In replica exchange simulations a fast diffusion of the replicas through the temperature space maximizes the efficiency of the statistical sampling. Here, we compare the diffusion speed as measured by the round trip rates for four exchange algorithms. We find different efficiency profiles with optimal average acceptance probabilities ranging from 8% to 41%. The best performance is determined by benchmark simulations for the most widely used algorithm, which alternately tries to exchange all even and all odd replica pairs. By analytical mathematics we show that the excellent performance of this exchange scheme is due to the high diffusivity of the underlying random walk. Graphical abstract The deterministic even//odd exchange scheme alternately tries to exchange even and odd replica pairs (A). The resulting random walk is characterized by a complex elementary process (B).