Abstract
A map f: {0 , 1} n→ {0 , 1} n has localityt 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∈ {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 · 3 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∈ {0 , 1} n\ {0 n} can be computed either (1) in constant locality or (2) in constant time using standard operations on words of length Ω (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 Sn and by Morgenstern (J Comb Theory Ser B 62(1):44–62, 1994) for the special linear group SL(2,F2n).
Original language | English |
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Pages (from-to) | 225-244 |
Number of pages | 20 |
Journal | Computational Complexity |
Volume | 27 |
Issue number | 2 |
DOIs | |
State | Published - 1 Jun 2018 |
Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2017, Springer International Publishing AG.
Keywords
- Cayley graph
- Derandomization
- Expander graph
- Hardness amplification
- Locality