  The Vietoris-Rips filtration for an $n$-point metric space is a sequence of
large simplicial complexes adding a topological structure to the otherwise
disconnected space. The persistent homology is a key tool in topological data
analysis and studies topological features of data that persist over many
scales. The fastest algorithm for computing persistent homology of a filtration
has time $O(M(u)+u^2\log^2 u)$, where $u$ is the number of updates (additions
or deletions of simplices), $M(u)=O(u^{2.376})$ is the time for multiplication
of $u\times u$ matrices. For a space of $n$ points given by their pairwise
distances, we approximate the Vietoris-Rips filtration by a zigzag filtration
consisting of $u=o(n)$ updates, which is sublinear in $n$. The constant depends
on a given error of approximation and on the doubling dimension of the metric
space. Then the persistent homology of this sublinear-size filtration can be
computed in time $o(n^2)$, which is subquadratic in $n$.
