An optimal algorithm for average distance in typical regular graphs

TL;DR

This paper presents a deterministic algorithm that approximates the average distance in typical constant degree regular graphs efficiently, and establishes fundamental limits on the trade-offs between approximation quality and query complexity.

Contribution

It introduces a novel algorithm for average distance approximation in typical regular graphs and proves tight lower bounds on query complexity for all such algorithms.

Findings

Algorithm achieves constant factor approximation with O(n) queries in typical regular graphs.

Impossibility results establish lower bounds on query complexity for given approximation factors.

Matching upper and lower bounds characterize the fundamental limits of the problem.

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 $\Omega(n^2)$ distances, any such algorithm whose approximation guarantee is less than $6$ must query $\Omega(n^{3/2})$ distances, any such algorithm whose approximation guarantee less than $8$ must query $\Omega(n^{4/3})$ distances, and so forth, and furthermore there exist algorithms achieving those parameters.