A map $${f : \{0,1\}^{n} \to \{0,1\}^{n}}$$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 $${x\in\{0,1\}^{n}}$$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 $${n=4\cdot3^{t}}$$n=4·3t, an explicit construction of bipartite Ramanujan graphs of degree 3 with 2n−1 nodes in each side such that the neighbors of a node $${x\in \{0,1\}^{n}{\setminus} \{0^{n}\}}$$x∈{0,1}n\{0n} can be computed either (1) in constant locality or (2) in constant time using standard operations on words of length $${\Omega(n)}$$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 $${S_{n}}$$Sn and by Morgenstern (J Comb Theory Ser B 62(1):44–62, 1994) for the special linear group SL$${(2,F_{2^{n}})}$$(2,F2n).

中文翻译：

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本地扩展器

映射 $${f : \{0,1\}^{n} \to \{0,1\}^{n}}$$f:{0,1}n→{0,1}n 有localityt 如果 f 的每个输出位仅取决于 t 个输入位。阿罗拉等人。（自动机、语言和编程讨论会，ICALP，2009 年）询问是否存在 2n 个节点上的有界扩展图，使得节点的邻居 $${x\in\{0,1\}^{n}} $$x∈{0,1}n 可以通过恒定位置的映射来计算。我们给出了这种具有局部性的图的显式构造。然后，我们给出了这种结构的三种应用：（1）具有恒定局部性的无损扩展器，（2）更有效地减少随机算法的错误，以及（3）更有效地对单向排列进行硬度放大。我们还给出，对于形式为 $${n=4\cdot3^{t}}$$n=4·3t 的 n，3 阶二分拉马努金图的显式构造，每边有 2n−1 个节点，使得节点的邻居 $${x\in \{0,1\}^{n}{\setminus} \{0^ {n}\}}$$x∈{0,1}n\{0n} 可以（1）在恒定位置或（2）在恒定时间内使用长度为 $${\Omega( n)}$$Ω(n)。我们的结果在 Cayley 扩展器图的黑盒时尚深度显式构造中使用，由 Kassabov (Invent Math 170(2):327–354, 2007) 用于对称群 $${S_{n}}$$Sn 和 Morgenstern (J Comb Theory Ser B 62(1):44–62, 1994) 用于特殊线性群 SL$${(2,F_{2^{n}})}$$(2,F2n)。