Paper 2019/377

Lower Bounds for Oblivious Near-Neighbor Search

Kasper Green Larsen, Tal Malkin, Omri Weinstein, and Kevin Yeo

Abstract

We prove an $\mathrm{\Omega }(d\lg n/(\lg \lg n{)}^2)$ 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 $d=\mathrm{\Theta }(\log n)$, our result implies an $\tilde{\mathrm{\Omega }}({\lg }^{2}n)$ 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).