A New Algorithm for Finding Closest Pair of Vectors.

Given $n$ vectors $x_0, x_1, ldots, x_{n-1}$ in ${0,1}^{m}$, how to find two vectors whose pairwise Hamming distance is minimum? This problem is known as the Closest Pair Problem. If these vectors are generated uniformly at random except two of them are correlated with Pearson-correlation coefficient $rho$, then the problem is called the Light Bulb Problem. In this work, we propose a novel coding-based scheme for the Close Pair Problem. We design both randomized and deterministic algorithms, which achieve the best-known running time when the minimum distance is very small compared to the length of input vectors. When applied to the Light Bulb Problem, our algorithms yields state-of-the-art deterministic running time when the Pearson-correlation coefficient $rho$ is very large.