  In the context of state-space models, skeleton-based smoothing algorithms
rely on a backward sampling step which by default has a $\mathcal O(N^2)$
complexity (where $N$ is the number of particles). Existing improvements in the
literature are unsatisfactory: a popular rejection sampling -- based approach,
as we shall show, might lead to badly behaved execution time; another rejection
sampler with stopping lacks complexity analysis; yet another MCMC-inspired
algorithm comes with no stability guarantee. We provide several results that
close these gaps. In particular, we prove a novel non-asymptotic stability
theorem, thus enabling smoothing with truly linear complexity and adequate
theoretical justification. We propose a general framework which unites most
skeleton-based smoothing algorithms in the literature and allows to
simultaneously prove their convergence and stability, both in online and
offline contexts. Furthermore, we derive, as a special case of that framework,
a new coupling-based smoothing algorithm applicable to models with intractable
transition densities. We elaborate practical recommendations and confirm those
with numerical experiments.
