Who was the first person/what was the first paper to invent/mention the persistence algorithm? (see page 6 of these lecture notes).
That'd be Delfinado and Edelsbrunner in An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere. Check the last section. Similar algorithms may have been known by others, though, since persistent homology was discovered independently in a couple other places and worked on for a while before they all found one another.