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We consider a communication problem in which an update of the source messageneeds to be conveyed to one or more distant receivers that are interested inmaintaining specific linear functions of the source message. The setting is onein which the updates are sparse in nature, and where neither the source nor thereceiver(s) is aware of the exact {\em difference vector}, but only know theamount of sparsity that is present in the difference-vector. Under thissetting, we are interested in devising linear encoding and decoding schemesthat minimize the communication cost involved. We show that the optimalsolution to this problem is closely related to the notion of maximallyrecoverable codes (MRCs), which were originally introduced in the context ofcoding for storage systems. In the context of storage, MRCs guarantee optimalerasure protection when the system is partially constrained to have localparity relations among the storage nodes. In our problem, we show that optimalsolutions exist if and only if MRCs of certain kind (identified by the desiredlinear functions) exist. We consider point-to-point and broadcast versions ofthe problem, and identify connections to MRCs under both these settings. Forthe point-to-point setting, we show that our linear-encoder based achievablescheme is optimal even when non-linear encoding is permitted. The theory isillustrated in the context of updating erasure coded storage nodes. We presentexamples based on modern storage codes such as the minimum bandwidthregenerating codes.

Communication Cost for Updating Linear Functions when Message Updates are Sparse: Connections to Maximally Recoverable Codes