Hardness of Approximate Nearest Neighbor Search

TL;DR

This paper establishes near-quadratic time lower bounds for approximate nearest neighbor search problems under various metrics, assuming SETH, and introduces novel reductions using Algebraic Geometry codes to prove these hardness results.

Contribution

It provides the first hardness results for approximate nearest neighbor search based on SETH, utilizing improved reductions with Algebraic Geometry codes.

Findings

Near-quadratic lower bounds for approximate Bichromatic Closest Pair

Implication of near-linear query time for approximate nearest neighbor search

First hardness results using Algebraic Geometry code-based reductions

Abstract

We prove conditional near-quadratic running time lower bounds for approximate Bichromatic Closest Pair with Euclidean, Manhattan, Hamming, or edit distance. Specifically, unless the Strong Exponential Time Hypothesis (SETH) is false, for every $\delta>0$ there exists a constant $\epsilon>0$ such that computing a $(1+\epsilon)$-approximation to the Bichromatic Closest Pair requires $n^{2-\delta}$ time. In particular, this implies a near-linear query time for Approximate Nearest Neighbor search with polynomial preprocessing time. Our reduction uses the Distributed PCP framework of [ARW'17], but obtains improved efficiency using Algebraic Geometry (AG) codes. Efficient PCPs from AG codes have been constructed in other settings before [BKKMS'16, BCGRS'17], but our construction is the first to yield new hardness results.