Interior-point regenerating codes on graphs

We consider the use of regenerating codes in distributed storage systems where connections between the nodes are constrained by a graph. In this setting the cost of node repair is determined by the graphical distance from the helper nodes to the failed node. In our recent work (arXiv:2108:00939) we considered the MSR case, showing that linear MSR codes are amenable to intermediate processing of the information, resulting in reduced repair bandwidth which also meets the lower bound on the minimum repair cost. Here we extend this study to the non-MSR case. We derive a lower bound on the repair bandwidth and formulate repair procedures with intermediate processing for several families of regenerating codes, with an emphasis on the recent constructions from multilinear algebra. We also consider intermediate processing for the problem of partial node repair.