  A fast algorithm for path sampling in path integral Monte Carlo simulations
is proposed. The algorithm utilizes the Levy-Ciesielski implementation of
Lie-Trotter products to achieve a mathematically proven computational cost of
n*log_2(n) with the number of time slices n, despite the fact that each path
variable is updated separately, for reasons of optimality. In this respect, we
demonstrate that updating a group of random variables simultaneously results in
loss of efficiency.
