Probabilistic Polynomials and Hamming Nearest Neighbors

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(√n log(1/ε))) and error at most ε. The degree dependence on n and ε 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 sub quadratic time algorithm for computing a (worst-case) batch of Hamming distances in super logarithmic 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 in Q, we wish to compute a v in D with minimum Hamming distance from u. We solve this problem in n(2-1/O(c(n) log2 c(n))) randomized time. Hence, the problem is in "truly sub quadratic" time for O(log n) dimensions, and in sub quadratic time for d = o((log2 n)/(log log 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.