  In this paper we examine the role of the so called $c$-parallel updating
schemes in relaxation from disordered states to the final ferromagnetic steady
state. We investigate two zero-temperature single-spin flip dynamics on a one
dimensional lattice of length $L$: inflow (i.e. generalized zero-temperature
Glauber dynamics) and outflow opinion dynamics. The varying $c$ allows us to
change the updating scheme from random sequential updating ($c=1/L$) to
deterministic synchronous updating (for $c=1$). We show how the mean relaxation
times depend on $c$ and scale with the system size $L$. Moreover, we
empirically find an analytical formula for the ratio between mean relaxation
times for inflow and outflow dynamics. Results obtained in this paper suggest
that in some sense the original zero-temperature Glauber dynamics is a critical
one among a broader class of inflow dynamics.
