A new class of Monte Carlo algorithms for atomistic simulation of genuine high polymer systems is proposed. Derivations of two of these algorithms, dubbed "directed internal bridging" (DIB) and "directed end bridging" (DEB), are presented. Their performance is analyzed in detail, using linear united-atom polyethylene of mean chain length C-1000 as an archetypal entangled polymer melt. It is shown in particular that the DEB algorithm is substantially faster than previous alternatives in equilibrating such melts on all length scales. Used in a suitable protocol of mixed Monte Carlo moves, it thus provides the most powerful means available to date for quantitative molecular simulation of such materials, and makes atomic level characterization of realistic high polymer melts a feasible proposition.