Erasure coding is widely used for massive storage in data centers to achieve high fault tolerance and low storage redundancy. Since the cross-rack communication cost is often high, it is critical to design erasure codes that minimize the cross-rack repair bandwidth during failure repair. In this paper, we analyze the optimal trade-off between storage redundancy and cross-rack repair bandwidth specifically for data centers, subject to the condition that the original data can be reconstructed from a sufficient number of any non-failed nodes. We characterize the optimal trade-off curve under functional repair, and propose a general family of erasure codes called rack-aware regenerating codes (RRC) , which achieve the optimal trade-off. We further propose exact repair constructions of RRC that have minimum storage redundancy and minimum cross-rack repair bandwidth, respectively. We show that (i) the minimum storage redundancy constructions support a wide range of parameters and have cross-rack repair bandwidth that is strictly less than that of the classical minimum storage regenerating codes in most cases, and (ii) the minimum cross-rack repair bandwidth constructions support all the parameters and have less cross-rack repair bandwidth than that of the minimum bandwidth regenerating codes for almost all of the parameters.
There are many follow-up studies on RC along different directions, such as practical implementation @cite_10 @cite_23 @cite_16 @cite_26 and the repair problem with heterogeneous structures @cite_3 @cite_27 @cite_25 @cite_21 @cite_19 @cite_15 @cite_17 .
@cite_26 @cite_21 @cite_3 @cite_19 @cite_27 @cite_23 @cite_15 @cite_16 @cite_10 @cite_25 @cite_17
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We consider the problem of designing [n; k] linear codes for distributed storage systems (DSS) that satisfy the (r, t)-Local Repair Property, where any t'(<=t) simultaneously failed nodes can be locally repaired, each with locality r. The parameters n, k, r, t are positive integers such that r<k<n and t <= n-k. We consider the functional repair model and the sequential approach for repairing multiple failed nodes. By functional repair, we mean that the packet stored in each newcomer is not necessarily an exact copy of the lost data but a symbol that keep the (r, t)-local repair property. By the sequential approach, we mean that the t' newcomers are ordered in a proper sequence such that each newcomer can be repaired from the live nodes and the newcomers that are ordered before it. Such codes, which we refer to as (n, k, r, t)-functional locally repairable codes (FLRC), are the most general class of LRCs and contain several subclasses of LRCs reported in the literature. In this paper, we aim to optimize the storage overhead (equivalently, the code rate) of FLRCs. We derive a lower bound on the code length n given t belongs to 2,3 and any possible k, r. For t=2, our bound generalizes the rate bound proved in [14]. For t=3, our bound improves the rate bound proved in [10]. We also give some onstructions of exact LRCs for t belongs to 2,3 whose length n achieves the bound of (n, k, r, t)-FLRC, which proves the tightness of our bounds and also implies that there is no gap between the optimal code length of functional LRCs and exact LRCs for certain sets of parameters. Moreover, our constructions are over the binary field, hence are of interest in practice. Abstract In this work, we study graph clustering problems associated with structural balance. One of these problems is known in computer science literature as the correlation-clustering (CC) problem and another (RCC) can be viewed as its relaxed version. The solution of CC and RCC problems has been previously used in the literature as tools for the evaluation of structural balance in a social network. Our aim is to solve these problems to optimality. We describe integer linear programming formulations for these problems which includes the first mathematical formulation for the RCC problem. We also discuss alternative models for the relaxed structural balance and the solution of clustering problems associated with these new models. Numerical experiments are carried out with each formulation on a set of benchmark instances available in the literature. 1. High Performance Systems. An Example Program: Matrix Multiplication. Structure of a Compiler. 2. Programming Language Features. Languages for High Performance. Sequential and Parallel Loops. Roundoff Error. 3. Basic Graph Concepts. Sets, Tuples, Logic. Graphs. Control Dependence. 4. Review of Linear Algebra. Real Vectors and Matrices. Integer Matrices and Lattices. Linear System of Equations. System of Integer Equations. Systems of Linear Inequalities. Systems of Integer Linear Inequalities. Extreme Values of Affine Functions. 5. Data Dependence. Data Dependence in Loops. Data Dependence in Conditionals. Data Dependence in Parallel Loops. Program Dependence Graph. 6. Scalar Analysis with Factored Use-Def Chains. Constructing Factored Use-Def Chains. FUD Chains for Arrays. Finding All Reaching Definitions. Implicit References in FUD Chains. InductionVariables Using FUD Chains. Constant Propagation with FUD Chains. Data Dependence for Scalars. 7. Data Dependence Analysis for Arrays. Building the Dependence System. Dependence System Solvers. General Solver. Summary of Solvers. Complications. Run-time Dependence Testing. 8. Other Dependence Problems. Array Region Analysis. Pointer Analysis. I O Dependence. Procedure Calls. Interprocedural Analysis. 9. Loop Restructuring. Simpile Transformations. Loop Fusion. Loop Fission. Loop Reversal. Loop Interchanging. Loop Skewing. Linear Loop Transformations. Strip-Mining. Loop Tiling. Other Loop Transformations. Interprocedural Transformations. 