{'fraction_non_alphanumeric': 0.022753128555176336, 'fraction_numerical': 0.0, 'mean_word_length': 5.423357664233577, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Periodicity in optimal hierarchical checkpointing schemes for adjoint computations

We reexamine the work of Aupy et al. on optimal algorithms for hierarchical adjoint computations, where two levels of memories are available. The previous optimal algorithm had a quadratic execution time. Here, with structural arguments, namely periodicity, on the optimal solution, we provide an optimal algorithm in constant time and space, with appropriate pre-processing. We also provide an asymptotically optimal algorithm for the online problem, when the adjoint chain size is not known before-hand. Again, these algorithms rely on the proof that the optimal solution for hierarchical adjoint computations is weakly periodic. We conjecture a closed-form formula for the period. Finally, we assess the convergence speed of the approximation ratio for the online problem through simulations.