Orthogonal Vectors Indexing

TL;DR

This paper investigates the space complexity of the Orthogonal Vectors (OV) indexing problem, proposing algorithms, tradeoffs, and connections to other problems, with implications for understanding conditional lower bounds in computational complexity.

Contribution

It introduces space-efficient algorithms for OV indexing, explores space-time tradeoffs, and links the problem to SetDisjointness, advancing understanding of its complexity.

Findings

Proposed space-efficient algorithms for OV indexing.

Established a tradeoff between space and query time.

Connected OV indexing to SetDisjointness and analyzed on random inputs.

Abstract

In the recent years, intensive research work has been dedicated to prove conditional lower bounds in order to reveal the inner structure of the class P. These conditional lower bounds are based on many popular conjectures on well-studied problems. One of the most heavily used conjectures is the celebrated Strong Exponential Time Hypothesis (SETH). It turns out that conditional hardness proved based on SETH goes, in many cases, through an intermediate problem - the Orthogonal Vectors (OV) problem. Almost all research work regarding conditional lower bound was concentrated on time complexity. Very little attention was directed toward space complexity. In a recent work, Goldstein et al.[WADS 2017] set the stage for proving conditional lower bounds regarding space and its interplay with time. In this spirit, it is tempting to investigate the space complexity of a data structure variant of OV which is called \emph{OV indexing}. In this problem $n$ boolean vectors of size $c\log{n}$ are given for preprocessing. As a query, a vector $v$ is given and we are required to verify if there is an input vector that is orthogonal to it or not. This OV indexing problem is interesting in its own, but it also likely to have strong implications on problems known to be conditionally hard, in terms of time complexity, based on OV. Having this in mind, we study OV indexing in this paper from many aspects. We give some space-efficient algorithms for the problem, show a tradeoff between space and query time, describe how to solve its reporting variant, shed light on an interesting connection between this problem and the well-studied SetDisjointness problem and demonstrate how it can be solved more efficiently on random input.