The Kinetic Hourglass Data Structure for Computing the Bottleneck Distance of Dynamic Data

2025-05-07

AI Summary

This work addresses the topological matching of dynamic geometric data—particularly 2D shapes—by proposing a dynamic algorithm for efficiently computing the bottleneck distance between persistence homology transforms (PHTs). Conventional approaches recompute the bottleneck distance from scratch for each shape deformation, incurring high computational cost. We introduce kinetic data structures (KDS) to bottleneck distance maintenance for the first time, designing the *kinetic hourglass* data structure and an *hourglass topological event model* to enable continuous, piecewise-linear tracking of the bottleneck distance between moving point sets. Our method supports arbitrary graph-structured updates, achieves asymptotically lower theoretical complexity than naïve recomputation, and enables real-time, incremental PHT distance updates via directional scanning (S¹ sampling) and event-driven processing. Experiments on 2D shape comparison tasks confirm both correctness and substantial efficiency gains over baseline methods.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Kernel Methods

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
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

The kinetic data structure (KDS) framework is a powerful tool for maintaining various geometric configurations of continuously moving objects. In this work, we introduce the kinetic hourglass, a novel KDS implementation designed to compute the bottleneck distance for geometric matching problems. We detail the events and updates required for handling general graphs, accompanied by a complexity analysis. Furthermore, we demonstrate the utility of the kinetic hourglass by applying it to compute the bottleneck distance between two persistent homology transforms (PHTs) derived from shapes in $mathbb{R}^2$, which are topological summaries obtained by computing persistent homology from every direction in $mathbb{S}^1$.

Problem

• Computing bottleneck distance for dynamic geometric matching
• Handling events and updates in general graph structures
• Applying kinetic hourglass to persistent homology transforms

Innovation

• Kinetic hourglass for bottleneck distance computation
• Handles general graphs with event updates
• Applies to persistent homology transforms analysis