1 research outputs found
Probabilistic Polynomials and Hamming Nearest Neighbors
We show how to compute any symmetric Boolean function on variables over
any field (as well as the integers) with a probabilistic polynomial of degree
and error at most . The degree
dependence on and 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
. Suppose we are given a database
of vectors in and a collection of query vectors
in the same dimension. For all , we wish to compute a
with minimum Hamming distance from . We solve this problem in randomized time. Hence, the problem is in "truly subquadratic"
time for dimensions, and in subquadratic time for . We apply the algorithm to computing pairs with maximum
inner product, closest pair in for vectors with bounded integer
entries, and pairs with maximum Jaccard coefficients.Comment: 16 pages. To appear in 56th Annual IEEE Symposium on Foundations of
Computer Science (FOCS 2015