  Persistent homology is a crucial invariant that is used in many areas to
understand data. The $O(N^4)$ run time is a hindrance to its use on most large
datasets. We give a parallelization method to utilize multi-core machines and
clusters. We implement the computation of the $0^{th}$ persistent homology with
OpenMP parallelization and observe a 1.75 fold performance increase by using 2
threads on a dual core machine. We also benchmark the computation using larger
numbers of threads and show that the thread computational overhead decreases
performance. With GPU parallelization, we analytically and empirically decrease
the run time scaling from $O(N^4)$ to $O(N^3)$ and even $O(N^2)$ where $N$ is
the number of data points, for a large enough GPU. Next, we analytically show
run time scaling $O(N)$ for an even larger GPU.
