Navigable Graphs for High-Dimensional Nearest Neighbor Search: Constructions and Limits

TL;DR

This paper investigates the construction and limitations of navigable graphs for high-dimensional nearest neighbor search, providing new bounds on graph sparsity and demonstrating the inherent complexity in high-dimensional spaces.

Contribution

It introduces a simple method to construct sparse navigable graphs in any dimension and establishes fundamental lower bounds on graph sparsity in high-dimensional settings.

Findings

Constructed navigable graphs with average degree $O(\sqrt{n \log n})$ for any point set.

Proved that random high-dimensional point sets cannot have sparse navigable graphs with average degree below $O(n^{1/2})$.

Used anti-concentration bounds to show limitations on graph sparsity in high dimensions.

Abstract

There has been significant recent interest in graph-based nearest neighbor search methods, many of which are centered on the construction of navigable graphs over high-dimensional point sets. A graph is navigable if we can successfully move from any starting node to any target node using a greedy routing strategy where we always move to the neighbor that is closest to the destination according to a given distance function. The complete graph is navigable for any point set, but the important question for applications is if sparser graphs can be constructed. While this question is fairly well understood in low-dimensions, we establish some of the first upper and lower bounds for high-dimensional point sets. First, we give a simple and efficient way to construct a navigable graph with average degree $O(\sqrt{n \log n })$ for any set of $n$ points, in any dimension, for any distance function. We compliment this result with a nearly matching lower bound: even under the Euclidean metric in $O(\log n)$ dimensions, a random point set has no navigable graph with average degree $O(n^{\alpha})$ for any $\alpha < 1/2$. Our lower bound relies on sharp anti-concentration bounds for binomial random variables, which we use to show that the near-neighborhoods of a set of random points do not overlap significantly, forcing any navigable graph to have many edges.