# Lower Bounds for Oblivious Near-Neighbor Search

## TL;DR

This paper establishes a new super-logarithmic lower bound on the dynamic cell-probe complexity of oblivious approximate-near-neighbor search in high-dimensional Hamming space, advancing understanding of data structure limitations.

## Contribution

It provides the first unconditional super-logarithmic lower bound for oblivious ANN data structures and shows how to obliviously dynamize static decomposable search problems efficiently.

## Key findings

- Proves an $ ilde{ ext{Omega}}( ext{lg}^2 n)$ lower bound for oblivious ANN.
- First super-logarithmic unconditional lower bound for ANN against general data structures.
- Shows oblivious dynamization of static decomposable search problems with $O( ext{log} n)$ overhead.

## Abstract

We prove an $\Omega(d \lg n/ (\lg\lg n)^2)$ lower bound on the dynamic cell-probe complexity of statistically $\mathit{oblivious}$ approximate-near-neighbor search ($\mathsf{ANN}$) over the $d$-dimensional Hamming cube. For the natural setting of $d = \Theta(\log n)$, our result implies an $\tilde{\Omega}(\lg^2 n)$ lower bound, which is a quadratic improvement over the highest (non-oblivious) cell-probe lower bound for $\mathsf{ANN}$. This is the first super-logarithmic $\mathit{unconditional}$ lower bound for $\mathsf{ANN}$ against general (non black-box) data structures. We also show that any oblivious $\mathit{static}$ data structure for decomposable search problems (like $\mathsf{ANN}$) can be obliviously dynamized with $O(\log n)$ overhead in update and query time, strengthening a classic result of Bentley and Saxe (Algorithmica, 1980).