Array Codes Research Articles

Compilation of Shape Operators on Sparse Arrays

We show how to build a compiler for a sparse array language that supports shape operators such as reshaping or concatenating arrays, in addition to compute operators. Existing sparse array programming systems implement generic shape operators for only some sparse data structures, reduce shape operators on other data structures to those, and do not support fusion. Our system compiles sparse array expressions to code that efficiently iterates over reshaped views of irregular sparse data structures, without needing to materialize temporary storage for intermediates. Our evaluation shows that our approach generates sparse array code competitive with popular sparse array libraries: our generated shape operators achieve geometric mean speed-ups of 1.66×–15.3× when compared to hand-written kernels in scipy.sparse and 1.67×–651× when compared to generic implementations in pydata/sparse. For operators that require data structure conversions in these libraries, our generated code achieves geometric mean speed-ups of 7.29×–13.0× when compared to scipy.sparse and 21.3×–511× when compared to pydata/sparse. Finally, our evaluation demonstrates that fusing shape and compute operators improves the performance of several expressions by geometric mean speed-ups of 1.22×–2.23×.

Constructions of Binary MDS Array Codes With Optimal Repair/Access Bandwidth

Constructions of Binary MDS Array Codes With Optimal Repair/Access Bandwidth

New Constructions of MDS Array Codes and Optimal Locally Repairable Array Codes

New Constructions of MDS Array Codes and Optimal Locally Repairable Array Codes

New Systematic MDS Array Codes With Two Parities

New Systematic MDS Array Codes With Two Parities

New Constructions of q-Ary MDS Array Codes With Multiple Parities and Their Effective Decoding

New Constructions of <i>q</i>-Ary MDS Array Codes With Multiple Parities and Their Effective Decoding

A Generic Transformation to Enable Optimal Repair/Access MDS Array Codes With Multiple Repair Degrees

In the literature, most of the known high-rate ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, k</i> ) MDS array codes with the optimal repair property only support a single repair degree (i.e., the number of helper nodes contacted during a repair process) <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> , where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> – 1. However, in practical storage systems, the number of available nodes changes frequently. Thus, it is preferred to construct ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, k</i> ) MDS array codes with multiple repair degrees and the optimal repair property for all nodes. To the best of our knowledge, only two high-rate MDS array codes have such properties in the literature, which were proposed by Ye and Barg (IEEE Trans. Inform. Theory, 63(10), 2001-2014, 2017). However, their sub-packetization levels are relatively large. In this paper, we present a generic construction method that can convert some MDS array codes with a single repair degree into ones with multiple repair degrees and optimal repair property for a set of nodes, while the repair efficiency/degrees of the remaining nodes can be kept. As an application of the generic construction method, an explicit construction of high-rate MDS array code with multiple repair degrees and the optimal access property for all nodes is obtained over a small finite field by choosing the code proposed by Vajha <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> . as the base code. Especially, the sub-packetization level is much smaller than that of the two codes proposed by Ye and Barg concerning the same parameters <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> .

Extended EVENODD+ and STAR+ With Asymptotic Optimal Update and Efficient Decoding

The increasing capacity of storage devices and the growing frequency of correlated failures demands better fault tolerance by using maximum distance separable (MDS) array codes with triple-parity. Although many constructions of triple-parity MDS array codes have been proposed, they either have large update complexity or have large encoding/decoding complexity or only support specified parameters. In this paper, we propose two classes of triple-parity MDS array codes, called extended EVENODD+ and STAR+ that both have asymptotically optimal update complexity and lower encoding/decoding complexity. We show that the existing extended EVENODD and STAR are a special case of our extended EVENODD+ and STAR+, respectively. Moreover, we show that our extended EVENODD+ and STAR+ have strictly less encoding/decoding/update complexity than that of extended EVENODD and STAR for most parameters.

A Vertical-Horizontal Framework for Building Rack-Aware Regenerating Codes

Rack-aware regenerating codes (RRCs) achieve the optimal repair bandwidth for single node failures in the hierarchical data center where nodes are organized into racks and the intra-rack communication is cost-free. In this work, a vertical-horizontal framework is proposed for building RRCs from MDS array codes and regenerating codes (RCs). Particularly for RRCs with the minimum storage (i.e., MSRR codes), the framework is further improved to achieve lower sub-packetization. As a key for building MSRR codes, MSR codes (i.e., RCs with the minimum storage) with improved sub-packetization level are also developed. Applying the newly derived MSR codes into the vertical-horizontal framework, MSRR codes with an improved sub-packetization level are explicitly constructed, and those achieving the lowest sub-packetization by far are proved to exist over sufficiently large fields.

