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We consider a set up where a file of size M is stored in n distributedstorage nodes, using an (n,k) minimum storage regenerating (MSR) code, i.e., amaximum distance separable (MDS) code that also allows efficient exact-repairof any failed node. The problem of interest in this paper is to minimize therepair bandwidth B for exact regeneration of a single failed node, i.e., theminimum data to be downloaded by a new node to replace the failed node by itsexact replica. Previous work has shown that a bandwidth of B=[M(n-1)]/[k(n-k)]is necessary and sufficient for functional (not exact) regeneration. It hasalso been shown that if k < = max(n/2, 3), then there is no extra cost of exactregeneration over functional regeneration. The practically relevant setting oflow-redundancy, i.e., k/n>1/2 remains open for k>3 and it has been shown thatthere is an extra bandwidth cost for exact repair over functional repair inthis case. In this work, we adopt into the distributed storage context anasymptotically optimal interference alignment scheme previously proposed byCadambe and Jafar for large wireless interference networks. With this scheme wesolve the problem of repair bandwidth minimization for (n,k) exact-MSR codesfor all (n,k) values including the previously open case of k > \max(n/2,3). Ourmain result is that, for any (n,k), and sufficiently large file sizes, there isno extra cost of exact regeneration over functional regeneration in terms ofthe repair bandwidth per bit of regenerated data. More precisely, we show thatin the limit as M approaches infinity, the ratio B/M = (n-1)/(k(n-k))$.

Distributed Data Storage with Minimum Storage Regenerating Codes - Exact and Functional Repair are Asymptotically Equally Efficient