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DOI: 10.4230/LIPIcs.ESA.2018.71

Large Low-Diameter Graphs are Good Expanders

Authors: Michael Dinitz, Michael Schapira, and Gal Shahaf

Published in: LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Abstract

We revisit the classical question of the relationship between the diameter of a graph and its expansion properties. One direction is well understood: expander graphs exhibit essentially the lowest possible diameter. We focus on the reverse direction, showing that "sufficiently large" graphs of fixed diameter and degree must be "good" expanders. We prove this statement for various definitions of "sufficiently large" (multiplicative/additive factor from the largest possible size), for different forms of expansion (edge, vertex, and spectral expansion), and for both directed and undirected graphs. A recurring theme is that the lower the diameter of the graph and (more importantly) the larger its size, the better the expansion guarantees. Aside from inherent theoretical interest, our motivation stems from the domain of network design. Both low-diameter networks and expanders are prominent approaches to designing high-performance networks in parallel computing, HPC, datacenter networking, and beyond. Our results establish that these two approaches are, in fact, inextricably intertwined. We leave the reader with many intriguing questions for future research.

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Michael Dinitz, Michael Schapira, and Gal Shahaf. Large Low-Diameter Graphs are Good Expanders. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 71:1-71:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dinitz_et_al:LIPIcs.ESA.2018.71,
  author =	{Dinitz, Michael and Schapira, Michael and Shahaf, Gal},
  title =	{{Large Low-Diameter Graphs are Good Expanders}},
  booktitle =	{26th Annual European Symposium on Algorithms (ESA 2018)},
  pages =	{71:1--71:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-081-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{112},
  editor =	{Azar, Yossi and Bast, Hannah and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.71},
  URN =		{urn:nbn:de:0030-drops-95348},
  doi =		{10.4230/LIPIcs.ESA.2018.71},
  annote =	{Keywords: Network design, Expander graphs, Spectral graph theory}
}

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DOI: 10.4230/LIPIcs.SNAPL.2015.209

New Directions for Network Verification

Authors: Aurojit Panda, Katerina Argyraki, Mooly Sagiv, Michael Schapira, and Scott Shenker

Published in: LIPIcs, Volume 32, 1st Summit on Advances in Programming Languages (SNAPL 2015)

Abstract

Network verification has recently gained popularity in the programming languages and verification community. Much of the recent work in this area has focused on verifying the behavior of simple networks, whose actions are dictated by static, immutable rules configured ahead of time. However, in reality, modern networks contain a variety of middleboxes, whose behavior is affected both by their configuration and by mutable state updated in response to packets received by them. In this position paper we critically review recent progress on network verification, propose some next steps towards a more complete form of network verification, dispel some myths about networks, provide a more formal description of our approach, and end with a discussion of the formal questions posed to this community by the network verification agenda.

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Aurojit Panda, Katerina Argyraki, Mooly Sagiv, Michael Schapira, and Scott Shenker. New Directions for Network Verification. In 1st Summit on Advances in Programming Languages (SNAPL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 32, pp. 209-220, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{panda_et_al:LIPIcs.SNAPL.2015.209,
  author =	{Panda, Aurojit and Argyraki, Katerina and Sagiv, Mooly and Schapira, Michael and Shenker, Scott},
  title =	{{New Directions for Network Verification}},
  booktitle =	{1st Summit on Advances in Programming Languages (SNAPL 2015)},
  pages =	{209--220},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-80-4},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{32},
  editor =	{Ball, Thomas and Bodík, Rastislav and Krishnamurthi, Shriram and Lerner, Benjamin S. and Morriset, Greg},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SNAPL.2015.209},
  URN =		{urn:nbn:de:0030-drops-50278},
  doi =		{10.4230/LIPIcs.SNAPL.2015.209},
  annote =	{Keywords: Middleboxes, Network Verification, Mutable Dataplane}
}

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Documents authored by Schapira, Michael

Large Low-Diameter Graphs are Good Expanders

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New Directions for Network Verification

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