PMDS Array Codes With Small Sub-Packetization, Small Repair Bandwidth/Rebuilding Access

Partial maximum distance separable (PMDS) codes are a kind of erasure codes where the nodes are divided into multiple groups with each forming an MDS code with a smaller code length, thus they allow repairing a failed node with only a few helper nodes and can correct all erasure patterns that are information-theoretically correctable. However, the repair of a failed node of PMDS codes still requires a large amount of communication if the group size is large. Recently, PMDS array codes with each local code being an MSR code were introduced to reduce the repair bandwidth further. However, they require extensive rebuilding access and unavoidably a significant sub packetization level. In this paper, we first propose two constructions of PMDS array codes with two global parities that have smaller sub-packetization levels and much smaller finite fields than the existing one. One construction can support an arbitrary number of local parities and has $(1+\epsilon)$-optimal repair bandwidth (i.e., $(1+\epsilon)$ times the optimal repair bandwidth), while the other one is limited to two local parities but has significantly smaller rebuilding access and its sub packetization level is only $2$. In addition, we present a construction of PMDS array code with three global parities, which has a smaller sub-packetization level as well as $(1+\epsilon)$-optimal repair bandwidth, the required finite field is significantly smaller than existing ones.

A Generalization of Array Codes With Local Properties and Efficient Encoding/Decoding

An <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,k)$ </tex-math></inline-formula> recoverable property array code is composed of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m\times n$ </tex-math></inline-formula> arrays such that any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> out of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> columns suffice to retrieve all the information symbols, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n &gt; k$ </tex-math></inline-formula> . Note that maximum distance separable (MDS) array code is a special <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,k)$ </tex-math></inline-formula> recoverable property array code of size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m\times n$ </tex-math></inline-formula> with the number of information symbols being <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$km$ </tex-math></inline-formula> . Expanded-Blaum-Roth (EBR) codes and Expanded-Independent-Parity (EIP) codes are two classes of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,k)$ </tex-math></inline-formula> recoverable property array codes that can repair any one symbol in a column by locally accessing some other symbols within the column, where the number of symbols <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> in a column is a prime number. By generalizing the constructions of EBR and EIP codes, we propose new <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,k)$ </tex-math></inline-formula> recoverable property array codes, such that any one symbol can be locally recovered and the number of symbols in a column can be not only a prime number but also a power of an odd prime number. Also, we present an efficient encoding/decoding method for the proposed generalized EBR (GEBR) and generalized EIP (GEIP) codes based on the LU factorization of a Vandermonde matrix. We show that the proposed decoding method has less computational complexity than existing methods. Furthermore, we show that the proposed GEBR codes have both a larger minimum symbol distance and a larger recovery ability of erased lines for some parameters when compared to EBR codes. We also present a necessary and sufficient condition of enabling EBR codes to recover any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> erased lines of a slope for any parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> , which was an open problem. Moreover, we show that EBR codes can recover any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> consecutive erased lines of any slope for any parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> .

A Modified Shamir Secret Sharing Scheme With Efficient Encoding

The Shamir secret sharing (SSS) scheme requires a Maximum Distance Separable (MDS) code, and in its most common implementation, a Reed-Solomon (RS) code is used. In this letter, we observe that the encoding procedure can be made simpler and faster by dropping the MDS condition and specifying the possible symbols that can be shared. In particular, the process can be made even faster by using array codes based on exclusive-or (XOR) operations instead of RS codes.

Array BP-XOR Codes for Hierarchically Distributed Matrix Multiplication

A novel fault-tolerant computation technique based on array Belief Propagation (BP)-decodable XOR (BP-XOR) codes is proposed for distributed matrix-matrix multiplication. The proposed scheme is shown to be configurable and suited for modern hierarchical compute architectures such as Graphical Processing Units (GPUs) equipped with multiple nodes, whereby each has many small independent processing units with increased core-to-core communications. The proposed scheme is shown to outperform a few of the well--known earlier strategies in terms of total end-to-end execution time while in presence of slow nodes, called $stragglers$. This performance advantage is due to the careful design of array codes which distributes the encoding operation over the cluster (slave) nodes at the expense of increased master-slave communication. An interesting trade-off between end-to-end latency and total communication cost is precisely described. In addition, to be able to address an identified problem of scaling stragglers, an asymptotic version of array BP-XOR codes based on projection geometry is proposed at the expense of some computation overhead. A thorough latency analysis is conducted for all schemes to demonstrate that the proposed scheme achieves order-optimal computation in both the sublinear as well as the linear regimes in the size of the computed product from an end-to-end delay perspective.

