Logarithmic growth of temporal and spatial complexity in single and double reversals

In the present paper memory efficient implementation of evolutions consisting of two and three alternative sweeps are considered. Two-sweeps evolutions result for instance by a reversal of a program execution needed in automatic differentiation. Three-sweeps evolutions arise through the solution of optimal control problems by Newton’s method or through the spatial discretization of PDEs with adaptive FE grids. The memory efficient implementation of various types of two sweeps evolutions with checkpointing techniques was already discussed in the literature. Nevertheless, in the present paper we introduce and prove new representation for the minimal temporal complexity of single reversals. This new representation gives so far unknown properties of optimal checkpointing strategies and thus provides new insights. After that, we extend the so far known results concerning the logarithmic complexity of single reversal to the double reversal case. In this context we prove in the present paper that the temporal complexity needed to implement a triple-sweep evolution grows as a second power of natural logarithm of a number of time steps representing a sweep length.