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An Erasure-Resilient and Compute-Efficient Coding Scheme

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An Erasure-Resilient and Compute-Efficient Coding Scheme An Erasure-Resilient and Compute-Efficient Coding Scheme for Storage Applications Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften vorgelegt beim Fachbereich Informatik und Mathematik der Johann Wolfgang Goethe-Universität in Frankfurt am Main von Sebastian Kalcher aus Lahn-Gießen Frankfurt 2013 (D30) vom Fachbereich Informatik und Mathematik der Johann Wolfgang Goethe-Universität als Dissertation angenommen. Dekan: Prof. Dr. Thorsten Theobald Gutachter: Prof. Dr. Volker Lindenstruth Prof. Dr. Udo Kebschull Datum der Disputation: 31.10.2013 Abstract Driven by rapid technological advancements, the amount of data that is created, captured, communica- ted, and stored worldwide has grown exponentially over the past decades. Along with this development it has become critical for many disciplines of science and business to being able to gather and analyze large amounts of data. The sheer volume of the data often exceeds the capabilities of classical storage systems, with the result that current large-scale storage systems are highly distributed and are comprised of a high number of individual storage components. As with any other electronic device, the reliability of storage hardware is governed by certain probability distributions, which in turn are influenced by the physical processes utilized to store the information. The traditional way to deal with the inherent unre- liability of combined storage systems is to replicate the data several times. Another popular approach to achieve failure tolerance is to calculate the block-wise parity in one or more dimensions. With better understanding of the different failure modes of storage components, it has become evident that so- phisticated high-level error detection and correction techniques are indispensable for the ever-growing distributed systems. The utilization of powerful cyclic error-correcting codes, however, comes with a high computational penalty, since the required operations over finite fields do not map very well onto current commodity processors. This thesis introduces a versatile coding scheme with fully adjustable fault-tolerance that is tailored specifically to modern processor architectures. To reduce stress on the memory subsystem the conventional table-based algorithm for multiplication over finite fields has been replaced with a polynomial version. This arithmetically intense algorithm is better suited to the wide SIMD units of the currently available general purpose processors, but also displays significant bene- fits when used with modern many-core accelerator devices (for instance the popular general purpose graphics processing units). A CPU implementation using SSE and a GPU version using CUDA are presented. The performance of the multiplication depends on the distribution of the polynomial coeffi- cients in the finite field elements. This property has been used to create suitable matrices that generate a linear systematic erasure-correcting code which shows a significantly increased multiplication perfor- mance for the relevant matrix elements. Several approaches to obtain the optimized generator matrices are elaborated and their implications are discussed. A Monte-Carlo-based construction method allows it to influence the specific shape of the generator matrices and thus to adapt them to special storage and archiving workloads. Extensive benchmarks on CPU and GPU demonstrate the superior performance and the future application scenarios of this novel erasure-resilient coding scheme. i Zusammenfassung Die sogenannte digitale Revolution, beginnend am Ende des 20. Jahrhunderts, hat unseren Zugang zu Informationen umfassend verändert. Viele Bereiche des gesellschaftlichen Lebens, der Wirtschaft und der Wissenschaft sind mittlerweile stark auf Informationstechnologie angewiesen. Zusammen mit der rasanten Entwicklung der Mikroprozessoren war die Evolution der magnetischen Speichertechnolo- gien einer der Schlüsselfaktoren für den Übergang in eine Informationsgesellschaft. Kostengünstige Datenspeicher mit hoher Kapazität bilden heute die Basis für die große Familie von allgegenwärtigen Geräten, die in die Klasse der personal computing devices eingeordnet werden, beispielsweise klassi- sche Arbeitsplatzrechner, tragbare Laptops, Mobiltelefone, Tablet-Computer, sowie Foto- und Video- kameras. Besonders der Durchbruch der Halbleiter-Speicher hat die Entwicklung immer kleinerer und robusterer Geräte aus dieser Klasse begünstigt. Aber auch die Entwicklung der professionellen Hoch- leistungscomputer und deren Speichersysteme wird durch diese Schlüsseltechnologien angetrieben. Besonders diese Art von Computern hat enorme Bedeutung für den wissenschaftlichen Erkenntnis- gewinn, aber