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We consider the (1+ϵ)-approximate nearest neighbor search problem: given a set X of n points in a d-dimensional space, build a data structure that, given any query point y, finds a point x∈X whose distance to y is at most (1+ϵ)minx∈X ‖x−y‖ for an accuracy parameter ϵ∈(0,1). Our main result is a data structure that occupies only O(ϵ^−2 n log(n)log(1/ϵ)) bits of space, assuming all point coordinates are integers in the range {−n^O(1)…n^O(1)}, i.e., the coordinates have O(logn) bits of precision. This improves over the best previously known space bound of O(ϵ^−2 n log(n)^2), obtained via the randomized dimensionality reduction method of Johnson and Lindenstrauss (1984). We also consider the more general problem of estimating all distances from a collection of query points to all data points X, and provide almost tight upper and lower bounds for the space complexity of this problem.
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This content will become publicly available on December 13, 2025
Navigable Graphs for High-Dimensional Nearest Neighbor Search: Constructions and Limits
There has been significant recent interest in \emph{graph-based nearest neighbor search methods}, many of which are centered on the construction of (approximately) {\em 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 the given distance function. The complete graph is obviously 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.
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- Award ID(s):
- 2106888
- PAR ID:
- 10577177
- Publisher / Repository:
- Advances in Neural Information Processing Systems
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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