparse Spanners, Emulators, and Diameter-reducing shortcuts. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 26:1-26:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.SWAT.2018.26

Abstract

We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We show that any O(n)-size shortcut set cannot bring the diameter below Omega(n^{1/6}), and that any O(m)-size shortcut set cannot bring it below Omega(n^{1/11}). These improve Hesse's [Hesse, 2003] lower bound of Omega(n^{1/17}). By combining these constructions with Abboud and Bodwin's [Abboud and Bodwin, 2017] edge-splitting technique, we get additive stretch lower bounds of +Omega(n^{1/13}) for O(n)-size spanners and +Omega(n^{1/18}) for O(n)-size emulators. These improve Abboud and Bodwin's +Omega(n^{1/22}) lower bounds.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
Keywords
  • additive spanners
  • emulators
  • shortcutting directed graphs

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References

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