Fast Queries of Fibered Barcodes

Abstract: The fibered barcode $\mathcal{F}(M)$ of a bipersistence module $M$ is the map sending each non-negatively sloped affine line $\ell \subset \mathbb{R}^2$ to the barcode of the restriction of $M$ along $\ell$. The simplicity, computability, and stability of $\mathcal{F}(M)$ make it a natural choice of invariant for data analysis applications. In an earlier preprint [arXiv:1512.00180], we introduced a framework for real-time interactive visualization of $\mathcal{F}(M)$, which allows the user to select a single line $\ell$ via a GUI and then plots the associated barcode. This visualization is a key feature of our software RIVET for the visualization and analysis of bipersistent homology. Such interactive visualization requires a framework for efficient queries of $\mathcal{F}(M)$, i.e., for quickly obtaining the barcode along a given line $\ell$. To enable such queries, we introduced a novel data structure based on planar line arrangements, called an augmented arrangement. The aim of the present paper is to give an updated and improved exposition of the parts of our preprint [arXiv:1512.00180] concerning the mathematics of the augmented arrangement and its computation. Notably, by taking the input to be a minimal presentation rather than a chain complex, we are able to substantially simplify our main algorithm and its complexity analysis.