{'fraction_non_alphanumeric': 0.02601156069364162, 'fraction_numerical': 0.0, 'mean_word_length': 4.944206008583691, '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}}
Centralized multi-node repair in distributed storage

In distributed storage systems, multiple storage node failures are frequent and efficiently recovering them is crucial for high system performance. In this work, we consider the problem of repairing multiple failures in a centralized way, which can be desirable in many data storage configurations. We first establish the tradeoff between the repair bandwidth and the storage size for functional repair. Using a graph-theoretic approach, the optimal tradeoff is identified as the solution to an integer optimization problem, for which we derive a closed-form expression. When the number of erasures e satisfies e ≥ k, k being the minimum number of nodes needed to reconstruct the entire data, the tradeoff reduces to a single point, for which we provide an explicit code construction. Expressions of the extreme points, namely the minimum storage multi-node repair (MSMR) and minimum bandwidth multi-node repair (MBMR) points, are also derived. Furthermore, we prove that functional MBMR point is not achievable for linear exact repair codes. Finally, for e | k and e | d, where d is the number of helper nodes during repair, we show that the functional repair tradeoff is not achievable under exact repair, except for maybe a small portion near the MSMR point, which parallels the results for single erasure repair by Shah et al.