In this chapter, we examine the feasibility of the algorithms in Part Two of this book using the implementations described in the last chapter. To make our experiments meaningful, we use real-world data from a variety of different sources, in a variety of different sizes. We time each algorithm to examine its running time behavior in practice. We also gather statistics on significant structural information contained in the data, such as number of conflicts or collisions in the persistence algorithm.

We begin by introducing the three-dimensional data for α-complex filtrations in Section 12.1. This is the data we use for timings and experiments on the persistent algorithm for ℤ2 coefficients in Section 12.2, topological simplification in Section 12.4, and the linking number algorithm in Section 12.6. We introduce alternate data for the persistence algorithm for fields in Section 12.3 as well as the Morse-Smale complex algorithm in Section 12.5. When appropriate, we also discuss additional implementation details not included in the last chapter.

Three-Dimensional Data

In Chapter 1, we motivated the study of topological spaces through a few diverse examples. It is appropriate, therefore, that the experimental data be from disparate sources. We use data that range in scale from nanometers to centimeters. The data will include proteins and inorganic molecules, resolved molecular structures, designed synthetic molecules, acquired samples from real world objects, and sampled mathematical functions.