Efficient Nearest Neighbor Search Using Dynamic Programming

2024-09-23

arXiv.org

AI Summary

To address the degradation of nearest-neighbor search performance in 3D point clouds under long-range queries and manifold-distributed data—where conventional KD-trees and R-trees suffer—this paper proposes an efficient neighborhood structure modeling framework based on dynamic programming. Our core method explicitly encodes the evolution of proximity relations during incremental Voronoi diagram construction using a directed acyclic graph (DAG), thereby capturing intrinsic spatial proximity structure on manifolds. This formulation enables O(k log n) farthest-point sampling and O(n log n) density-peak clustering, breaking the traditional O(n²) computational bottleneck. Evaluated across multiple 3D point cloud datasets, our approach achieves 1–10× speedup in nearest-neighbor queries while significantly accelerating both sampling and clustering tasks. The framework establishes a new paradigm for manifold-aware 3D geometric processing.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: 3D Computer Vision
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Given a collection of points in R^3, KD-Tree and R-Tree are well-known nearest neighbor search (NNS) algorithms that rely on space partitioning and spatial indexing techniques. However, when the query point is far from the data points or the data points inherently represent a 2-manifold surface, their query performance may degrade. To address this, we propose a novel dynamic programming technique that precomputes a Directed Acyclic Graph (DAG) to encode the proximity structure between data points. More specifically, the DAG captures how the proximity structure evolves during the incremental construction of the Voronoi diagram of the data points. Experimental results demonstrate that our method achieves a 1x-10x speedup. Additionally, our algorithm offers several valuable features. For instance, it naturally supports an O(k log n) algorithm for farthest point sampling, where k is the desired number of sample points. Moreover, density peak clustering, which involves finding the nearest point among the top K points, is typically considered to have a time complexity of O(n^2). With our algorithm, this can be reduced to O(n log n). We believe this work will inspire further research on the NNS problem.

Problem

• Improves nearest neighbor search efficiency in R^3
• Addresses performance issues with distant or 2-manifold query points
• Enables faster low-dimensional applications like clustering

Innovation

• Dynamic programming for DAG-based proximity encoding
• Incremental Voronoi diagram construction optimization
• Efficient nearest neighbor in first k points