Array Codes for Functional PIR and Batch Codes

A functional PIR array code is a coding scheme which encodes some s information bits into a t × m array such that every linear combination of the s information bits has k mutually disjoint recovering sets. Every recovering set consists of some of the array’s columns while it is allowed to read at most l encoded bits from every column in order to receive the requested linear combination of the information bits. Functional batch array codes impose a stronger property where every multiset request of k linear combinations has k mutually disjoint recovering sets. Locality functional array codes demand that the size of every recovering set is restrained to be at most r. Given the values of s, k, t, l, r, the goal of this paper is to study the optimal value of the number of columns m such that these codes exist. Several lower bounds are presented as well as explicit constructions for several of these parameters.

A Generic Transformation for Optimal Node Repair in MDS Array Codes Over F2

For high-rate linear systematic maximum distance separable (MDS) codes, most early constructions could initially optimally repair all the systematic nodes but not all the parity nodes. Fortunately, this issue was first solved by Li et al. in (IEEE Trans. Inform. Theory, 64(9), 6257-6267, 2018), where a transformation that can convert any nonbinary MDS array code into another one with desired properties was proposed. However, the transformation does not work for binary MDS array codes. In this paper, we address this issue by proposing another generic transformation that can convert any $[n, k]$ binary MDS array code into a new one, which endows any $r=n-k\ge2$ chosen nodes with optimal repair bandwidth and optimal rebuilding access properties, and at the same time, preserves the normalized repair bandwidth/rebuilding access for the remaining $k$ nodes under some conditions. As two immediate applications, we show that 1) by applying the transformation multiple times, any binary MDS array code can be converted into one with optimal rebuilding access for all nodes, 2) any binary MDS array code with optimal repair bandwidth or optimal rebuilding access for the systematic nodes can be converted into one with the corresponding optimality property for all nodes.

Update Bandwidth for Distributed Storage

In this paper, we consider the update bandwidth in distributed storage systems (DSSs). The update bandwidth, which measures the transmission efficiency of the update process in DSSs, is defined as the average amount of data symbols transferred in the network when the data symbols stored in a node are updated. This paper contains the following contributions. First, we establish the closed-form expression of the minimum update bandwidth attainable by irregular array codes. Second, after defining a class of irregular array codes, called Minimum Update Bandwidth (MUB) codes, which achieve the minimum update bandwidth of irregular array codes, we determine the smallest code redundancy attainable by MUB codes. Third, the code parameters, with which the minimum code redundancy of irregular array codes and the smallest code redundancy of MUB codes can be equal, are identified, which allows us to define MR-MUB codes as a class of irregular array codes that simultaneously achieve the minimum code redundancy and the minimum update bandwidth. Fourth, we introduce explicit code constructions of MR-MUB codes and MUB codes with the smallest code redundancy. Fifth, we establish a lower bound of the update complexity of MR-MUB codes, which can be used to prove that the minimum update complexity of irregular array codes may not be achieved by MR-MUB codes. Last, we construct a class of $(n = k + 2, k)$ vertical maximum-distance separable (MDS) array codes that can achieve all of the minimum code redundancy, the minimum update bandwidth and the optimal repair bandwidth of irregular array codes.

Cross-Rack Updates in Vertical Array Codes Storage Systems with High Fault Tolerance

Cross-Rack Updates in Vertical Array Codes Storage Systems with High Fault Tolerance

Research on the application of binary-like coding and Hough circle detection technology in PCB traceability system

With the rapid development of smart devices, printed circuit board (PCB), as the core components of smart devices, not only proposed more and more demand for the quality and precision of their production, but also the controllability of their management and the traceability of manufacturing. The traditional processes for PCB including: acid–base corrosion, water washing, grinding and so on. Due to the traditional spray bar code, thunder engraving two-dimensional code and other methods cannot be used for PCB board identification, but the drilling method has been utilized. At present, the major methods for drilling identification and matrix drilling identification are time consuming, and low identification speed. To tackle this problem, in this paper, we combine the advantages of binary coding and the industry characteristics of PCB production, proposed a kind of binary-like code to identify the position of PCB production, in which, each digit of the binary code can be identified by four holes of a square with equal distance, and can represent the numbers from 0 to 9. Besides, fewer letters and the starting position of the code on the basis of few punching points. This method will greatly improve the efficiency of drilling identification. The system uses the photoelectric analysis module to automatically scan the hole array code on the penal (PNL) board to obtain the original image. Then, the CCD image is preprocessed, mainly including Gaussian filter for image smoothing and unsharp mask operator for image enhancement. Then, Hough detection algorithm is utilized to locate the hole, which can effectively identify the holes. Secondly, the horizontal and vertical directions of these points can be obtained according to the distance between the center points of each hole. Furthermore, the image is righted, and the starting position can be obtained. At the same time, Hough detection algorithm is used to segment the image threshold, and each character is decoded into a corresponding number. The simulation results show that the method is simple, accurate and effective. Finally, the identified digital coding sequences are compared with the database of online code reading application system, and the reliable identification and automatic identification trace-ability of PCB production are realized, and the automatic identification time is less than 1 s.

