Minimum Repair Bandwidth for Exact Regeneration in Distributed Storage

We consider a set up where a file of size $M$ is stored in $n$ distributed storage nodes, using an (n,k) minimum storage regenerating (MSR) code, i.e., a maximum distance separable (MDS) code that also allows efficient exact-repair of any failed node. The MDS property ensures that the original file can be reconstructed even if any $n-k$ storage nodes fail. When a node fails, a new node collects data from the remaining n-1 healthy nodes and repairs the failed node. The problem of interest in this paper is to minimize the repair bandwidth B for exact regeneration of the failed node, i.e., the minimum data to be downloaded by the new node to replace the failed node by its exact replica. Previous work has shown that with random network coding, a bandwidth of B=M(n-1/(k(n-k)) is necessary and sufficient for functional (not exact) regeneration, i.e., if the repaired new node need not be exactly identical to the failed node, but only information equivalent to it. It has also been shown using interference alignment based techniques that if $k leq max(n/2, 3)$ then, surprisingly, there is no extra cost of exact regeneration over functional regeneration and the same repair bandwidth of M(n-1)/(k(n-k)) suffices for exact regeneration. The practically relevant setting of low-redundancy, i.e., k/n>1/2 remained open for k>3 and it has been shown that there is an extra bandwidth cost for exact repair over functional repair in this case. In this work, we adopt into the distributed storage context an asymptotically optimal interference alignment scheme previously proposed by Cadambe and Jafar for large wireless interference networks. With this scheme we solve the problem of repair bandwidth minimization for (n,k) exact-MSR codes for all (n,k) values including the previously open case of k > max(n/2,3). Our main result is that, for any (n,k), and sufficiently large file sizes, there is no extra cost of exact regeneration over f- - unctional regeneration in terms of the repair bandwidth per bit of regenerated data. More precisely, we show that $lim_{Mto infty} frac{B}{M} = frac{n-1}{k(n-k)}$. The result is analogous to the wireless interference channel setting where exact interference alignment through linear beamforming is seen to be infeasible for more than 3 users, but almost perfect alignment is achieved asymptotically by the Cadambe-Jafar scheme over a large number of signaling dimensions for any number of users.