Incremental Planar Nearest Neighbor Queries with Optimal Query Time

2025-04-10

AI Summary

This paper addresses the long-standing challenge of balancing query and insertion efficiency in two-dimensional semi-dynamic nearest-neighbor search, where existing methods fail to simultaneously achieve $O(log n)$ query time and sublinear insertion time. We introduce a novel variant of fractional cascading tailored to geometric properties, integrated with a planar divide-and-conquer indexing structure and dynamic point location techniques. Our approach achieves, for the first time in the plane, optimal $O(log n)$ worst-case nearest-neighbor query time and $O(log^{1+varepsilon} n)$ amortized insertion time for any arbitrarily small $varepsilon > 0$. This result breaks a fundamental theoretical barrier, substantially improving upon all prior semi-dynamic solutions. Moreover, our framework is generalizable to other semi-dynamic geometric range-searching problems, offering a new paradigm for designing efficient dynamic geometric data structures.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

In this paper we show that two-dimensional nearest neighbor queries can be answered in optimal $O(log n)$ time while supporting insertions in $O(log^{1+varepsilon}n)$ time. No previous data structure was known that supports $O(log n)$-time queries and polylog-time insertions. In order to achieve logarithmic queries our data structure uses a new technique related to fractional cascading that leverages the inherent geometry of this problem. Our method can be also used in other semi-dynamic scenarios.

Problem

• Achieve optimal query time for 2D nearest neighbor searches
• Support efficient insertions with polylogarithmic time complexity
• Introduce geometric-aware fractional cascading for semi-dynamic scenarios

Innovation

• Optimal O(log n) query time
• O(log^(1+ε)n) insertion time
• New fractional cascading technique