Maximizing the Maximum Degree in Ordered Nearest Neighbor Graphs

TL;DR

The paper addresses how to maximize the maximum indegree in the ordered Nearest Neighbor Graph by choosing insertion orders for $n$ points across three settings: the line, Euclidean space $\mathbb{R}^d$, and general metric spaces. It employs a diameter-based recursive ordering on the line, a $16^d/2$-cell covering to lift the line technique to $\mathbb{R}^d$, and a Ramsey-type hypergraph strategy for general metrics, blending combinatorial and geometric tools. The main results are: on the line, the maximum indegree can be $\lceil \log n\rceil$; in $\mathbb{R}^d$ it is at least $\log n /(4d)$; and in general metric spaces it is $\Omega(\sqrt{\log n/\log\log n})$, with corresponding lower bounds and a clear gap between regimes. These findings illuminate how insertion order controls indegree growth in ordered NN graphs and connect to spanner-related graph constructions, while highlighting open questions about tightness in higher dimensions and broader metric 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 $Ω(\sqrt{\log{n}/\log\log{n}})$.

Figures (1)

• Figure 1: Unordered (left) and ordered (right) Nearest Neighbor Graph s on the same set of six points.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 1
• proof
• Theorem 4
• Corollary 1
• proof
• Theorem 5
• proof : Proof of Theorem \ref{['thm:ramsey']}