Abstract
We show how to compute any symmetric Boolean function on n variables over any field (as well as the integers) with a probabilistic polynomial of degree O(sqrt(n*log(1/eps)) and error at most eps. The degree dependence on n and eps is optimal, matching a lower bound of Razborov (1987) and Smolensky (1987) for the MAJORITY function. The proof is constructive: a low-degree polynomial can be efficiently sampled from the distribution.
This polynomial construction is combined with other algebraic ideas to give the first subquadratic time algorithm for computing a (worst-case) batch of Hamming distances in superlogarithmic dimensions, exactly. To illustrate, let c(n):N→N. Suppose we are given a database D of n vectors in {0,1}^(c(n)*log n) and a collection of n query vectors Q in the same dimension. For all u∈Q, we wish to compute a v∈D with minimum Hamming distance from u. We solve this problem in n^(2−1/O(c(n)*log^2 c(n))) randomized time. Hence, the problem is in "truly subquadratic" time for O(log n) dimensions, and in subquadratic time for d=o((log^2 n)/(loglog n)^2). We apply the algorithm to computing pairs with maximum inner product, closest pair in l1 for vectors with bounded integer entries, and pairs with maximum Jaccard coefficients.