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 $n\times n$ matrices in $M(n)$ time, our algorithm runs in $O(M(n) log n)$ time if $M(n) = O(n^2)$, and $O(M(n))$ time otherwise, for a sequence of n additions and deletions. In particular, the running time is $O(n^2.376)$, by result of Coppersmith and Winograd. The fastest previously known algorithm for this problem takes $O(n^3)$ time in the worst case.