On heterogeneous distributed storage systems: bounds and code constructions
In Distributed Storage Systems (DSSs), usually, data is stored using encoded packets on different chunk servers. In this thesis, we have considered heterogeneous DSSs in which each node may store a different number of packets and each having different repair bandwidth. In particular, a data collector can reconstruct the file at time t using some specific nodes in the system, and for arbitrary node failure, the system can be repaired by some set of arbitrary nodes. Using min-cut bound, we investigate the fundamental trade-off between storage and repair cost for our model of heterogeneous DSS. In particular, the problem is formulated as a biobjective optimization linear programming problem. For an arbitrary DSS, it is shown that the calculated min-cut bound is tight. For a DSS with symmetric parameters, a well known class of Distributed Replication-based Simple Storage (DRESS) codes is Fractional Repetition (FR) code. In such systems, the replicas of data packets encoded by Maximum Distance Separable (MDS) code, are stored on distributed nodes. Most of the available constructions for the FR codes are based on combinatorial designs and Graph theory. In this thesis, FR codes with generalized parameters (such as replication factor of each packet are not same and storage capacity of each node are also not same) are considered, and it is called as Generalized Fractional Repetition (GFR) code. For the GFR code, we propose an elegant sequence-based approach for the construction of the GFR code called Flower codes. Further, it is shown that any GFR code is equivalent to a Flower code. The condition for the universally good GFR code is given on such sequences. For some sequences, the universally good GFR codes are explored. In general, for the GFR codes with non-uniform parameters, bounds on the GFR code rate and DSS code rate are studied. Further, we have shown that a GFR code corresponds to a hypergraph. Using the correspondence, properties and bounds of a hypergraph are directly mapped to the associated GFR code. In general, necessary and sufficient conditions for the existence of a GFR code is obtained using the correspondence. It is also shown that any GFR code associated with a linear hypergraph is universally good.
- PhD Theses