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 \emph{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 \emph{Light Bulb Problem}. In this work, we propose a novel coding-based scheme for the Closest Pair Problem. We design both randomized and deterministic algorithms, which achieve the best-known running time when the length of input vectors $m$ is small and the minimum distance is very small compared to $m$. Specifically, the running time of our randomized algorithm is $O(n\log^{2}n\cdot 2^{c m} \cdot \mathrm{poly}(m))$ and the running time of our deterministic algorithm is $O(n\log{n}\cdot 2^{c' m} \cdot \mathrm{poly}(m))$, where $c$ and $c'$ are constants depending only on the (relative) distance of the closest pair. When applied to the Light Bulb Problem, our result yields state-of-the-art deterministic running time when the Pearson-correlation coefficient $\rho$ is very large. Specifically, when $\rho \geq 0.9933$, our deterministic algorithm runs faster than the previously best deterministic algorithm (Alman, SOSA 2019).