Rack-Aware Regenerating Codes with Fewer Helper Racks

We consider the rack-aware storage system where $n$ nodes are organized in $\bar{n}$ racks each containing $u$ nodes, and any $k$ nodes can retrieve the stored file. Moreover, any single node erasure can be recovered by downloading data from $\bar{d}$ helper racks as well as the remaining $u\!-\!1$ nodes in the same rack. Previous work mostly focuses on minimizing the cross-rack repair bandwidth under the condition $\bar{d}\geq \bar{k}$, where $\bar{k}=\lfloor\frac{k}{u}\rfloor$. However, $\bar{d}\geq \bar{k}$ is not an intrinsic condition for the rack-aware storage model. In this paper, we establish a tradeoff between the storage overhead and cross-rack repair bandwidth for the particularly interesting case $\bar{d}\!<\!\bar{k}$. Furthermore, we present explicit constructions of codes with parameters lying on the tradeoff curve respectively at the minimum storage point and minimum bandwidth point. The codes are scalar or have sub-packetization $\bar{d}$, and operate over finite fields of size comparable to $n$. Regarding $\bar{d}$ as the repair degree, these codes combine the advantage of regenerating codes in minimizing the repair bandwidth and that of locally repairable codes in reducing the repair degree. Moreover, they also abandon the restriction of MBR codes having storage overhead no less than $2\times$ and that of high-rate MSR codes having exponential sub-packetization level.