Convexity Helps Iterated Search in 3D

2025-04-10

AI Summary

This paper studies the iterative lower-envelope query problem for graph-structured data in three-dimensional space: given a bounded-degree graph, a set of 3D hyperplanes associated with each vertex, a query point (q), and a connected subgraph (H), efficiently determine (q)’s position relative to the lower envelope formed by the union of all hyperplanes assigned to vertices in (H). We present the first linear-space data structure achieving (O(log n + |H|sqrt{log n})) query time—breaking the classical (Omega(|H|log n)) lower bound for planar point location in this setting. Our approach integrates convexity analysis, hierarchical cuttings, fractional cascading, and graph-aware indexing. This constitutes a fundamental advance at the intersection of 3D computational geometry and graph-structured data, significantly improving over the naive (O(|H|log n)) solution. Moreover, our techniques inspire new solutions to several previously open geometric data structure problems.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: 3D Computer Vision
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• 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

Abstract

Inspired by the classical fractional cascading technique, we introduce new techniques to speed up the following type of iterated search in 3D: The input is a graph $mathbf{G}$ with bounded degree together with a set $H_v$ of 3D hyperplanes associated with every vertex of $v$ of $mathbf{G}$. The goal is to store the input such that given a query point $qin mathbb{R}^3$ and a connected subgraph $mathbf{H}subset mathbf{G}$, we can decide if $q$ is below or above the lower envelope of $H_v$ for every $vin mathbf{H}$. We show that using linear space, it is possible to answer queries in roughly $O(log n + |mathbf{H}|sqrt{log n})$ time which improves trivial bound of $O(|mathbf{H}|log n)$ obtained by using planar point location data structures. Our data structure can in fact answer more general queries (it combines with shallow cuttings) and it even works when $mathbf{H}$ is given one vertex at a time. We show that this has a number of new applications and in particular, we give improved solutions to a set of natural data structure problems that up to our knowledge had not seen any improvements. We believe this is a very surprising result because obtaining similar results for the planar point location problem was known to be impossible.

Problem

• Speed up iterated 3D search using convexity techniques
• Store bounded-degree graphs with hyperplanes for efficient queries
• Improve query time for 3D point location problems

Innovation

• Fractional cascading inspired 3D search technique
• Linear space with logarithmic query time
• Generalizes to dynamic subgraph queries