SAMPLING LOWER BOUNDS: BOOLEAN AVERAGE-CASE AND PERMUTATIONS

We show that for every small AC(0) circuit C : {0, 1}(l) -> {0, 1}(m) there exists a multiset S of 2(m-m Omega(1)) restrictions that preserve the output distribution of C and, moreover, polarize minentropy: the restriction of C to any r is an element of S either is constant or has polynomial min-entropy. This structural result is then applied to exhibit an explicit boolean function h : {0,1}(n) -> {0, 1} such that for every small AC(0) circuit C : {0, 1}(l) -> {0, 1}(n+1) the output distribution of C for a uniform input has statistical distance exponentially close to 1/2 from the distribution (U, h(U)) for U uniform in {0, 1}(n). Previous such "sampling lower bounds" either gave exponentially small statistical distance or applied to functions h with large output length. We also show that the output distribution of a d-local map f : [n](l) -> [n](n) for a uniform input has statistical distance at least 1- 2.exp(-n/log(exp(O(d))) n) from a uniform permutation of [n]. Here d-local means that each output symbol in [n] = {1, 2, ..., n} depends only on d of the .e input symbols in [n]. This separates AC(0) sampling from local, because small AC(0) circuits can sample almost uniform permutations. As an application, we prove that any cell-probe data structure for storing permutations pi of n elements such that pi(i) can be retrieved with d nonadaptive probes must use space >= log(2) n! n/log(exp(O)(d)))n.