Rack-aware storage systems organize the storage nodes in racks such that the cross-rack communication cost is much more expensive than the intra-rack communication cost. In this paper, we primarily investigate the cross-rack update bandwidth defined as the average amount of symbols transferred across different racks during an update process of one single node. It is critical to design erasure codes that minimize the cross-rack update bandwidth. Our main contributions are as follows. First, we establish the model of cross-rack update bandwidth of irregular array codes over rack-aware storage systems. Second, we derive the tight lower bound on cross-rack update bandwidth, and define minimum cross-rack update bandwidth (MCUB) codes as the irregular array codes that can achieve our tight lower bound. Third, we derive the tight lower bound on redundancy defined as the total number of parity symbols for MCUB codes, and define minimum redundancy MCUB (MR-MCUB) codes as the MCUB codes that can achieve the redundancy lower bound. Fourth, we present explicit constructions of MR-MCUB codes that achieve both the minimum cross-rack update bandwidth and the minimum redundancy, which means that the two lower bounds are tight. Last, we define intra-rack update bandwidth as the average amount of symbols incurred within one rack in an update process, and derive the lower bound of intra-rack update bandwidth of MCUB codes. Moreover, we show that our MCUB codes constructions can also achieve the lower bound of intra-rack update bandwidth.

In data centers, storage nodes are typically organized in racks and rack-aware regenerating codes (RRCs) can achieve the optimal trade-off between storage capacity and cross-rack repair bandwidth. In this paper, we present explicit constructions for multiple erasure tolerance of rack-aware regenerating codes (MET-RRCs), i.e., RRCs with optimal cross-rack repair bandwidth for multi-node failures. We refine the MET-RRC framework in [2] under more extensive parameters and extend the existing construction methods of RRCs. Specifically, we leverage the parity-check matrix structure to present a general framework for multiple erasure tolerance of minimum storage rack-aware regenerating (MET-MSRR) codes. The existing parity-check matrix construction of MSRR codes can be elucidated with this framework. Moreover, we present the construction for multiple erasure tolerance of minimum bandwidth rack-aware regenerating (MET-MBRR) codes using the product-matrix framework.

Erasure codes have been widely employed in distributed storage systems to provide high data reliability at a cost of small redundancy. Modern distributed storage systems usually organize the storage nodes in racks, in which the cross-rack communication cost is much more expensive than the intra-rack communication cost. When the original data symbols stored in a single node are updated, it is critical to design erasure codes that can update the corresponding coded symbols with the cross-rack update bandwidth defined as the average amount of symbols transferred across different racks as small as possible. In this paper, we first derive a tight lower bound on the cross-rack update bandwidth under the condition of$(n, k)$reconstruction property that is any$k$out of the$n$nodes can retrieve all the data symbols. Moreover, we derive the lower bound on redundancy subject to the minimum cross-rack update bandwidth. Furthermore, we propose explicit constructions that can achieve both the minimum cross-rack update bandwidth and the minimum redundancy.

In distributed storage systems, storage nodes are organized in racks in which the cross-rack communication cost is much more expensive than the intra-rack communication cost. It is critical to design erasure codes that minimize the cross-rack update bandwidth which is defined as the total amount of symbols transferred across different racks during an update process. In this paper, we analyze the cross-rack update bandwidth of erasure codes for distributed storage systems. We derive a lower bound on the cross-rack update bandwidth and show that the proposed lower bound is achievable under certain parameters.