Zigzag persistent homology in matrix multiplication time
We present a new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions. Provided that there is an algorithm that multiplies two n n matrices in M(n) time, our algorithm runs in O(M(n) + n2 log2 n) time for a sequence of n additions and deletions. In particular, the running time is O(n2:376), by result of Coppersmith and Winograd. The fastest previously known algorithm for this problem takes O(n3) time in the worst case.