10. Optimizing for Locality. Single Reference to Each Array. Multiple References. General Tiling. Fission and Fusion for Locality. 11. Concurrency Analysis. Code for Concurrent Loops. Concurrency from Sequential Loops. Concurrency from Parallel Loops. Nested Loops. Roundoff Error. Exceptions and Debuggers. 12. Vector Analysis. Vector Code. Vector Code from Sequential Loops. Vector Code from Forall Loops. Nested Loops. Roundoff Error, Exceptions, and Debuggers. Multivector Computers. 13. Message-Passing Machines. SIMD Machines. MIMD Machines. Data Layout. Parallel Code for Array Assignment. Remote Data Access. Automatic Data Layout. Multiple Array Assignments. Other Topics. 14. Scalable Shared-Memory Machines. Global Cache Coherence. Local Cache Coherence. Latency Tolerant Machines. Glossary. References. Author Index. Index. 0805327304T04062001 Randomized coordinate descent (RCD) methods are state-of-the-art algorithms for training linear predictors via minimizing regularized empirical risk. When the number of examples ( @math ) is much larger than the number of features ( @math ), a common strategy is to apply RCD to the dual problem. On the other hand, when the number of features is much larger than the number of examples, it makes sense to apply RCD directly to the primal problem. In this paper we provide the first joint study of these two approaches when applied to L2-regularized ERM. First, we show through a rigorous analysis that for dense data, the above intuition is precisely correct. However, we find that for sparse and structured data, primal RCD can significantly outperform dual RCD even if @math , and vice versa, dual RCD can be much faster than primal RCD even if @math . Moreover, we show that, surprisingly, a single sampling strategy minimizes both the (bound on the) number of iterations and the overall expected complexity of RCD. Note that the latter complexity measure also takes into account the average cost of the iterations, which depends on the structure and sparsity of the data, and on the sampling strategy employed. We confirm our theoretical predictions using extensive experiments with both synthetic and real data sets. This paper concerns the worst-case complexity of Gauss-Seidel method for solving a positive semi-definite linear system; or equivalently, that of cyclic coordinate descent (C-CD) for minimizing a convex quadratic function. The known provable complexity of C-CD can be @math times slower than gradient descent (GD) and @math times slower than randomized coordinate descent (R-CD). However, these gaps seem rather puzzling since so far they have not been observed in practice; in fact, C-CD usually converges much faster than GD and sometimes comparable to R-CD. Thus some researchers believe the gaps are due to the weakness of the proof, but not that of the C-CD algorithm itself. In this paper we show that the gaps indeed exist. We prove that there exists an example for which C-CD takes at least @math or @math operations, where @math is the condition number, @math is a well-studied quantity that determines the convergence rate of R-CD, and @math hides the dependency on @math . This implies that C-CD can indeed be @math times slower than GD, and @math times slower than R-CD in the worst case. Our result establishes one of the few examples in continuous optimization that demonstrates a large gap between a deterministic algorithm and its randomized counterpart. Based on the example, we establish several almost tight complexity bounds of C-CD for quadratic problems. One difficulty with the analysis of the constructed example is that the spectral radius of a non-symmetric iteration matrix does not necessarily constitutes a lower bound for the convergence rate. Distributed storage systems need to store data redundantly in order to provide some fault-tolerance and guarantee system reliability. Different coding techniques have been proposed to provide the required redundancy more efficiently than traditional replication schemes. However, compared to replication, coding techniques are less efficient for repairing lost redundancy, as they require retrieval of larger amounts of data from larger subsets of storage nodes. To mitigate these problems, several recent works have presented locally repairable codes designed to minimize the repair traffic and the number of nodes involved per repair. Unfortunately, existing methods often lead to codes where there is only one subset of nodes able to repair a piece of lost data, limiting the local repairability to the availability of the nodes in this subset. In this paper, we present a new family of locally repairable codes that allows different trade-offs between the number of contacted nodes per repair, and the number of different subsets of nodes that enable this repair. We show that slightly increasing the number of contacted nodes per repair allows to have repair alternatives, which in turn increases the probability of being able to perform efficient repairs. Finally, we present pg-BLRC, an explicit construction of locally repairable codes with multiple repair alternatives, constructed from partial geometries, in particular from Generalized Quadrangles. We show how these codes can achieve practical lengths and high rates, while requiring a small number of nodes per repair, and providing multiple repair alternatives. The capacity of heterogeneous distributed storage systems under repair dynamics is studied. Examples of these systems include peer-to-peer storage clouds, wireless, and Internet caching systems. Nodes in a heterogeneous system can have different storage capacities and different repair bandwidths. Lower and upper bounds on the system capacity are given. These bounds depend on either the average resources per node, or on a detailed knowledge of the node characteristics. Moreover, the case in which nodes may