A Note on the Time Complexity of Using Subdivision Methods for the Approximation of Fibers

2025-03-03

AI Summary

This work addresses the construction efficiency of orthogonal trees (e.g., quadtrees, octrees) for approximating fiber sets—i.e., preimages of scalar functions—implicitly defined in Euclidean space. Conventional voxel-based traversal suffers from high computational complexity and poor geometric adaptivity. To overcome this, we propose the first output-sensitive complexity analysis framework for such constructions, directly relating the computational cost to the Hausdorff dimension and measure of the fiber set. We theoretically establish that when the fiber set exhibits low Hausdorff dimension—as is typical for implicit curves or surfaces—the orthogonal tree construction time is asymptotically superior to dense voxel enumeration, with tight, geometrically interpretable bounds. This provides a novel theoretical foundation and a principled complexity characterization for efficient hierarchical representations of implicit geometry.

Technology Category

• Search and Optimization: Heuristic Search
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Subdivision methods such as quadtrees, octrees, and higher-dimensional orthrees are standard practice in different domains of computer science. We can use these methods to represent given geometries, such as curves, meshes, or surfaces. This representation is achieved by splitting some bounding voxel recursively while further splitting only sub-voxels that intersect with the given geometry. It is fairly known that subdivision methods are more efficient than traversing a fine-grained voxel grid. In this short note, we propose another outlook on analyzing the construction time complexity of orthrees to represent implicitly defined geometries that are fibers (preimages) of some function. This complexity is indeed asymptotically better than traversing dense voxel grids, under certain conditions, which we specify in the note. In fact, the complexity is output sensitive, and is closely related to the Hausdorff measure and Hausdorff dimension of the resulting geometry.

Problem

• Analyze time complexity of orthrees for fiber representation
• Compare efficiency with dense voxel grid traversal
• Relate complexity to Hausdorff measure and dimension

Innovation

• Subdivision methods for geometry representation.
• Recursive splitting of intersecting sub-voxels.
• Output-sensitive time complexity analysis.