Explicit Expanding Expanders

Michael Dinitz*, Michael Schapira, Asaf Valadarsky

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

Deterministic constructions of expander graphs have been an important topic of research in computer science and mathematics, with many well-studied constructions of infinite families of expanders. In some applications, though, an infinite family is not enough: we need expanders which are “close” to each other. We study the following question: Construct an an infinite sequence of expanders G0, G1, … , such that for every two consecutive graphs Gi and Gi + 1, Gi + 1 can be obtained from Gi by adding a single vertex and inserting/removing a small number of edges, which we call the expansion cost of transitioning from Gi to Gi + 1. This question is very natural, e.g., in the context of datacenter networks, where the vertices represent racks of servers, and the expansion cost captures the amount of rewiring needed when adding another rack to the network. We present an explicit construction of d-regular expanders with expansion cost at most 5d2, for any d≥ 6. Our construction leverages the notion of a “2-lift” of a graph. This operation was first analyzed by Bilu and Linial (Combinatorica 26(5): 495–519, 2006), who repeatedly applied 2-lifts to construct an infinite family of expanders which double in size from one expander to the next. Our construction can be viewed as a way to “interpolate” between Bilu–Linial expanders with low expansion cost while preserving good edge expansion throughout. While our main motivation is centralized (datacenter networks), we also get the best-known distributed expander construction in the “self-healing” model.

Original languageAmerican English
Pages (from-to)1225-1245
Number of pages21
JournalAlgorithmica
Volume78
Issue number4
DOIs
StatePublished - 1 Aug 2017

Bibliographical note

Publisher Copyright:
© 2016, Springer Science+Business Media New York.

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