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Unformatted text preview: Introduction to expander graphs Michael A. Nielsen 1, * 1 School of Physical Sciences, The University of Queensland, Brisbane, Queensland 4072, Australia (Dated: June 22, 2005) I. INTRODUCTION TO EXPANDERS Expander graphs are one of the deepest tools of theoret ical computer science and discrete mathematics, popping up in all sorts of contexts since their introduction in the 1970s. Heres a list of some of the things that expander graphs can be used to do. Dont worry if not all the items on the list make sense: the main thing to take away is the sheer range of areas in which expanders can be applied. Reduce the need for randomness: That is, ex panders can be used to reduce the number of ran dom bits needed to make a probabilistic algorithm work with some desired probability. Find good errorcorrecting codes: Expanders can be used to construct errorcorrecting codes for pro tecting information against noise. Most astonish ingly for information theorists, expanders can be used to find errorcorrecting codes which are effi ciently encodable and decodable, with a nonzero rate of transmission. This is astonishing because finding codes with these properties was one of the holy grails of coding theory for decades after Shan nons pioneering work on coding and information theory back in the 1940s. A new proof of PCP: One of the deepest results in computer science is the PCP theorem, which tells us that for all languages L in NP there is a randomized polyonomialtime proof verifier which need only check a constant number of bits in a pur ported proof that x L or x 6 L , in order to de termine (with high probability of success) whether the proof is correct or not. This result, originally established in the earlier 1990s, has recently been given a new proof based on expanders. Whats remarkable is that none of the topics on this list appear to be related, a priori , to any of the other topics, nor do they appear to be related to graph the ory. Expander graphs are one of these powerful unifying tools, surprisingly common in science, that can be used to gain insight into an an astonishing range of apparently disparate phenomena. Im not an expert on expanders. Im writing these notes to help myself (and hopefully others) to under stand a little bit about expanders and how they can be * nielsen@physics.uq.edu.au and www.qinfo.org/people/nielsen applied. Im not learning about expanders with any spe cific intended application in mind, but rather because they seem to behind some of the deepest insights weve had in recent years into information and computation. What is an expander graph? Informally, its a graph G = ( V,E ) in which every subset S of vertices expands quickly, in the sense that it is connected to many vertices in the set S of complementary vertices. Making this def inition precise is the main goal of the remainder of this section....
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