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 \(\varOmega (\sqrt{\log {n}/\log \log {n}})\).
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Notes
- 1.
For instance, in the notation of Sect. 4, it is not possible that two triples \(\{1,2,3\}\) and \(\{1,3,4\}\) are blue, while \(\{1,2,4\}\) is green, because otherwise we would have \(v_1v_2 < v_1v_3 < v_1v_4 < v_1v_2\), a contradiction.
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Acknowledgments
The authors would like to thank the organizers of the Focused Week on Geometric Spanners (Oct 23–Oct 29, 2023) at the Erdős Center, Budapest, where this joint work began. Research partially supported by ERC Advanced Grant ‘GeoScape’ No. 882971, by the Erdős Center and by the Ministry of Innovation and Technology NRDI Office within the framework of the Artificial Intelligence National Laboratory (RRF-2.3.1-21-2022-00004). Research was also partially supported by the Overseas Research Fellowship of IIIT-Delhi, India.
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Ágoston, P., Dumitrescu, A., Sagdeev, A., Singh, K., Zeng, J. (2025). Maximizing the Maximum Degree in Ordered Nearest Neighbor Graphs. In: Gaur, D., Mathew, R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2025. Lecture Notes in Computer Science, vol 15536. Springer, Cham. https://doi.org/10.1007/978-3-031-83438-7_30
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