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

Linear Run Time of Persistent Homology Computation with GPU Parallelization