  We describe a framework to significantly reduce the computational effort to
evaluate large deviation functions of time integrated observables within
nonequilibrium steady states. We do this by incorporating an auxiliary dynamics
into trajectory based Monte Carlo calculations, through a transformation of the
system's propagator using an approximate guiding function. This procedure
importance samples the trajectories that most contribute to the large deviation
function, mitigating the exponentially complexity of such calculations. We
illustrate the method by studying driven diffusions and interacting lattice
models in one and two dimensions. Our work offers an avenue to calculate large
deviation functions for high dimensional systems driven far from equilibrium.
