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arXiv · 2406.08913

Maximizing the Maximum Degree in Ordered Nearest Neighbor Graphs

Péter Ágoston·Adrian Dumitrescu·Arsenii Sagdeev·Karamjeet Singh·Ji Zeng
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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 $Ω(\sqrt{\log{n}/\log\log{n}})$.

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Péter Ágoston, Adrian Dumitrescu, Arsenii Sagdeev, Karamjeet Singh, Ji Zeng. 2025-10-13. Maximizing the Maximum Degree in Ordered Nearest Neighbor Graphs. https://doi.org/10.1016/j.comgeo.2025.102229

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