Journal of the ACM (JACM)
Combinatorica
Better expanders and superconcentrators
Journal of Algorithms
Sorting and selecting in rounds
SIAM Journal on Computing
Expanders obtained from affine transformations
Combinatorica - Theory of Computing
Expanders, randomness, or time versus space
Journal of Computer and System Sciences - Structure in Complexity Theory Conference, June 2-5, 1986
Sorting in c log n parallel steps
Combinatorica
Small-bias probability spaces: efficient constructions and applications
SIAM Journal on Computing
Existence and explicit constructions of q+1 regular Ramanujan graphs for every prime power q
Journal of Combinatorial Theory Series B
Pseudorandomness for network algorithms
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
P = BPP if E requires exponential circuits: derandomizing the XOR lemma
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Randomness conductors and constant-degree lossless expanders
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Expanders from symmetric codes
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Randomness-efficient low degree tests and short PCPs via epsilon-biased sets
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Semi-Direct Product in Groups and Zig-Zag Product in Graphs: Connections and Applications
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
IEEE Transactions on Information Theory - Part 1
Linear-time encodable and decodable error-correcting codes
IEEE Transactions on Information Theory - Part 1
Simple permutations mix even better
Random Structures & Algorithms
Symmetric LDPC codes and local testing
Property testing
Symmetric LDPC codes and local testing
Property testing
Quantum expanders from any classical Cayley graph expander
Quantum Information & Computation
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We assume that for some fixed large enough integer d, the symmetric group Sd can be generated as an expander using d1/30 generators. Under this assumption, we explicitly construct an infinite family of groups Gn, and explicit sets of generators Yn ⊂ Gn, such that all generating sets have bounded size (at most d1/7), and the associated Cayley graphs are all expanders. The groups Gn above are very simple, and completely different from previous known examples of expanding groups. Indeed, Gn is (essentially) all symmetries of the d-regular tree of depth n. The proof is completely elementary, using only simple combinatorics and linear algebra. The recursive structure of the groups Gn (iterated wreath products of the alternating group Ad) allows for an inductive proof of expansion, using the group theoretic analogue [4] of the zig-zag graph product of [38]. The explicit construction of the generating sets Yn uses an efficient algorithm for solving certain equations over these groups, which relies on the work of [33] on the commutator width of perfect groups.We stress that our assumption above on weak expansion in the symmetric group is an open problem. We conjecture that it holds for all d. We discuss known results related to its likelihood in the paper.