  We discuss the non-local Variant Monte Carlo algorithm which has been
successfully employed in the study of driven elastic strings in disordered
media at the depinning threshold. Here we prove two theorems, which establish
that the algorithm satisfies the crucial no-passing rule and that, after some
initial time, the string exclusively moves forward. The Variant Monte Carlo
algorithm overcomes the shortcomings of local methods, as we show by analyzing
the depinning threshold of a single-pin problem.
