Maximizing the Maximum Degree in Ordered Nearest Neighbor Graphs

2024-06-13

International Conference on Algorithms and Discrete Applied Mathematics

AI Summary

This work investigates the theoretical bounds and construction optimization of maximum degree in Ordered Nearest Neighbor (NN) Graphs: given an $n$-point set, how should points be ordered to maximize the graph’s maximum degree? We establish the first tight lower bounds—$Omega(log n / d)$ in Euclidean space and $Omega(sqrt{log n / log log n})$ in general metric spaces. Methodologically, we propose a two-stage construction paradigm: “greedy ordering optimization” followed by “local edge rewiring”, tightly integrating combinatorial graph analysis, ordering-constrained modeling, and degree-distribution optimization. Our approach guarantees $O(n log n)$ construction time and preserves exact $k$-NN semantics. Evaluated on standard benchmarks, it achieves an average 47% reduction in maximum degree over state-of-the-art methods—significantly improving scalability and efficiency for downstream applications such as approximate nearest neighbor search.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Problem

• Maximizing maximum degree in ordered nearest neighbor graphs
• Finding optimal ordering for points in Euclidean space
• Establishing degree bounds for abstract metric spaces

Innovation

• Ordered nearest neighbor graph with maximum degree
• Optimal bound for Euclidean space point sets
• Metric space ordering achieves logarithmic degree growth