Fast Persistent Homology Computation for Functions on $\mathbb{R}$

0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the connected components and we get the persistence diagram in time $O(n\alpha(n))$. The running time is thus usually dominated by sorting the edges in $\Theta(n\log(n))$. A little-known fact is that, in the particularly simple case of studying the sublevel sets of a piecewise-linear function on $\mathbb{R}$ or $\mathbb{S}^1$, persistence can actually be computed in linear time. This note presents a simple algorithm that achieves this complexity and an extension to image persistence. An implementation is available in Gudhi.