  The goal of the paper is to lay the foundation for the qualitative analogue
of the classical, quantitative sparse graph limit theory. In the first part of
the paper we introduce the qualitative analogues of the Benjamini-Schramm and
local-global graph limit theories for sparse graphs. The natural limit objects
are continuous actions of finitely generated groups on totally disconnected
compact metric spaces. We prove that the space of weak equivalent classes of
free Cantor actions is compact and contains a smallest element, as in the
measurable case. We will introduce and study various notions of almost
finiteness, the qualitative analogue of hyperfiniteness, for classes of bounded
degree graphs. We prove the almost finiteness of a new class of \'etale
groupoids associated to Cantor actions and construct an example of a
nonamenable, almost finite totally disconnected \'etale groupoid, answering a
query of Suzuki. Motivated by the notions and results on qualitative graph
limits, in the second part of our paper we give a precise definition of
constant-time distributed algorithms on sparse graphs. We construct such
constant-time algorithms for various approximation problems for hyperfinite and
almost finite graph classes. We also prove the Hausdorff convergence of the
spectra of convergent graph sequences in the strongly almost finite category.