auch ein breites wirtschaftliches Anwendungsspektrum. Das gewaltige Wachstum der Menge an Informationen, die weltweit erzeugt (oder gewonnen), übertragen und gespeichert wird ist daher wenig überraschend. Schätzungen zu Folge stieg die Gesamtmenge aller gespeicherten Infor- mationen (auf allen gängigen analogen und digitalen Medien) von 2,6 Exabytes1 in 1986 auf 295 Exabytes in 2007 [1]. Bemerkenswerterweise trugen die digitalen Speichertechnologien erst ab dem Jahr 2000 signifikant zur gesamten Speicherkapazität bei. Nur drei Jahre später wurden geschätzte 90% aller neuen Daten auf magnetischen Medien gespeichert [2] und im Jahr 2007 überholten die digitalen Speicher die analogen endgültig: 52% aller existenten Informationen waren auf Festplatten, 28% auf optischen Medien und 11% auf magnetischen Bändern gespeichert. Neuere Studien aus dem Jahr 2011 quantifizierten die Gesamtmenge aller vom Menschen jemals erzeugten und gespeicherten Informationen auf gewaltige 1,8 Zettabytes [3][4]. Die datenintensiven Wissenschaften gelten neben dem Experiment, der Theorie und der Simulation mittlerweile als vierte essentielle wissenschaftliche Disziplin. Das prominenteste Beispiel kommt aus Gebiet der Teilchenphysik: Der Large Hadron Col- lider (LHC) der Europäische Organisation für Kernforschung (CERN) in Genf [5] beheimatet sechs unterschiedlich große Experimente die zusammen einige zehn Petabyte an zu speichernden Daten pro Jahr aufnehmen. Aber nicht nur derartige Großforschungsprojekte stellen hohe Anforderungen an die Datenverarbeitungs- und Speichersysteme: Die Entwicklung bildgebenden Sensoren mit Auflösungen Gigapixel-Bereich und immer höher auflösenden Teleskopen stellt die astronomische Forschung vor ähnliche Herausforderungen. Sogar die relativ kompakten DNA-Sequenzierungsgeräte aktueller Bau- art produzieren bereits Rohdatenmengen im Bereich einiger Terabyte pro Tag. Mit den immer größer 1Ein Exabyte steht für 1018 Bytes. iii werdenden Datenmengen müssen natürlich auch die Hochleistungsrechner mit immer größeren Spei- chersystemen ausgestattet werden, um die Analyse der gewonnen Informationen zu ermöglichen. Doch die Verarbeitung von großen Datenmengen hat nicht nur im wissenschaftlichen Umfeld enorm an Be- deutung gewonnen. Besonders das Geschäftsfeld der Internet Suchmaschinen wäre ohne die Fähigkeit enorme Datenmengen zu sammeln, zu analysieren und zu speichern heute undenkbar. Weiter Beispiele sind elektronische Handelsplattformen und weltumspannende soziale Netzwerke. Der größte Teil all dieser Daten wird zum Online-Zugriff auf großen Sammlungen von Festplatten vorgehalten, deren Verwaltung und Pflege die Besitzer vor große Herausforderungen stellt: Obwohl die Zuverlässigkeit einer einzelnen Festplatte durchaus beachtlich ist, ergeben sich bei Systemen aus Zehn- tausenden Festplatten hohe Raten an Fehlerereignissen. Große wissenschaftliche Experimente können aus Kostengründen oft nicht beliebig wiederholt werden und schon eine kurze Unerreichbarkeit im In- ternethandel kann enorme Verluste erzeugen. Typischerweise werden daher Kopien der Daten erstellt und diese auf (geographisch) verschiedene Rechenzentren verteilt. Da diese Strategien der mehrfachen Replikation relativ kostenintensiv sind, wurden effizientere Verfahren entwickelt. In ihrem berühmten Artikel aus dem Jahre 1988 beschreiben Patterson, Gibson und Katz [6], wie man die Zuverlässigkeit von Sammlungen von Festplatten mit verschiedenen Techniken erhöhen kann. Die Berechnung redun- danter horizontaler Paritätsinformationen zur Erzeugung einer einfachen Fehlertoleranz hat sich daraus zum Standard für die folgenden Jahrzehnte entwickelt (RAID). Später kamen diverse Erweiterungen hinzu, die sogar den Ausfall von zwei Festplatten tolerieren können. All diese Verfahren stellen jedoch nur die Spezialfälle der umfassenden mathematischen Theorie der linearen fehlerkorrigierenden Codes dar. Im Rahmen dieser Theorie nutzt man die Werkzeuge der linearen Algebra um das Kodieren und das Dekodieren zu formalisieren. Als besonders leistungsstark haben sich die zyklischen linearen Co- des über endliche Körper erwiesen. Die weitverbreiteten Reed-Solomon-Codes [7] erlauben es, zu einer Menge an n Nachrichtensymbolen eine frei wählbare Menge an k Redundanzsymbolen zu berechnen. Die original Nachricht kann auch dann noch rekonstruiert werden, wenn bis zu k Symbole verloren ge- gangen sind. Eine Variante dieser Codes eignet sich besonders für den Einsatz in Speichersystemen [8]. Für die n Nachrichtensymbole d = d0;d1;:::;dn−1, müssen k Redundanzsymbole c = c1;c2;:::;ck−1 berechnet werden. Mit Hilfe einer speziellen Generatormatrix G = (gi; j), die von einer Vandermonde- Matrix abgeleitet wird, lässt sich die Kodierung durch eine Matrix-Vektor-Multiplikation darstellen: 0 1 d0 B d C B 1 C 0 1 B . C d0 B . C B C ! B d1 C Bd C d B C B n−1C GB . C = B
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