A sampling lower bound for permutations

A map f : [n]` → [n]n has locality d if each output symbol in [n] = {1, 2, . . . , n} depends only on d of the ` input symbols in [n]. We show that the output distribution of a d-local map has statistical distance at least 1−2 · exp(−n/ logcd n) from a uniform permutation of [n]. This seems to be the first lower bound for the well-studied problem of generating permutations. Because poly(n)-size AC0 circuits can sample almost uniform permutations, the result also separates AC0 sampling from local. As an application, we prove that any cell-probe data structure for storing permutations π of n elements such that π(i) can be retrieved with d non-adaptive probes must use space ≥ log2 n! + n/ log d n . This is arguably the first lower bound for a natural data structure problem that is obtained from a sampling lower bound.