arXiv subjects

Alexandros G. Dimakis

Publications and source records attributed to Alexandros G. Dimakis.

At least 145 records · Page 8Linked to original sources

arXiv2010

Distributed Storage Codes Meet Multiple-Access Wiretap Channels

We consider {\it i)} the overhead minimization of maximum-distance separable (MDS) storage codes for the repair of a single failed node and {\it ii)} the total secure degrees-of-freedom (S-DoF) maximization in a multiple-access compound wiretap channel. We show that the two problems are connected. Specifically, the overhead minimization for a single node failure of an {\it optimal} MDS code, i.e. one that can achieve the information theoretic overhead minimum, is equivalent to maximizing the S-DoF in a multiple-access compound wiretap channel. Additionally, we show that maximizing the S-DoF in a multiple-access compound wiretap channel is equivalent to minimizing the overhead of an MDS code for the repair of a departed node. An optimal MDS code maps to a full S-DoF channel and a full S-DoF channel maps to an MDS code with minimum repair overhead for one failed node. We also state a general framework for code-to-channel and channel-to-code mappings and performance bounds between the two settings. The underlying theme for all connections presented is interference alignment (IA). The connections between the two problems become apparent when we restate IA as an optimization problem. Specifically, we formulate the overhead minimization and the S-DoF maximization as rank constrained, sum-rank and max-rank minimization problems respectively. The derived connections allow us to map repair strategies of recently discovered repair codes to beamforming matrices and characterize the maximum S-DoF for the single antenna multiple-access compound wiretap channel.

cs.IT↗

arXiv2010

Rebuilding for Array Codes in Distributed Storage Systems

In distributed storage systems that use coding, the issue of minimizing the communication required to rebuild a storage node after a failure arises. We consider the problem of repairing an erased node in a distributed storage system that uses an EVENODD code. EVENODD codes are maximum distance separable (MDS) array codes that are used to protect against erasures, and only require XOR operations for encoding and decoding. We show that when there are two redundancy nodes, to rebuild one erased systematic node, only 3/4 of the information needs to be transmitted. Interestingly, in many cases, the required disk I/O is also minimized.

cs.IT↗

arXiv2010

Symmetric Allocations for Distributed Storage

We consider the problem of optimally allocating a given total storage budget in a distributed storage system. A source has a data object which it can code and store over a set of storage nodes; it is allowed to store any amount of coded data in each node, as long as the total amount of storage used does not exceed the given budget. A data collector subsequently attempts to recover the original data object by accessing each of the nodes independently with some constant probability. By using an appropriate code, successful recovery occurs when the total amount of data in the accessed nodes is at least the size of the original data object. The goal is to find an optimal storage allocation that maximizes the probability of successful recovery. This optimization problem is challenging because of its discrete nature and nonconvexity, despite its simple formulation. Symmetric allocations (in which all nonempty nodes store the same amount of data), though intuitive, may be suboptimal; the problem is nontrivial even if we optimize over only symmetric allocations. Our main result shows that the symmetric allocation that spreads the budget maximally over all nodes is asymptotically optimal in a regime of interest. Specifically, we derive an upper bound for the suboptimality of this allocation and show that the performance gap vanishes asymptotically in the specified regime. Further, we explicitly find the optimal symmetric allocation for a variety of cases. Our results can be applied to distributed storage systems and other problems dealing with reliability under uncertainty, including delay tolerant networks (DTNs) and content delivery networks (CDNs).

cs.IT↗

arXiv2010

Efficient Algorithms for Renewable Energy Allocation to Delay Tolerant Consumers

We investigate the problem of allocating energy from renewable sources to flexible consumers in electricity markets. We assume there is a renewable energy supplier that provides energy according to a time-varying (and possibly unpredictable) supply process. The plant must serve consumers within a specified delay window, and incurs a cost of drawing energy from other (possibly non-renewable) sources if its own supply is not sufficient to meet the deadlines. We formulate two stochastic optimization problems: The first seeks to minimize the time average cost of using the other sources (and hence strives for the most efficient utilization of the renewable source). The second allows the renewable source to dynamically set a price for its service, and seeks to maximize the resulting time average profit. These problems are solved via the Lyapunov optimization technique. Our resulting algorithms do not require knowledge of the statistics of the time-varying supply and demand processes and are robust to arbitrary sample path variations.

