  The bisection method for kinetically constrained models (KCM) of Cancrini,
Martinelli, Roberto and Toninelli is a vital technique applied also beyond KCM.
In this note we present a new way of performing it, based on a novel two-block
dynamics with a probabilistic proof instead of the original spectral one. We
illustrate the method by very directly proving an upper bound on the relaxation
time of KCM like the one for the East model in a strikingly general setting.
Namely, we treat KCM on finite or infinite one-dimensional volumes, with any
boundary condition, conditioned on any of the irreducible components of the
state space, with arbitrary site-dependent state spaces and, most importantly,
arbitrary inhomogeneous rules.
