Salim El Rouayheb

On the Index Coding Problem and Its Relation to Network Coding and Matroid Theory

The index coding problem has recently attracted a significant attention from the research community due to its theoretical significance and applications in wireless ad hoc networks. An instance of the index coding problem includes a sender that holds a set of information messages X={x1,...,xk} and a set of receivers R. Each receiver (x,H) in R needs to obtain a message x X and has prior side information consisting of a subset H of X . The sender uses a noiseless communication channel to broadcast encoding of messages in X to all clients. The objective is to find an encoding scheme that minimizes the number of transmissions required to satisfy the demands of all the receivers. In this paper, we analyze the relation between the index coding problem, the more general network coding problem, and the problem of finding a linear representation of a matroid. In particular, we show that any instance of the network coding and matroid representation problems can be efficiently reduced to an instance of the index coding problem. Our reduction implies that many important properties of the network coding and matroid representation problems carry over to the index coding problem. Specifically, we show that vector linear codes outperform scalar linear index codes and that vector linear codes are insufficient for achieving the optimum number of transmissions.

Fractional repetition codes for repair in distributed storage systems

We introduce a new class of exact Minimum-Bandwidth Regenerating (MBR) codes for distributed storage systems, characterized by a low-complexity uncoded repair process that can tolerate multiple node failures. These codes consist of the concatenation of two components: an outer MDS code followed by an inner repetition code. We refer to the inner code as a Fractional Repetition code since it consists of splitting the data of each node into several packets and storing multiple replicas of each on different nodes in the system. Our model for repair is table-based, and thus, differs from the random access model adopted in the literature. We present constructions of Fractional Repetition codes based on regular graphs and Steiner systems for a large set of system parameters. The resulting codes are guaranteed to achieve the storage capacity for random access repair. The considered model motivates a new definition of capacity for distributed storage systems, that we call Fractional Repetition capacity. We provide upper bounds on this capacity, while a precise expression remains an open problem.

On coding for cooperative data exchange

We consider the problem of data exchange by a group of closely-located wireless nodes. In this problem each node holds a set of packets and needs to obtain all the packets held by other nodes. Each of the nodes can broadcast the packets in its possession (or a combination thereof) via a noiseless broadcast channel of capacity one packet per channel use. The goal is to minimize the total number of transmissions needed to satisfy the demands of all the nodes, assuming that they can cooperate with each other and are fully aware of the packet sets available to other nodes. This problem arises in several practical settings, such as peer-to-peer systems and wireless data broadcast. In this paper, we establish upper and lower bounds on the optimal number of transmissions and present an efficient algorithm with provable performance guarantees. The effectiveness of our algorithms is established through numerical simulations.

A randomized algorithm and performance bounds for coded cooperative data exchange

We consider scenarios where wireless clients are missing some packets, but they collectively know every packet. The clients collaborate to exchange missing packets over an error-free broadcast channel with capacity of one packet per channel use. First, we present an algorithm that allows each client to obtain missing packets, with minimum number of transmissions. The algorithm employs random linear coding over a sufficiently large field. Next, we show that the field size can be reduced while maintaining the same number of transmissions. Finally, we establish lower and upper bounds on the minimum number of transmissions that are easily computable and often tight as demonstrated by numerical simulations.

DRESS codes for the storage cloud: Simple randomized constructions

We introduce an efficient family of exact regenerating codes for data storage in large-scale distributed systems. We refer to these new codes as Distributed Replication-based Exact Simple Storage (DRESS) codes. A key property of DRESS codes is their very efficient distributed and uncoded repair and growth processes that have minimum bandwidth, reads and computational overheads. This property is essential for large-scale systems with high reliability and availability requirements. DRESS codes will first encode the file using a Maximum Distance Separable (MDS) code, then place multiple replicas of the coded packets on different nodes in the system. We propose a simple and flexible randomized scheme for placing those replicas based on the balls-and-bins model. Our construction showcases the power of the probabilistic approach in constructing regenerating codes that can be efficiently repaired and grown.

On secure distributed data storage under repair dynamics

We address the problem of securing distributed storage systems against passive eavesdroppers that can observe a limited number of storage nodes. An important aspect of these systems is node failures over time, which demand a repair mechanism aimed at maintaining a targeted high level of system reliability. If an eavesdropper observes a node that is added to the system to replace a failed node, it will have access to all the data downloaded during repair, which can potentially compromise the entire information in the system.We are interested in determining the secrecy capacity of distributed storage systems under repair dynamics, i.e., the maximum amount of data that can be securely stored and made available to a legitimate user without revealing any information to any eavesdropper. We derive a general upper bound on the secrecy capacity and show that this bound is tight for the bandwidth-limited regime which is of importance in scenarios such as peer-to-peer distributed storage systems. We also provide a simple explicit code construction that achieves the capacity for this regime.

An Equivalence Between Network Coding and Index Coding

We show that the network coding and index coding problems are equivalent. This equivalence holds in the general setting which includes linear and nonlinear codes. Specifically, we present a reduction that maps a network coding instance to an index coding instance while preserving feasibility, i.e., the network coding instance has a feasible solution if and only if the corresponding index coding instance is feasible. In addition, we show that one can determine the capacity region of a given network coding instance with colocated sources by studying the capacity region of a corresponding index coding instance. Previous connections between network and index coding were restricted to the linear case.

Deterministic algorithm for the cooperative data exchange problem

In this paper we study the problem of data exchange, where each node in the system has a number of linear combinations of the data packets. Communicating over a public channel, the goal is for all nodes to reconstruct the entire set of the data packets in minimal total number of bits exchanged over the channel. We present a novel divide and conquer based architecture that determines the number of bits each node should transmit. This along with the well known fact, that it is sufficient for the nodes to broadcast linear combinations of their local information, provides a polynomial time deterministic algorithm for reconstructing the entire set of the data packets at all nodes in minimal amount of total communication.

Data secrecy in distributed storage systems under exact repair

The problem of securing data against eavesdropping in distributed storage systems is studied. The focus is on systems that use linear codes and implement exact repair to recover from node failures. The maximum file size that can be stored securely is determined for systems in which all the available nodes help in repair (i.e., repair degree d = n -1, where n is the total number of nodes) and for any number of compromised nodes. Similar results in the literature are restricted to the case of at most two compromised nodes. Moreover, new explicit upper bounds are given on the maximum secure file size for systems with d <; n - 1. The key ingredients for the contribution of this paper are new results on subspace intersection for the data downloaded during repair. The new bounds imply the interesting fact that the maximum amount of data that can be stored securely decreases exponentially with the number of compromised nodes. Whether this exponential decrease is fundamental or is a consequence of the exactness and linearity constraints remains an open question.

Robust network codes for unicast connections: a case study

We consider the problem of establishing reliable unicast connections across a communication network with nonuniform edge capacities. Our goal is to provide instantaneous recovery from single edge failures. With instantaneous recovery, the destination node can decode the packets sent by the source node even if one of the network edges fails, without the need of retransmission or rerouting. It has been recognized that the network coding technique offers significant advantages for this problem over standard solutions such as disjoint path routing and diversity coding. We focus on two cases of practical interest: 1) backup protection of a single flow that can be split into two subflows; and 2) shared backup protection of two unicast flows. We present an efficient network coding algorithm that operates over a small finite field (GF(2)). The small size of the underlying field results in a significant reduction in the computational and communication overhead associated with the practical implementation of the network coding technique. Our algorithm exploits the unique structure of minimum coding networks, i.e., networks that do not contain redundant edges. We also consider the related capacity reservation problem and present an approximation algorithm that finds a solution whose cost is at most two times more than the optimum.