We study the average-case version of the Orthogonal Vectors problem, in which one is given as input $n$ vectors from $\{0,1\}^d$ which are chosen randomly so that each coordinate is $1$ independently with probability $p$. Kane and Williams [ITCS 2019] showed how to solve this problem in time $O(n^{2 - \delta_p})$ for a constant $\delta_p > 0$ that depends only on $p$. However, it was previously unclear how to solve the problem faster in the hardest parameter regime where $p$ may depend on $d$. The best prior algorithm was the best worst-case algorithm by Abboud, Williams and Yu [SODA 2014], which in dimension $d = c \cdot \log n$, solves the problem in time $n^{2 - \Omega(1/\log c)}$. In this paper, we give a new algorithm which improves this to $n^{2 - \Omega(\log\log c /\log c)}$ in the average case for any parameter $p$. As in the prior work, our algorithm uses the polynomial method. We make use of a very simple polynomial over the reals, and use a new method to analyze its performance based on computing how its value degrades as the input vectors get farther from orthogonal. To demonstrate the generality of our approach, we also solve the average-case version of the closest pair problem in the same running time.

翻译：我们研究正交向量问题的平均情况版本，其中输入为从$\{0,1\}^d$中随机选取的$n$个向量，每个坐标以概率$p$独立取值为$1$。Kane与Williams [ITCS 2019] 证明了该问题可在$O(n^{2 - \delta_p})$时间内求解，其中常数$\delta_p > 0$仅依赖于$p$。然而，在$p$可能依赖于$d$的最困难参数区域中，如何获得更快的算法此前尚不明确。先前最佳算法是Abboud、Williams与Yu [SODA 2014] 提出的最坏情况算法，其在维度$d = c \cdot \log n$下以$n^{2 - \Omega(1/\log c)}$时间求解该问题。本文提出一种新算法，对于任意参数$p$，在平均情况下将时间复杂度改进为$n^{2 - \Omega(\log\log c /\log c)}$。与先前工作类似，我们的算法采用多项式方法。我们构造了一个非常简单的实数域多项式，并提出一种基于计算输入向量偏离正交性时多项式值衰减程度的新分析方法。为证明方法的普适性，我们还在相同运行时间内解决了平均情况下的最近对问题。