Two Classes of Binary MDS Array Codes With Asymptotically Optimal Repair for Any Single Column

An $m\times (k+r)$ binary maximum distance separable (MDS) array code contains $k$ information columns and $r$ parity columns with each entry being a bit, where any $k$ out of $k+r$ columns can recover the $k$ information columns. When there is a failed column, it is critical to minimize the repair bandwidth that is the total number of bits downloaded from $d$ out of $k+r-1$ surviving columns in repairing the failed column. In this article, we first propose two explicit constructions of binary MDS array codes that have asymptotically optimal repair bandwidth for any information column, where $r\geq 2$ and $d=k+r-1$ for the first construction, and $r\geq 4$ is an even number and $d=k+\frac {r}{2}-1$ for the second construction. By applying a generic transformation for the proposed two classes of binary MDS array codes, we then obtain two classes of new binary MDS array codes that also have optimal repair bandwidth for any parity column and asymptotically optimal repair bandwidth for any information column.

Analysis of the accuracy of actuation electronics for the laser interferometer space antenna.

Electrostatic actuation of a free-floating test-mass was tested in the Laser Interferometer Space Antenna (LISA) Pathfinder mission, and it will be integrated into the LISA. We have investigated the LISA Pathfinder actuation accuracy with respect to the precision of fractional digits in the field programmable gate array (FPGA) code of actuation electronics. The LISA Pathfinder data showed that the rounding errors in the FPGA code result in an erroneous force that contaminated the main mission observable, and this error was compensated in the post-processing of the LISA Pathfinder data. To avoid a similar issue for the LISA, the LISA actuation accuracy can be improved by increasing the number of fractional digits in the FPGA code. However, this is restricted by some hardware limitations. In this paper, we investigate the necessary enlargement of the FPGA to fulfill the LISA acceleration requirements and propose a design optimization for LISA actuation electronics.

Abstract 4696: Endometrial receptivity markers in patients with uterine fibroids and their correlation with air pollution exposure

Abstract Uterine leiomyomas (LM) are the most frequent benign tumors in women of reproductive age. They are associated with infertility, pelvic pain, abnormal uterine bleeding and represent a public health challenge. LM are over 4 times more common in African-descendent women. Additionally, this tumor constitutes a major health and financial burden in several countries. Similar to many pathologies, environmental exposure contributes to their origin and development. Air pollution acts as a disruptor in the hormonal activity and in other key pathways leading to enhanced tumor progression and making possible pregnancies difficult. In this study, we evaluated LM samples from women resident in urban (high exposure area) and no-urban areas (low or negative exposure area) for the expression of genes related to endometrial receptivity. We selected a total of 100 patients and among them twenty fill the specific profile of those exposed or not exposed to air pollution. mRNA expression analysis was performed using an array composed by 84 receptivity-related genes (Array Code: CLAH27251A, Qiagen, USA). Gene expression analyses showed PRLR, LITAF, COL6A3, HOXA11, KLRC1, UBE2A e IGFBP5 with higher fold expression (fold &amp;gt;2) in the exposed group. On the other hand, PENK, CALB2, DUOX1, COMP, CXCL14, CTNNA2, TCN1, S100P, WNT5A, ERBB2, MTNR1B and EFNA1 were dowregulated in these samples (fold &amp;lt; - 4). Several other genes exhibited differential regulation between exposed and non-exposed group of patients, but with lower significance. Our results corroborate the expected for these populations, since it is known that women resident in areas with high air pollution have reproductive complications and higher tendency to tumors development. However, this work is the first evidence of the impairment of endometrial receptivity in uterine LM patients associated to air pollution in Brazil. Further studies are ongoing to validate and elucidate the specific roles of these genes in uterine LM origin and development, as well as their reproductive function in this population. Citation Format: Dara O. Ferreira, Giovana D. Maffazioli, Edmund C. Baracat, Katia C. Carvalho. Endometrial receptivity markers in patients with uterine fibroids and their correlation with air pollution exposure [abstract]. In: Proceedings of the Annual Meeting of the American Association for Cancer Research 2020; 2020 Apr 27-28 and Jun 22-24. Philadelphia (PA): AACR; Cancer Res 2020;80(16 Suppl):Abstract nr 4696.