Graph-based time-space trade-offs for approximate near neighbors

We take a first step towards a rigorous asymptotic analysis of graph-based approaches for finding (approximate) nearest neighbors in high-dimensional spaces, by analyzing the complexity of (randomized) greedy walks on the approximate near neighbor graph. For random data sets of size $n = 2^{o(d)}$ on the $d$-dimensional Euclidean unit sphere, using near neighbor graphs we can provably solve the approximate nearest neighbor problem with approximation factor $c > 1$ in query time $n^{\rho_q + o(1)}$ and space $n^{1 + \rho_s + o(1)}$, for arbitrary $\rho_q, \rho_s \geq 0$ satisfying \begin{align} (2c^2 - 1) \rho_q + 2 c^2 (c^2 - 1) \sqrt{\rho_s (1 - \rho_s)} \geq c^4. \end{align} Graph-based near neighbor searching is especially competitive with hash-based methods for small $c$ and near-linear memory, and in this regime the asymptotic scaling of a greedy graph-based search matches the recent optimal hash-based trade-offs of Andoni-Laarhoven-Razenshteyn-Waingarten [SODA'17]. We further study how the trade-offs scale when the data set is of size $n = 2^{\Theta(d)}$, and analyze asymptotic complexities when applying these results to lattice sieving.