This paper studies the fundamental problem of how to reroute $k$ unsplittable flows of a certain demand in a capacitated network from their current paths to their respective new paths, in a congestion-free manner and fast. This scheduling problem has applications in traffic engineering in communication networks and has recently received much attention in software-defined networks, in which updates are distributed over an asynchronous network by a software controller. However, existing algorithms for this problem either have a super-polynomial runtime or only compute feasible schedules, which do not provide any guarantees on the length of the rerouting schedule. This paper presents the first polynomial-time algorithm for computing shortest update schedules to reroute flows in a congestion-free manner. We contribute an almost tight characterization of the polynomial-time tractability of the problem: We present the first polynomial-time solution for this problem for two flows, but also show that even the question whether a feasible update schedule exists, is already NP-hard for six flows. In fact, the presented algorithm runs in linear time and is hence not only optimal in terms of scheduling but also asymptotically optimal in terms of runtime.