Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T17:29:29.942Z Has data issue: false hasContentIssue false
  • English
  • Français

An Elementary Construction of Constant-Degree Expanders

Published online by Cambridge University Press:  01 May 2008

NOGA ALON ,
ODED SCHWARTZ  and
ASAF SHAPIRA
NOGA ALON
Affiliation:
Schools of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: [email protected])
ODED SCHWARTZ
Affiliation:
School of Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: [email protected], [email protected])
ASAF SHAPIRA
Affiliation:
School of Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We describe a short and easy-to-analyse construction of constant-degree expanders. The construction relies on the replacement product, applied by Reingold, Vadhan and Wigderson (2002) to give an iterative construction of bounded-degree expanders. Here we give a simpler construction, which applies the replacement product (only twice!) to turn the Cayley expanders of Alon and Roichman (1994), whose degree is polylog n, into constant-degree expanders. This enables us to prove the required expansion using a simple new combinatorial analysis of the replacement product (instead of the spectral analysis used by Reingold, Vadhan and Wigderson).

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 3 , May 2008 , pp. 319 - 327
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ajtai, M. (1994) Recursive construction for 3-regular expanders. Combinatorica 14 379–416.CrossRefGoogle Scholar
[2]Alon, N. (1986) Eigenvalues and expanders. Combinatorica 6 83–96.CrossRefGoogle Scholar
[3]Alon, N. and Milman, V. D. (1984) Eigenvalues, expanders and superconcentrators. In Proc. 25th Annual Symp. on Foundations of Computer Science (FOCS), Singer Island, Florida, IEEE, pp. 320–322. Also: λ1, isoperimetric inequalities for graphs and superconcentrators. J. Combin. Theory Ser. B 38 73–88 (1985).Google Scholar
[4]Alon, N. and Roichman, Y. (1994) Random Cayley graphs and expanders. Random Struct. Alg. 5 271–284.CrossRefGoogle Scholar
[5]Bilu, Y. and Linial, N. (2006) Lifts, discrepancy and nearly optimal spectral gaps. Combinatorica 26 495–519.CrossRefGoogle Scholar
[6]Dodziuk, J. (1984) Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Amer. Math. Soc. 284 787–794.CrossRefGoogle Scholar
[7]Gabber, O. and Galil, Z. (1981) Explicit constructions of linear-sized superconcentrators. J. Comput. Syst. Sci. 22 407–420.CrossRefGoogle Scholar
[8]Gromov, M. (1983) Filling Riemannian manifolds. J. Differential Geometry 18 1–147.CrossRefGoogle Scholar
[9]Hoory, S., Linial, N. and Wigderson, A. (2006) Expander graphs and their applications. Bull. Amer. Math. Soc. 43 439–561.CrossRefGoogle Scholar
[10]Lubotzky, A., Phillips, R. and Sarnak, P. (1988) Ramanujan graphs. Combinatorica 8 261–277.CrossRefGoogle Scholar
[11]Margulis, G. (1973) Explicit constructions of expanders. Problemy Peredacl Informacii 9 71–80.Google Scholar
[12]Margulis, G. A. (1988) Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators. Problems of Information Transmission 24 39–46.Google Scholar
[13]Pinsker, M. (1973) On the complexity of a concentrator. In 7th Annual Teletraffic Conference, pp. 1–4.Google Scholar
[14]Reingold, O., Vadhan, S. and Wigderson, A. (2002) Entropy waves, the zig-zag graph product, and new constant-degree expanders. Ann. of Math. 155 157–187.CrossRefGoogle Scholar