  The computation of Vietoris-Rips persistence barcodes is both
execution-intensive and memory-intensive. In this paper, we study the
computational structure of Vietoris-Rips persistence barcodes, and identify
several unique mathematical properties and algorithmic opportunities with
connections to the GPU. Mathematically and empirically, we look into the
properties of apparent pairs, which are independently identifiable persistence
pairs comprising up to 99% of persistence pairs. We give theoretical upper and
lower bounds of the apparent pair rate and model the average case. We also
design massively parallel algorithms to take advantage of the very large number
of simplices that can be processed independently of each other. Having
identified these opportunities, we develop a GPU-accelerated software for
computing Vietoris-Rips persistence barcodes, called Ripser++. The software
achieves up to 30x speedup over the total execution time of the original Ripser
and also reduces CPU-memory usage by up to 2.0x. We believe our
GPU-acceleration based efforts open a new chapter for the advancement of
topological data analysis in the post-Moore's Law era.
