Local Expanders

Abstract

A map ${f : \{0,1\}^{n} \to \{0,1\}^{n}}$ has locality t if every output bit of f depends only on t input bits. Arora et al. (Colloquium on automata, languages and programming, ICALP, 2009) asked if there exist bounded-degree expander graphs on 2ⁿ nodes such that the neighbors of a node ${x\in\{0,1\}^{n}}$ can be computed by maps of constant locality. We give an explicit construction of such graphs with locality one. We then give three applications of this construction: (1) lossless expanders with constant locality, (2) more efficient error reduction for randomized algorithms, and (3) more efficient hardness amplification of one-way permutations. We also give, for n of the form ${n=4\cdot3^{t}}$, an explicit construction of bipartite Ramanujan graphs of degree 3 with 2ⁿ−1 nodes in each side such that the neighbors of a node ${x\in \{0,1\}^{n}{\setminus} \{0^{n}\}}$ can be computed either (1) in constant locality or (2) in constant time using standard operations on words of length ${\Omega(n)}$. Our results use in black-box fashion deep explicit constructions of Cayley expander graphs, by Kassabov (Invent Math 170(2):327–354, 2007) for the symmetric group ${S_{n}}$ and by Morgenstern (J Comb Theory Ser B 62(1):44–62, 1994) for the special linear group SL${(2,F_{2^{n}})}$.