math.OC↗

arXiv2010

Security in Distributed Storage Systems by Communicating a Logarithmic Number of Bits

We investigate the problem of maintaining an encoded distributed storage system when some nodes contain adversarial errors. Using the error-correction capabilities that are built into the existing redundancy of the system, we propose a simple linear hashing scheme to detect errors in the storage nodes. Our main result is that for storing a data object of total size $\size$ using an $(n,k)$ MDS code over a finite field $\F_q$, up to $t_1=\lfloor(n-k)/2\rfloor$ errors can be detected, with probability of failure smaller than $1/ \size$, by communicating only $O(n(n-k)\log \size)$ bits to a trusted verifier. Our result constructs small projections of the data that preserve the errors with high probability and builds on a pseudorandom generator that fools linear functions. The transmission rate achieved by our scheme is asymptotically equal to the min-cut capacity between the source and any receiver.

cs.CR↗

arXiv2010

A Survey on Network Codes for Distributed Storage

Distributed storage systems often introduce redundancy to increase reliability. When coding is used, the repair problem arises: if a node storing encoded information fails, in order to maintain the same level of reliability we need to create encoded information at a new node. This amounts to a partial recovery of the code, whereas conventional erasure coding focuses on the complete recovery of the information from a subset of encoded packets. The consideration of the repair network traffic gives rise to new design challenges. Recently, network coding techniques have been instrumental in addressing these challenges, establishing that maintenance bandwidth can be reduced by orders of magnitude compared to standard erasure codes. This paper provides an overview of the research results on this topic.

cs.IT↗

arXiv2010

Gossip Algorithms for Distributed Signal Processing

Gossip algorithms are attractive for in-network processing in sensor networks because they do not require any specialized routing, there is no bottleneck or single point of failure, and they are robust to unreliable wireless network conditions. Recently, there has been a surge of activity in the computer science, control, signal processing, and information theory communities, developing faster and more robust gossip algorithms and deriving theoretical performance guarantees. This article presents an overview of recent work in the area. We describe convergence rate results, which are related to the number of transmitted messages and thus the amount of energy consumed in the network for gossiping. We discuss issues related to gossiping over wireless links, including the effects of quantization and noise, and we illustrate the use of gossip algorithms for canonical signal processing tasks including distributed estimation, source localization, and compression.

cs.DC↗

arXiv2009

On the Delay of Network Coding over Line Networks

We analyze a simple network where a source and a receiver are connected by a line of erasure channels of different reliabilities. Recent prior work has shown that random linear network coding can achieve the min-cut capacity and therefore the asymptotic rate is determined by the worst link of the line network. In this paper we investigate the delay for transmitting a batch of packets, which is a function of all the erasure probabilities and the number of packets in the batch. We show a monotonicity result on the delay function and derive simple expressions which characterize the expected delay behavior of line networks. Further, we use a martingale bounded differences argument to show that the actual delay is tightly concentrated around its expectation.

cs.IT↗

arXiv2009

Searching for Minimum Storage Regenerating Codes

Regenerating codes allow distributed storage systems to recover from the loss of a storage node while transmitting the minimum possible amount of data across the network. We present a systematic computer search for optimal systematic regenerating codes. To search the space of potential codes, we reduce the potential search space in several ways. We impose an additional symmetry condition on codes that we consider. We specify codes in a simple alternative way, using additional recovered coefficients rather than transmission coefficients and place codes into equivalence classes to avoid redundant checking. Our main finding is a few optimal systematic minimum storage regenerating codes for $n=5$ and $k=3$, over several finite fields. No such codes were previously known and the matching of the information theoretic cut-set bound was an open problem.

