Maximizing the Maximum Degree in Ordered Nearest Neighbor Graphs

TL;DR

This paper investigates the maximum degree of ordered nearest neighbor graphs in Euclidean and metric spaces, establishing bounds that are tight up to constant factors and revealing how the structure depends on the space's properties.

Contribution

It proves tight lower bounds on the maximum degree of ordered nearest neighbor graphs in Euclidean and metric spaces, highlighting the influence of space geometry on graph complexity.

Findings

In Euclidean space, the maximum degree can be at least d imes rac{\, log n}{4d} for some order.

In general metric spaces, the maximum degree can be  .5 imes \, rac{ log n}{ log log n} for some order.

These bounds are asymptotically tight, showing fundamental limits of graph degree in these settings.

Abstract

For an ordered point set in a Euclidean space or, more generally, in an abstract metric space, the ordered Nearest Neighbor Graph is obtained by connecting each of the points to its closest predecessor by a directed edge. We show that for every set of $n$ points in $\mathbb{R}^d$, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree at least $\log{n}/(4d)$. Apart from the $1/(4d)$ factor, this bound is the best possible. As for the abstract setting, we show that for every $n$-element metric space, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree $\Omega(\sqrt{\log{n}/\log\log{n}})$.