Sparse Navigable Graphs for Nearest Neighbor Search: Algorithms and Hardness

TL;DR

This paper studies the construction of sparse navigable graphs for efficient nearest neighbor search, revealing computational hardness, establishing approximation algorithms, and providing lower bounds on query complexity.

Contribution

It introduces approximation algorithms for sparse navigable graphs, proves their near-optimality, and establishes hardness results linking the problem to Set Cover.

Findings

DiskANN can produce solutions much less sparse than optimal.

An $O(n^3)$-time $( ext{ln} n + 1)$-approximation algorithm is developed.

Any $o(n)$-approximation requires examining $ ext{Omega}(n^2)$ distances.

Abstract

We initiate the study of approximation algorithms and computational barriers for constructing sparse $\alpha$-navigable graphs [IX23, DGM+24], a core primitive underlying recent advances in graph-based nearest neighbor search. Given an $n$-point dataset $P$ with an associated metric $\mathsf{d}$ and a parameter $\alpha \geq 1$, the goal is to efficiently build the sparsest graph $G=(P, E)$ that is $\alpha$-navigable: for every distinct $s, t \in P$, there exists an edge $(s, u) \in E$ with $\mathsf{d}(u, t) < \mathsf{d}(s, t)/\alpha$. We consider two natural sparsity objectives: minimizing the maximum out-degree and minimizing the total size. We first show a strong negative result: the slow-preprocessing version of DiskANN (analyzed in [IX23] for low-doubling metrics) can yield solutions whose sparsity is $\widetilde{\Omega}(n)$ times larger than optimal, even on Euclidean instances. We then show a tight approximation-preserving equivalence between the Sparsest Navigable Graph problem and the classic Set Cover problem, obtaining an $O(n^3)$-time $(\ln n + 1)$-approximation algorithm, as well as establishing NP-hardness of achieving an $o(\ln n)$-approximation. Building on this equivalence, we develop faster $O(\ln n)$-approximation algorithms. The first runs in $\widetilde{O}(n \cdot \mathrm{OPT})$ time and is thus much faster when the optimal solution is sparse. The second, based on fast matrix multiplication, is a bicriteria algorithm that computes an $O(\ln n)$-approximation to the sparsest $2\alpha$-navigable graph, running in $\widetilde{O}(n^{\omega})$ time. Finally, we complement our upper bounds with a query complexity lower bound, showing that any $o(n)$-approximation requires examining $\Omega(n^2)$ distances. This result shows that in the regime where $\mathrm{OPT} = \widetilde{O}(n)$, our $\widetilde{O}(n \cdot \mathrm{OPT})$-time algorithm is essentially best possible.