cs.IT↗

arXiv2009

Near-Optimal Detection in MIMO Systems using Gibbs Sampling

In this paper we study a Markov Chain Monte Carlo (MCMC) Gibbs sampler for solving the integer least-squares problem. In digital communication the problem is equivalent to performing Maximum Likelihood (ML) detection in Multiple-Input Multiple-Output (MIMO) systems. While the use of MCMC methods for such problems has already been proposed, our method is novel in that we optimize the "temperature" parameter so that in steady state, i.e. after the Markov chain has mixed, there is only polynomially (rather than exponentially) small probability of encountering the optimal solution. More precisely, we obtain the largest value of the temperature parameter for this to occur, since the higher the temperature, the faster the mixing. This is in contrast to simulated annealing techniques where, rather than being held fixed, the temperature parameter is tended to zero. Simulations suggest that the resulting Gibbs sampler provides a computationally efficient way of achieving approximative ML detection in MIMO systems having a huge number of transmit and receive dimensions. In fact, they further suggest that the Markov chain is rapidly mixing. Thus, it has been observed that even in cases were ML detection using, e.g. sphere decoding becomes infeasible, the Gibbs sampler can still offer a near-optimal solution using much less computations.

cs.IT↗

arXiv2009

LP Decoding meets LP Decoding: A Connection between Channel Coding and Compressed Sensing

This is a tale of two linear programming decoders, namely channel coding linear programming decoding (CC-LPD) and compressed sensing linear programming decoding (CS-LPD). So far, they have evolved quite independently. The aim of the present paper is to show that there is a tight connection between, on the one hand, CS-LPD based on a zero-one measurement matrix over the reals and, on the other hand, CC-LPD of the binary linear code that is obtained by viewing this measurement matrix as a binary parity-check matrix. This connection allows one to translate performance guarantees from one setup to the other.

cs.IT↗

arXiv2009

Sparse Recovery of Positive Signals with Minimal Expansion

We investigate the sparse recovery problem of reconstructing a high-dimensional non-negative sparse vector from lower dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes are crucial in applications, such as DNA microarrays and sensor networks, where dense measurements are not practically feasible. One possible construction uses the adjacency matrices of expander graphs, which often leads to recovery algorithms much more efficient than $\ell_1$ minimization. However, to date, constructions based on expanders have required very high expansion coefficients which can potentially make the construction of such graphs difficult and the size of the recoverable sets small. In this paper, we construct sparse measurement matrices for the recovery of non-negative vectors, using perturbations of the adjacency matrix of an expander graph with much smaller expansion coefficient. We present a necessary and sufficient condition for $\ell_1$ optimization to successfully recover the unknown vector and obtain expressions for the recovery threshold. For certain classes of measurement matrices, this necessary and sufficient condition is further equivalent to the existence of a "unique" vector in the constraint set, which opens the door to alternative algorithms to $\ell_1$ minimization. We further show that the minimal expansion we use is necessary for any graph for which sparse recovery is possible and that therefore our construction is tight. We finally present a novel recovery algorithm that exploits expansion and is much faster than $\ell_1$ optimization. Finally, we demonstrate through theoretical bounds, as well as simulation, that our method is robust to noise and approximate sparsity.

cs.IT↗

arXiv2008

Probabilistic Analysis of Linear Programming Decoding

We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous non-asymptotic results for LDPC codes, and in particular exceeds the best previous finite-length result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bit-flipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a generalized matching. An interesting by-product of our analysis is to establish the existence of ``probabilistic expansion'' in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, for sets much larger than in the classical worst-case setting.

cs.IT↗

arXiv2008

Network Coding for Distributed Storage Systems

Distributed storage systems provide reliable access to data through redundancy spread over individually unreliable nodes. Application scenarios include data centers, peer-to-peer storage systems, and storage in wireless networks. Storing data using an erasure code, in fragments spread across nodes, requires less redundancy than simple replication for the same level of reliability. However, since fragments must be periodically replaced as nodes fail, a key question is how to generate encoded fragments in a distributed way while transferring as little data as possible across the network. For an erasure coded system, a common practice to repair from a node failure is for a new node to download subsets of data stored at a number of surviving nodes, reconstruct a lost coded block using the downloaded data, and store it at the new node. We show that this procedure is sub-optimal. We introduce the notion of regenerating codes, which allow a new node to download \emph{functions} of the stored data from the surviving nodes. We show that regenerating codes can significantly reduce the repair bandwidth. Further, we show that there is a fundamental tradeoff between storage and repair bandwidth which we theoretically characterize using flow arguments on an appropriately constructed graph. By invoking constructive results in network coding, we introduce regenerating codes that can achieve any point in this optimal tradeoff.

