“Twist” Optimization Example: same boundary matrix as before unreduced matrix
Runtime Complexity However, in practice, the runtime is often better than cubic. Note that this complexity is just for the matrix reduction, and doesn’t include building the filtration or boundary matrix.
Bit Tree … … … … 12364 …… ………… The bit tree supports nearly constant time insertion, deletion, and lookup. The “active” column is converted to a bit tree, then columns are added to it, and then it is converted back to a sparse structure. Bauer et. al. recorded significant speed improvements when using the bit tree for column additions.
References Ulrich Bauer, et. al. “PHAT – Persistent Homology Algorithms Toolbox.” http://phat.googlecode.com Herbert Edelsbrunner and John Harer. “Persistent homology: a survey.” in Surveys on discrete and computational geometry: twenty years later. AMS (2008). Afra Zomorodian and Gunnar Carlsson. “Computing persistent homology.” Discrete and Computational Geometry. Vol. 33, no. 2 (2005), p. 249 – 274.