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Register a new userComputing Linear Transformations with Unreliable Components
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Added by
Yaoqing Yang
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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We consider the problem of computing a binary linear transformation using unreliable components when all circuit components are unreliable. Two noise models of unreliable components are considered: probabilistic errors and permanent errors. We introduce the ENCODED technique that ensures that the error probability of the computation of the linear transformation is kept bounded below a small constant independent of the size of the linear transformation even when all logic gates in the computation are noisy. Further, we show that the scheme requires fewer operations (in order sense) than its uncoded counterpart. By deriving a lower bound, we show that in some cases, the scheme is order-optimal. Using these results, we examine the gain in energy-efficiency from use of voltage-scaling scheme where gate-energy is reduced by lowering the supply voltage. We use a gate energy-reliability model to show that tuning gate-energy appropriately at different stages of the computation (dynamic voltage scaling), in conjunction with ENCODED, can lead to order-sense energy-savings over the classical uncoded approach. Finally, we also examine the problem of computing a linear transformation when noiseless decoders can be used, providing upper and lower bounds to the problem.
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In recent years, several integer programming (IP) approaches were developed for maximum-likelihood decoding and minimum distance computation for binary linear codes. Two aspects in particular have been demonstrated to improve the performance of IP solvers as well as adaptive linear programming decoders: the dynamic generation of forbidden-set (FS) inequalities, a family of valid cutting planes, and the utilization of so-called redundant parity-checks (RPCs). However, to date, it had remained unclear how to solve the exact RPC separation problem (i.e., to determine whether or not there exists any violated FS inequality w.r.t. any known or unknown parity-check). In this note, we prove NP-hardness of this problem. Moreover, we formulate an IP model that combines the search for most violated FS cuts with the generation of RPCs, and report on computational experiments. Empirically, for various instances of the minimum distance problem, it turns out that while utilizing the exact separation IP does not appear to provide a computational advantage, it can apparently be avoided altogether by combining heuristics to generate RPC-based cuts.
In a heterogeneous unreliable multiaccess network, wherein terminals share a common wireless channel with distinctive error probabilities, existing works have showed that a persistent round-robin (RR-P) scheduling policy (i.e., greedy policy) can be arbitrarily worse than the optimum in terms of Age of Information (AoI) under standard Automatic Repeat reQuest (ARQ), and one must resort to Whittles index approach for optimal AoI. In this paper, practical Hybrid ARQ (HARQ) schemes which are widely-used in todays wireless networks are considered. We show that RR-P is very close to optimum with asymptotically many terminals in this case, by explicitly deriving tight, closed-form AoI gaps between optimum and achievable AoI by RR-P. In particular, it is rigorously proved that for RR-P, under HARQ models concerning fading channels (resp. finite-blocklength regime), the relative AoI gap compared with the optimum is within a constant of $(sqrt{e}-1)^2/4sqrt{e} cong 6.4%$ (resp. $6.2%$ with error exponential decay rate of $0.5$). In addition, RR-P enjoys the distinct advantage of implementation simplicity with channel-unaware and easy-to-decentralize operations, making it favorable in practice.
Age of Information (AoI), defined as the time elapsed since the generation of the latest received update, is a promising performance metric to measure data freshness for real-time status monitoring. In many applications, status information needs to be extracted through computing, which can be processed at an edge server enabled by mobile edge computing (MEC). In this paper, we aim to minimize the average AoI within a given deadline by jointly scheduling the transmissions and computations of a series of update packets with deterministic transmission and computing times. The main analytical results are summarized as follows. Firstly, the minimum deadline to guarantee the successful transmission and computing of all packets is given. Secondly, a emph{no-wait computing} policy which intuitively attains the minimum AoI is introduced, and the feasibility condition of the policy is derived. Finally, a closed-form optimal scheduling policy is obtained on the condition that the deadline exceeds a certain threshold. The behavior of the optimal transmission and computing policy is illustrated by numerical results with different values of the deadline, which validates the analytical results.
We analyse and compare the complexity of several algorithms for computing modular polynomials. We show that an algorithm relying on floating point evaluation of modular functions and on interpolation, which has received little attention in the literature, has a complexity that is essentially (up to logarithmic factors) linear in the size of the computed polynomials. In particular, it obtains the classical modular polynomials $Phi_ell$ of prime level $ell$ in time O (ell^3 log^4 ell log log ell). Besides treating modular polynomials for $Gamma^0 (ell)$, which are an important ingredient in many algorithms dealing with isogenies of elliptic curves, the algorithm is easily adapted to more general situations. Composite levels are handled just as easily as prime levels, as well as polynomials between a modular function and its transform of prime level, such as the Schlafli polynomials and their generalisations. Our distributed implementation of the algorithm confirms the theoretical analysis by computing modular equations of record level around 10000 in less than two weeks on ten processors.
This paper addresses the problem of constructing MDS codes that enable exact repair of each code block with small repair bandwidth, which refers to the total amount of information flow from the remaining code blocks during the repair process. This problem naturally arises in the context of distributed storage systems as the node repair problem [7]. The constructions of exact-repairable MDS codes with optimal repair-bandwidth require working with large sub-packetization levels, which restricts their employment in practice. This paper presents constructions for MDS codes that simultaneously provide both small repair bandwidth and small sub-packetization level. In particular, this paper presents two general approaches to construct exact-repairable MDS codes that aim at significantly reducing the required sub-packetization level at the cost of slightly sub-optimal repair bandwidth. The first approach gives MDS codes that have repair bandwidth at most twice the optimal repair-bandwidth. Additionally, these codes also have the smallest possible sub-packetization level $ell = O(r)$, where $r$ denotes the number of parity blocks. This approach is then generalized to design codes that have their repair bandwidth approaching the optimal repair-bandwidth at the cost of graceful increment in the required sub-packetization level. The second approach transforms an MDS code with optimal repair-bandwidth and large sub-packetization level into a longer MDS code with small sub-packetization level and near-optimal repair bandwidth. For a given $r$, the obtained codes have their sub-packetization level scaling logarithmically with the code length. In addition, the obtained codes require field size only linear in the code length and ensure load balancing among the intact code blocks in terms of the information downloaded from these blocks during the exact reconstruction of a code block.
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