cs.NI↗

arXiv2007

Guessing Facets: Polytope Structure and Improved LP Decoding

In this paper we investigate the structure of the fundamental polytope used in the Linear Programming decoding introduced by Feldman, Karger and Wainwright. We begin by showing that for expander codes, every fractional pseudocodeword always has at least a constant fraction of non-integral bits. We then prove that for expander codes, the active set of any fractional pseudocodeword is smaller by a constant fraction than the active set of any codeword. We further exploit these geometrical properties to devise an improved decoding algorithm with the same complexity order as LP decoding that provably performs better, for any blocklength. It proceeds by guessing facets of the polytope, and then resolving the linear program on these facets. While the LP decoder succeeds only if the ML codeword has the highest likelihood over all pseudocodewords, we prove that the proposed algorithm, when applied to suitable expander codes, succeeds unless there exist a certain number of pseudocodewords, all adjacent to the ML codeword on the LP decoding polytope, and with higher likelihood than the ML codeword. We then describe an extended algorithm, still with polynomial complexity, that succeeds as long as there are at most polynomially many pseudocodewords above the ML codeword.

cs.IT↗

arXiv2007

Geographic Gossip: Efficient Averaging for Sensor Networks

Gossip algorithms for distributed computation are attractive due to their simplicity, distributed nature, and robustness in noisy and uncertain environments. However, using standard gossip algorithms can lead to a significant waste in energy by repeatedly recirculating redundant information. For realistic sensor network model topologies like grids and random geometric graphs, the inefficiency of gossip schemes is related to the slow mixing times of random walks on the communication graph. We propose and analyze an alternative gossiping scheme that exploits geographic information. By utilizing geographic routing combined with a simple resampling method, we demonstrate substantial gains over previously proposed gossip protocols. For regular graphs such as the ring or grid, our algorithm improves standard gossip by factors of $n$ and $\sqrt{n}$ respectively. For the more challenging case of random geometric graphs, our algorithm computes the true average to accuracy $ε$ using $O(\frac{n^{1.5}}{\sqrt{\log n}} \log ε^{-1})$ radio transmissions, which yields a $\sqrt{\frac{n}{\log n}}$ factor improvement over standard gossip algorithms. We illustrate these theoretical results with experimental comparisons between our algorithm and standard methods as applied to various classes of random fields.

cs.IT↗

arXiv2007

Network Coding for Distributed Storage Systems

Peer-to-peer distributed storage systems provide reliable access to data through redundancy spread over nodes across the Internet. A key goal is to minimize the amount of bandwidth used to maintain that redundancy. Storing a file using an erasure code, in fragments spread across nodes, promises to require less redundancy and hence less maintenance bandwidth than simple replication to provide the same level of reliability. However, since fragments must be periodically replaced as nodes fail, a key question is how to generate a new fragment in a distributed way while transferring as little data as possible across the network. In this paper, we introduce a general technique to analyze storage architectures that combine any form of coding and replication, as well as presenting two new schemes for maintaining redundancy using erasure codes. First, we show how to optimally generate MDS fragments directly from existing fragments in the system. Second, we introduce a new scheme called Regenerating Codes which use slightly larger fragments than MDS but have lower overall bandwidth use. We also show through simulation that in realistic environments, Regenerating Codes can reduce maintenance bandwidth use by 25 percent or more compared with the best previous design--a hybrid of replication and erasure codes--while simplifying system architecture.

cs.IT↗

arXiv2006

Guessing Facets: Polytope Structure and Improved LP Decoder

A new approach for decoding binary linear codes by solving a linear program (LP) over a relaxed codeword polytope was recently proposed by Feldman et al. In this paper we investigate the structure of the polytope used in the LP relaxation decoding. We begin by showing that for expander codes, every fractional pseudocodeword always has at least a constant fraction of non-integral bits. We then prove that for expander codes, the active set of any fractional pseudocodeword is smaller by a constant fraction than the active set of any codeword. We exploit this fact to devise a decoding algorithm that provably outperforms the LP decoder for finite blocklengths. It proceeds by guessing facets of the polytope, and resolving the linear program on these facets. While the LP decoder succeeds only if the ML codeword has the highest likelihood over all pseudocodewords, we prove that for expander codes the proposed algorithm succeeds even with a constant number of pseudocodewords of higher likelihood. Moreover, the complexity of the proposed algorithm is only a constant factor larger than that of the LP decoder.

cs.IT↗