Distributed Persistent Homology for 2D Alpha Complexes

TL;DR

A new algorithm to parallelise the computation of persistent homology of 2D alpha complexes and shows how to compute the persistence Mayer-Vietoris spectral sequence from these covers and how to obtain persistent homology from it.

Abstract

We introduce a new algorithm to parallelise the computation of persistent homology of 2D alpha complexes. Our algorithm distributes the input point cloud among the cores which then compute a cover based on a rectilinear grid. We show how to compute the persistence Mayer-Vietoris spectral sequence from these covers and how to obtain persistent homology from it. For this, we introduce second-page collapse conditions and explain how to solve the extension problem. Finally, we give an overview of an implementation in C++ using Open MPI and discuss some experimental results.

Figures (19)

• Figure 1: Delaunay triangulation (DT) of some random points
• Figure 2: Shortest distance (green) between two vertices of a non-Gabriel edge versus actual point of first intersection (red) of the corresponding Voronoi cells
• Figure 3: Depiction of $\mathrm{A}_{r}(\mathbb{X})$ for increasing values.
• Figure 4: Persistent homology barcodes in dimension 0 (red) and in dimension 1 (blue) of $\mathrm{A}(\mathbb{X})$. One red bar "never ends", which we indicate with a vertical yellow line with the "inf" label.
• Figure 5: Barcode of $\mathrm{PH}_1(\mathrm{A}(\mathbb{X}))$ and cycle representatives using matching colours.

Theorems & Definitions (24)

• Lemma 2.1: Lemma of Delaunay
• Theorem 2.2: Nerve theorem
• Example 2.3
• Example 2.4
• Example 2.5
• Definition 2.6
• Example 2.7
• Definition 2.8
• Definition 2.9
• Example 2.10