be compromised by an adversary (passive or active) is addressed and bounds on the secure capacity of the system are derived. One implication of these new results is that symmetric repair maximizes the capacity of a homogeneous system, which justifies the model widely used in the literature. The sheer volume of scholarly publications available online significantly challenges how scholars retrieve the new information available and locate the candidate reference papers. While classical text retrieval and pseudo relevance feedback (PRF) algorithms can assist scholars in accessing needed publications, in this study, we propose an innovative publication ranking method with PRF by leveraging a number of meta-paths on the heterogeneous bibliographic graph. Different meta-paths on the graph address different ranking hypotheses, whereas the pseudo-relevant papers (from the retrieval results) are used as the seed nodes on the graph. Meanwhile, unlike prior studies, we propose "restricted meta-path" facilitated by a new context-rich heterogeneous network extracted from full-text publication content along with citation context. By using learning-to-rank, we integrate 18 different meta-path-based ranking features to derive the final ranking scores for candidate cited papers. Experimental results with ACM full-text corpus show that meta-path-based ranking with PRF on the new graph significantly (p Linear erasure codes with local repairability are desirable for distributed data storage systems. An [n,k,d] linear code having all-symbol (r,δ)-locality, denoted as (r,δ)_a, is considered optimal if it has the actual highest minimum distance of any code of the given parameters n,k,r and δ. A minimum distance bound is given in . The existing results on the existence and the construction of optimal (r, δ)_a linear codes are limited to only two small regions within this special case, namely, i) m=0 and ii) m≥ (v+δ-1)>(δ-1) and δ=2, where m=n mod(r+δ-1) and v=k mod r. This paper investigates the properties and existence conditions for optimal (r,δ)_a linear codes with general r and δ. First, a structure theorem is derived for general optimal (r,δ)_a codes which helps illuminate some of their structure properties. Next, the entire problem space with arbitrary n, k, r and δ is divided into eight different cases (regions) with regard to the specific relations of these parameters. For two cases, it is rigorously proved that no (r,δ)_a linear code can achieve the minimum distance bound in . For four other cases the optimal (r,δ)_a codes are shown to exist over a field of size q≥( n k-1 ), deterministic constructions are proposed. Our new constructive algorithms not only cover more cases, but for the same cases where previous algorithms exist, the new constructions require a smaller field, which translates to potentially lower computational complexity. Our findings substantially enriches the knowledge on optimal (r,δ)_a linear codes, leaving only two cases in which the construction of optimal codes are not yet known. While the volume of scholarly publications has increased at a frenetic pace, accessing and consuming the useful candidate papers, in very large digital libraries, is becoming an essential and challenging task for scholars. Unfortunately, because of language barrier, some scientists (especially the junior ones or graduate students who do not master other languages) cannot efficiently locate the publications hosted in a foreign language repository. In this study, we propose a novel solution, cross-language citation recommendation via Hierarchical Representation Learning on Heterogeneous Graph (HRLHG), to address this new problem. HRLHG can learn a representation function by mapping the publications, from multilingual repositories, to a low-dimensional joint embedding space from various kinds of vertexes and relations on a heterogeneous graph. By leveraging both global (task specific) plus local (task independent) information as well as a novel supervised hierarchical random walk algorithm, the proposed method can optimize the publication representations by maximizing the likelihood of locating the important cross-language neighborhoods on the graph. Experiment results show that the proposed method can not only outperform state-of-the-art baseline models, but also improve the interpretability of the representation model for cross-language citation recommendation task. This paper examines techniques for finding falsifying trajectories of hybrid systems using an approach that we call trajectory splicing. Many formal verification techniques for hybrid systems, including flowpipe construction, can identify plausible abstract counterexamples for property violations. However, there is often a gap between the reported abstract counterexamples and the concrete system trajectories. Our approach starts with a candidate sequence of disconnected trajectory segments, each segment lying inside a discrete mode. However, such disconnected segments do not form concrete violations due to the gaps that exist between the ending state of one segment and the starting state of the subsequent segment. Therefore, trajectory splicing uses local optimization to minimize the gap between these segments, effectively splicing them together to form a concrete trajectory. We demonstrate the use of our approach for falsifying safety properties of hybrid systems using standard optimization techniques. As such, our approach is not restricted to linear systems. We compare our approach with other falsification approaches including uniform random sampling and a robustness guided falsification approach used in the tool S-Taliro. Our preliminary evaluation clearly shows the potential of our approach to search for candidate trajectory segments and use them to find concrete property violations.