Codes for Graph Erasures

Motivated by systems where the information is represented by a graph, such as neural networks, associative memories, and distributed systems, we present in this work a new class of codes, called codes over graphs. Under this paradigm, the information is stored on the edges of an undirected graph, and a code over graphs is a set of graphs. A node failure is the event where all edges in the neighborhood of the failed node have been erased. We say that a code over graphs can tolerate $\rho$ node failures if it can correct the erased edges of any $\rho$ failed nodes in the graph. While the construction of such codes can be easily accomplished by MDS codes, their field size has to be at least $O(n^2)$, when $n$ is the number of nodes in the graph. In this work we present several constructions of codes over graphs with smaller field size. In particular, we present optimal codes over graphs correcting two node failures over the binary field, when the number of nodes in the graph is a prime number. We also present a construction of codes over graphs correcting $\rho$ node failures for all $\rho$ over a field of size at least $(n+1)/2-1$, and show how to improve this construction for optimal codes when $\rho=2,3$.Comment: To appear in IEEE International Symposium on Information Theor

Code Constructions for Distributed Storage With Low Repair Bandwidth and Low Repair Complexity

We present the construction of a family of erasure correcting codes for distributed storage that achieve low repair bandwidth and complexity at the expense of a lower fault tolerance. The construction is based on two classes of codes, where the primary goal of the first class of codes is to provide fault tolerance, while the second class aims at reducing the repair bandwidth and repair complexity. The repair procedure is a two- step procedure where parts of the failed node are repaired in the first step using the first code. The downloaded symbols during the first step are cached in the memory and used to repair the remaining erased data symbols at minimal additional read cost during the second step. The first class of codes is based on MDS codes modified using piggybacks, while the second class is designed to reduce the number of additional symbols that need to be downloaded to repair the remaining erased symbols. We numerically show that the proposed codes achieve better repair bandwidth compared to MDS codes, codes constructed using piggybacks, and local reconstruction/Pyramid codes, while a better repair complexity is achieved when compared to MDS, Zigzag, Pyramid codes, and codes constructed using piggybacks.Comment: To appear in IEEE Transactions on Communication