Locally Repairable Codes with Unequal Locality Requirements

When a node in a distributed storage system fails, it needs to be promptly repaired to maintain system integrity. While typical erasure codes can provide a significant storage advantage over replication, they suffer from poor repair efficiency. Locally repairable codes (LRCs) tackle this issue by reducing the number of nodes participating in the repair process (locality), with the cost of reduced minimum distance. In this paper, we study the tradeoff between locality and minimum distance of LRCs with local codes that have an arbitrary distance requirement. Unlike existing methods where a common locality requirement is imposed on every node, we allow the locality requirements vary arbitrarily from node to node. Such a property can be an advantage for distributed storage systems with non-homogeneous characteristics. We present Singleton-type distance upper bounds and also provide an optimal code construction with respect to these bounds. In addition, the feasible rate region is characterized by a dimension upper bound that does not depend on the distance. In line with the literature, we first derive bounds based on the notion of locality profile, which refers to the symbol localities specified in a minimum sense. Since the notion of locality profile is less desirable than the locality requirement, which all conventional problem formulations are also based on, we provide locality requirement-based bounds by exhaustively comparing over the relevant locality profiles. Furthermore, and most importantly, we derive bounds with direct expressions in terms of the locality requirement.