  We study a directed flipping process that underlies the performance of the
random edge simplex algorithm. In this stochastic process, which takes place on
a one-dimensional lattice whose sites may be either occupied or vacant,
occupied sites become vacant at a constant rate and simultaneously cause all
sites to the right to change their state. This random process exhibits rich
phenomenology. First, there is a front, defined by the position of the
left-most occupied site, that propagates at a nontrivial velocity. Second, the
front involves a depletion zone with an excess of vacant sites. The total
excess D_k increases logarithmically, D_k ~ ln k, with the distance k from the
front. Third, the front exhibits rejuvenation -- young fronts are vigorous but
old fronts are sluggish. We investigate these phenomena using a quasi-static
approximation, direct solutions of small systems, and numerical simulations.
