Locally repairable codes over a network

Locally repairable (LR) codes are used in distributed storage systems to minimize the number of storage nodes involved in node repair. While existing constructions of LR codes do not take the topology of the storage network into account, this work focuses on designing LR codes over a network. A new concept called node locality is introduced. It is shown that the decision problem of determining whether a binary linear LR code exists, subject to the constraints of code rate, symbol locality, node locality, and repair cost, is NP-complete. The corresponding optimization version, which aims to maximize the code rate, is also considered. It is proved that the problem can be reduced to the minimum k-set cover problem, and can be solved in polynomial time for the special case where the symbol locality is one. For the general case where the symbol locality is greater than or equal to two, the problem is NP-hard and can be approximately solved by a greedy algorithm.