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An optimal algorithm for average distance in typical regular graphs

Alexandros Eskenazis, Manor Mendel, Assaf Naor

TL;DR

This work advances deterministic average-distance approximation in graphs by bridging metric-geometry tools with sublinear query models. It proves that in typical constant-degree random graphs, one can deterministically obtain a constant-factor approximation to the average distance with $O(n)$ distance queries, while establishing sharp lower bounds that preclude similar guarantees for all constant-degree graphs with sublinear queries. The authors develop a CAT$(0)$-transform closure theorem, showing that metric transforms of CAT$(0)$ spaces embed with bounded distortion into CAT$(0)$ spaces, and they leverage Hadamard-space extrapolation, Euclidean cone constructions, and high-girth expanders to design universal approximators and to analyze embedding distortions. The results connect sublinear algorithms for distance computations with nonlinear geometric phenomena, yielding new insights into rigidity, embedding theory, and the limits of deterministic approximation in metric spaces. Collectively, the findings reveal a rich interplay between algorithmic lower bounds, expander-based constructions, and the geometry of nonpositive curvature, with implications for both theory and practical distance-computation tasks in large graphs.

Abstract

We design a deterministic algorithm that, given $n$ points in a \emph{typical} constant degree regular~graph, queries $O(n)$ distances to output a constant factor approximation to the average distance among those points, thus answering a question posed in~\cite{MN14}. Our algorithm uses the method of~\cite{MN14} to construct a sequence of constant degree graphs that are expanders with respect to certain nonpositively curved metric spaces, together with a new rigidity theorem for metric transforms of nonpositively curved metric spaces. The fact that our algorithm works for typical (uniformly random) constant degree regular graphs rather than for all constant degree graphs is unavoidable, thanks to the following impossibility result that we obtain: For every fixed $k\in \N$, the approximation factor of any algorithm for average distance that works for all constant degree graphs and queries $o(n^{1+1/k})$ distances must necessarily be at least $2(k+1)$. This matches the upper bound attained by the algorithm that was designed for general finite metric spaces in~\cite{BGS}. Thus, any algorithm for average distance in constant degree graphs whose approximation guarantee is less than $4$ must query $Ω(n^2)$ distances, any such algorithm whose approximation guarantee is less than $6$ must query $Ω(n^{3/2})$ distances, any such algorithm whose approximation guarantee less than $8$ must query $Ω(n^{4/3})$ distances, and so forth, and furthermore there exist algorithms achieving those parameters.

An optimal algorithm for average distance in typical regular graphs

TL;DR

This work advances deterministic average-distance approximation in graphs by bridging metric-geometry tools with sublinear query models. It proves that in typical constant-degree random graphs, one can deterministically obtain a constant-factor approximation to the average distance with distance queries, while establishing sharp lower bounds that preclude similar guarantees for all constant-degree graphs with sublinear queries. The authors develop a CAT-transform closure theorem, showing that metric transforms of CAT spaces embed with bounded distortion into CAT spaces, and they leverage Hadamard-space extrapolation, Euclidean cone constructions, and high-girth expanders to design universal approximators and to analyze embedding distortions. The results connect sublinear algorithms for distance computations with nonlinear geometric phenomena, yielding new insights into rigidity, embedding theory, and the limits of deterministic approximation in metric spaces. Collectively, the findings reveal a rich interplay between algorithmic lower bounds, expander-based constructions, and the geometry of nonpositive curvature, with implications for both theory and practical distance-computation tasks in large graphs.

Abstract

We design a deterministic algorithm that, given points in a \emph{typical} constant degree regular~graph, queries distances to output a constant factor approximation to the average distance among those points, thus answering a question posed in~\cite{MN14}. Our algorithm uses the method of~\cite{MN14} to construct a sequence of constant degree graphs that are expanders with respect to certain nonpositively curved metric spaces, together with a new rigidity theorem for metric transforms of nonpositively curved metric spaces. The fact that our algorithm works for typical (uniformly random) constant degree regular graphs rather than for all constant degree graphs is unavoidable, thanks to the following impossibility result that we obtain: For every fixed , the approximation factor of any algorithm for average distance that works for all constant degree graphs and queries distances must necessarily be at least . This matches the upper bound attained by the algorithm that was designed for general finite metric spaces in~\cite{BGS}. Thus, any algorithm for average distance in constant degree graphs whose approximation guarantee is less than must query distances, any such algorithm whose approximation guarantee is less than must query distances, any such algorithm whose approximation guarantee less than must query distances, and so forth, and furthermore there exist algorithms achieving those parameters.
Paper Structure (16 sections, 51 theorems, 109 equations, 3 figures)

This paper contains 16 sections, 51 theorems, 109 equations, 3 figures.

Table of Contents

  1. Extended Abstract
  2. Introduction
  3. Overview of the proof of Theorem \ref{['EA:thm:impossibility-DA']}
  4. Overview of the proof of Theorem \ref{['EA:thm:cat(0)-transform-closure-intro']}
  5. HADAMARD SUBSPACES ARE CLOSED UNDER METRIC TRANSFORMS (DRAFT)
  6. Introduction
  7. Metric transforms and rigidity
  8. Counterexample to nonlinear Kwapień formulations
  9. Matoušek extrapolation for Hadamard spaces
  10. Expanders with respect to random regular graphs
  11. Estimation of the average distance in random regular graphs
  12. Distortion growth of embedding into l_2 and l_1
  13. Euclidean cones, metric transforms, and embeddings
  14. Alexandrov spaces
  15. CAT(0) spaces
  16. ...and 1 more sections

Key Result

Theorem 1.1

For $k,n\in \mathbb{N}$ with $k\geqslant 2$ there is an $n$-vertex $(2k)$-universal approximator of size $O(kn^{1+1/k})$.

Figures (3)

  • Figure 1: The blue dots correspond to the available algorithms per \ref{['EA:thm:BGS-approximator']}. The red lines correspond to the impossibility statement of \ref{['EA:thm:impossibility-DA']}. Notice that the graph's horizontal axis decreases as one moves from left to right, and it introduces a macroscopic gap between $O(n^{1+1/k})$ queries and $o(n^{1+1/k})$ queries.
  • Figure 2: Notions of embedding in \ref{['rem:embedding-notions']} and their relative strength for embedding into Hilbert space.
  • Figure 3: The universal approximators of \ref{['thm:BGS-approximator']} in blue dots vs. the lower bounds of \ref{['thm:impossibility-DA']} and \ref{['prop:UA-lb14']} in red lines. Note that the chart's horizontal axis decreases as one moves from left to right, and it introduces a macroscopic gap between $O(n^{1+1/k})$ queries and $o(n^{1+1/k})$ queries.

Theorems & Definitions (83)

  • Definition 1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Proposition 1
  • Theorem 1.3
  • Remark 2
  • Theorem 1.4
  • Remark 3
  • Theorem 2.1
  • ...and 73 more