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

TL;DR

This paper introduces the kinetic hourglass, a new kinetic data structure for efficiently computing the bottleneck distance in dynamic geometric matching problems, with applications to topological shape analysis.

Contribution

The paper presents the kinetic hourglass, a novel KDS for dynamic bottleneck distance computation, including event handling, complexity analysis, and practical application to persistent homology transforms.

Findings

Efficiently maintains bottleneck distance for moving geometric data.

Applicable to persistent homology transforms of shapes in 2D.

Provides complexity bounds and implementation details.

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 $\mathbb{S}^1$.