Probabilistic Polynomials and Hamming Nearest Neighbors

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( 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 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) logn 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)/(log log n) 2 ). We apply the algorithm to computing pairs with maximum inner product, closest pair in ℓ 1 for vectors with bounded integer entries, and pairs with maximum Jaccard coefficients.

An Equivalence Class for Orthogonal Vectors

“…Prior work showed OV is equivalent to Dominating Pair 6 [Cha17] and other simple set problems [BCH16]; our results add several interesting new members into the equivalence class. All problems listed above were already known to be OV-hard [Wil05, AW15,Rub18]. Our main contribution here is to show that they can all be reduced back to OV.…”

Lower bounds for text indexing with mismatches and differences

“…Many of the problems below have been considered both in the field of algorithms on strings and in the field of computational geometry, and sometimes they are known under different Bichromatic closest pair with mismatches Ω(n 2−δ ) time ( [3,24]) Bichromatic closest pair with differences Ω(n 2−δ ) time ( [24], Corollary 6) Dictionary look-up with k mismatches Ω(n 1−δ ) query time ( [24]) Dictionary look-up with k differences Ω(n 1−δ ) query time ([24], Corollary 7)…”

“…For any δ > 0, there exists m = m(δ) such that SAT on m-CNF formulas with n variables cannot be solved in time O(2 (1−δ)n ). [3] showed that if there is δ > 0 such that for all constant c > 0, the bichromatic closest pair with mismatches problem, with d = c log n, can be solved in O(n 2−δ ) time, then SETH is false. By a standard reduction, this implies that no algorithm can process a dictionary of n strings of length d in polynomial time, and subsequently answer dictionary look-up queries with k = Θ(d) mismatches in O(n 1−δ ) time.…”

Hashing-Based-Estimators for Kernel Density in High Dimensions

“…Batch Setting. The study of the offline or batch setting for Nearest Neighbor Search has received renewed interest over the past years and has brought a wealth of techniques into light [40,2]. Given the connection between KDE and the Nearest Neighbor Search problem, it would be of practical and theoretical interest to design data-structures that offer provable speedups for the offline setting of the KDE problem.…”