  The Markov chain Monte Carlo method is a versatile tool in statistical
physics to evaluate multi-dimensional integrals numerically. For the method to
work effectively, we must consider the following key issues: the choice of
ensemble, the selection of candidate states, the optimization of transition
kernel, algorithm for choosing a configuration according to the transition
probabilities. We show that the unconventional approaches based on the
geometric allocation of probabilities or weights can improve the dynamics and
scaling of the Monte Carlo simulation in several aspects. Particularly, the
approach using the irreversible kernel can reduce or sometimes completely
eliminate the rejection of trial move in the Markov chain. We also discuss how
the space-time interchange technique together with Walker's method of aliases
can reduce the computational time especially for the case where the number of
candidates is large, such as models with long-range interactions.
