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

Haya Diwan, Jinrui Gou, Cameron Musco, Christopher Musco, Torsten Suel

TL;DR

The paper investigates the sparsity limits of navigable graphs for high-dimensional nearest-neighbor search under greedy routing. It proves a universal upper bound construction achieving an average out-degree of $O(\sqrt{n \log n})$ with a two-step small-world routing guarantee, valid for any distance function, and simultaneously establishes dimension-dependent lower bounds showing that, in Euclidean space with $d=\Omega(\log n)$, any navigable graph must have average degree at least $\Omega(n^{1/2-\delta})$ for any fixed $\delta>0$. The results rely on a constructive union of a near-neighbor graph and a sparse random (or deterministically chosen) edge set, and on anti-concentration bounds for binomial-type neighborhoods to bound how much near neighborhoods can overlap. Together, they delineate the trade-off between sparsity and guaranteed navigability in high dimensions, with additional insights on maximum-degree limitations and potential directions for end-to-end approximation guarantees in graph-based NN search.

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 $O(n^α)$ for any $α< 1/2$. 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.

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

TL;DR

The paper investigates the sparsity limits of navigable graphs for high-dimensional nearest-neighbor search under greedy routing. It proves a universal upper bound construction achieving an average out-degree of with a two-step small-world routing guarantee, valid for any distance function, and simultaneously establishes dimension-dependent lower bounds showing that, in Euclidean space with , any navigable graph must have average degree at least for any fixed . The results rely on a constructive union of a near-neighbor graph and a sparse random (or deterministically chosen) edge set, and on anti-concentration bounds for binomial-type neighborhoods to bound how much near neighborhoods can overlap. Together, they delineate the trade-off between sparsity and guaranteed navigability in high dimensions, with additional insights on maximum-degree limitations and potential directions for end-to-end approximation guarantees in graph-based NN search.

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 for any set of points, in any dimension, for any distance function. We compliment this result with a nearly matching lower bound: even under the Euclidean metric in dimensions, a random point set has no navigable graph with average degree for any . 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.
Paper Structure (11 sections, 4 theorems, 21 equations, 2 figures, 1 algorithm)

This paper contains 11 sections, 4 theorems, 21 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Constructing Sparse Navigable Graphs
  3. Our Results
  4. Outlook
  5. Preliminaries
  6. Upper Bound
  7. Lower Bounds
  8. Average Degree Lower Bound
  9. Probabilistic Claims about Near Neighborhoods
  10. Maximum Degree Lower Bound
  11. Omitted Proofs

Key Result

Theorem 1

For any input domain $\mathcal{X}$, point set $x_1, \ldots, x_n \in \mathcal{X}$, and distance function $D: \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}^{\ge 0}$ such that $D(x_i,x_i) = 0$ for all $i$ and $D(x_i,x_j) > 0$ for $x_j \neq x_i$, it is possible to efficiently construct a direc

Figures (2)

  • Figure 1: Example of a navigable graph $G = (V,E)$ on $5$ nodes. A double arrow indicates that both $(i,j)\in E$ and $(j,i)\in E$. We can check that $G$ is navigable by referring to \ref{['fig:permutation_view']}.
  • Figure 2: Distance-based permutations for the data set in Figure \ref{['fig:navigable_graph_full']}. As an example, in the plot above, we have $N_1(1) = 1, N_2(1) = 2, N_3(1) = 5, N_4(1) = 3, N_5(1) = 4$, we have $N_1(2) = 2, N_2(2) = 1, N_3(2) = 3, N_4(2) = 5, N_5(2) = 4$, etc. For the permutations, we draw all edges in the graph $G$ from \ref{['fig:navigable_graph_full']} that point "left" in the permutation. In particular, we show edges that connect any $N_\ell(t)$ to $N_k(t)$ with $k < \ell$. We can check that property \ref{['eq:permutation_property']} holds, so the graph is navigable.

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Definition 3: Navigable Graph
  • proof
  • Theorem 4
  • Claim 5: No Repeated Points
  • Definition 6: Fixed Radius Near-Neighborhood
  • Claim 7: Required Connections Within Neighborhoods
  • proof
  • Claim 8: Neighborhoods are Large
  • ...and 11 more