  We propose the clock Monte Carlo technique for sampling each successive chain
step in constant time. It is built on a recently proposed factorized transition
filter and its core features include its O(1) computational complexity and its
generality. We elaborate how it leads to the clock factorized Metropolis (clock
FMet) method, and discuss its application in other update schemes. By grouping
interaction terms into boxes of tunable sizes, we further formulate a variant
of the clock FMet algorithm, with the limiting case of a single box reducing to
the standard Metropolis method. A theoretical analysis shows that an overall
acceleration of ${\rm O}(N^\kappa)$ ($0 \! \leq \! \kappa \! \leq \! 1$) can be
achieved compared to the Metropolis method, where $N$ is the system size and
the $\kappa$ value depends on the nature of the energy extensivity. As a
systematic test, we simulate long-range O$(n)$ spin models in a wide parameter
regime: for $n \! = \! 1,2,3$, with disordered algebraically decaying or
oscillatory Ruderman-Kittel-Kasuya-Yoshida-type interactions and with and
without external fields, and in spatial dimensions from $d \! = \! 1, 2, 3$ to
mean-field. The O(1) computational complexity is demonstrated, and the expected
acceleration is confirmed. Its flexibility and its independence from the
interaction range guarantee that the clock method would find decisive
applications in systems with many interaction terms.
