Faster Algorithms for Average-Case Orthogonal Vectors and Closest Pair Problems

TL;DR

This paper introduces faster average-case algorithms for the Orthogonal Vectors and Closest Pair problems, improving previous bounds by leveraging a simple polynomial method and analyzing its degradation with input distance.

Contribution

It presents a novel algorithm that improves average-case complexity for these problems using a new polynomial analysis approach, extending prior worst-case results.

Findings

Achieves faster average-case algorithms with complexity $n^{2 - \, \Omega(\log\log c / \log c)}$

Uses a simple polynomial method with a new analysis technique

Extends results to both Orthogonal Vectors and Closest Pair problems

Abstract

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.