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Streams and Incremental Processing


Stream processing defines a pipeline of operators that transform, combine, or reduce (even to a single scalar) large amounts of data. Characteristically, data is accessed strictly linearly rather than randomly and repeatedly -- and processed uniformly. The upside of the limited expressiveness is the opportunity to process large amount of data efficiently, in constant and small space.



Stream-wise data processing, already present in one of the first programming languages (COBOL), has been coming to forefront with the popularity of big data and MapReduce. The uniformity and the predictability of data access help efficiently handle vast amount of data, far more than fits within the memory of a single processor.

Stream processing also exhibits the painful abstraction vs. performance trade-off. Manually written loops and state-machines offer the highest performance and the least memory overhead, but are not reusable or extensible. Libraries of freely composable stream components let programmers quickly assemble an obviously correct stream application, but suffer from the high overhead of abstractions, mainly due to the repeated creation and disposal of intermediate data structures such as closures, captured continuations, objects and collections -- let alone intermediate streams. Eliminating such intermediate structures is broadly known as stream fusion.

According to a survey of stream processing, ``Within computer science the term stream has been attributed to P.J. Landin formulated during the development of operational constructs presented as part of his work on the correspondence between ALGOL 60 and the lambda-calculus. Indeed, we note that P.J. Landin's original use for streams was to model the histories of loop variables, but he also observed that streams could have been used as a model for I/O in ALGOL 60.''

R. Stephens: A Survey Of Stream Processing
Acta Informatica, July 1997, Volume 34, Issue 7, pp 491–541

Stream Seminar
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Stream Fusion, to Completeness
The Related Word section of the paper


Stream Fusion, to Completeness

Stream processing is mainstream (again): Widely-used stream libraries are now available for virtually all modern OO and functional languages, from Java to C# to Scala to OCaml to Haskell. Yet expressivity and performance are still lacking. For instance, the popular, well-optimized Java 8 streams do not support the zip operator and are still an order of magnitude slower than hand-written loops.

We present the first approach that represents the full generality of stream processing and eliminates overheads, via the use of staging. It is based on an unusually rich semantic model of stream interaction. We support any combination of zipping, nesting (or flat-mapping), sub-ranging, filtering, mapping---of finite or infinite streams. Our model captures idiosyncrasies that a programmer uses in optimizing stream pipelines, such as rate differences and the choice of a ``for'' vs. ``while'' loops. Our approach delivers hand-written--like code, but automatically. It explicitly avoids the reliance on black-box optimizers and sufficiently-smart compilers, offering highest, guaranteed and portable performance.

Our approach relies on high-level concepts that are then readily mapped into an implementation. Accordingly, we have two distinct implementations: an OCaml stream library, staged via MetaOCaml, and a Scala library for the JVM, staged via LMS. In both cases, we derive libraries richer and simultaneously many tens of times faster than past work. We greatly exceed in performance the standard stream libraries available in Java, Scala and OCaml, including the well-optimized Java 8 streams.

Joint work with Aggelos Biboudis, Nick Palladinos and Yannis Smaragdakis.

strymonas.pdf [614K]
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The complete paper, whose shorter version (without Appendices) is published in the Proceedings of POPL 2017.

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The complete code for both MetaOCaml and Scala/Java versions of the strymonas library, and the complete code for all benchmarks. The code received the Artifact Evaluated badge from the POPL 2017 artifact evaluation committee.


Streams in Linear Algebra

The linear matrix library LinAlg is built upon matrix streams, which provide sequential view/access to a matrix or its parts. Many of Linear Algebra algorithms turn out to require only sequential access to a matrix or its rows/columns, which is simpler and faster than random access. All streams in LinAlg are light-weight: they use no heap storage and leave no garbage.

A stream-wise access to a collection is an important access method, which may even be supported by hardware. For example, Pentium III floating-point extension (Internet Streaming SIMD Extension) lets programmers designate arrays as streams and provides instructions to handle such data efficiently (Internet Streaming SIMD Extensions, Shreekant (Ticky) Thakkar and Tom Huff, Computer, Vol. 32, No. 12, December 1999, pp. 26-34). Streaming is a typical memory access model of DSPs: that's why DSP almost never incorporate a data cache (See "DSP Processors Hit the Mainstream," Jennifer Eyre and Jeff Bier, Computer, Vol. 31, No. 8, August 1998, pp. 51-59). A memory architecture designed in an article "Smarter Memory: Improving Bandwidth for Streamed References" (IEEE Computer, July 1998, pp.54-63) achieves low overall latencies because the CPU is told by a compiler that a stream operation is to follow. LinAlg offers this streaming access model to an application programmer.

Matrix streams may stride a matrix by an arbitrary amount. This lets us traverse a matrix along the diagonal, by columns, by rows, etc. Streams can be constructed of a Matrix itself, or from other matrix views (MatrixColumn, MatrixRow, MatrixDiag). In the latter case, the streams are confined only to specific portions of the matrix.

Many functions of LinAlg are written in terms of streams, for example, the computation of vector norms, the addition of a vector to the diagonal or the anti-diagonal of a matrix, Aitken-Lagrange interpolation. Singular value decomposition SVD demonstrates many applications of streams: e.g., multiplying a matrix by a rotation matrix avoids random access to matrix elements and the corresponding range checks and offset calculations. The stream code is also more lucid. One may create a stream that spans over a part of another stream. We use substreams, for example, to efficiently reflect the upper triangle of a square matrix onto the lower one, yielding a symmetric matrix. The SVD computation uses subranging extensively, e.g., for left Householder transformations.

LinAlg's streams may span an arbitrary rectangular block of a matrix, including the whole matrix, a single matrix element, a matrix row or a column, or a part thereof. Assigning a block of one matrix to a block of another takes only one line -- which, due to inlining, is just as efficient as the direct loop with pointer manipulation.

LinAlg: Basic Linear Algebra and Optimization classlib


How to zip folds: A library of fold transformers (streams)

We show how to merge two folds `elementwise' into the resulting fold. Furthermore, we present a library of potentially infinite ``lists'' represented as folds -- or, streams, or success-failure-continuation--based generators. Whereas the standard Haskell Prelude functions such as map and take transform lists, we transform folds. We implement the range of progressively more complex transformers -- from map, filter, takeWhile to take, drop, dropWhile, and finally, zip and zipWith. The standard list API is also provided.

Emphatically we never convert a stream to a list and so we never use recursion or recursive types. All iterative processing is driven by the fold itself. We only need higher-ranked types, because lists cannot be fully implemented in simply typed lambda-calculus.

The implementation of zip also solves the problem of ``parallel loops''. One can think of a fold as an accumulating loop and realize a nested loop as a nested fold. Representing a parallel loop as a fold is a challenge, answered at the end of the article. This becomes especially interesting in the case of general backtracking computations, or backtracking computations in direct style, with delimited continuations modeling `lists'.

The current version is 1.6, Jun 16, 2008.
zip-folds.lhs [13K]
Complete literate Haskell code
An earlier version was posted on the Haskell mailing list on Tue, 11 Oct 2005 17:25:24 -0700 (PDT). That version implemented zip with the help of a recursive type. The present version, inspired by a question from Chung-chieh Shan, introduces no extra data types, no recursion, and rather relies on the already defined functions to deconstruct a fold. Version 1.6 adds two examples of surprisingly simple expressions of list intersperse and Fibonacci in terms of fold.

Beyond Church encoding: Boehm-Berarducci isomorphism of algebraic data types and polymorphic lambda-terms

LogicT: backtracking monad transformer with fair operations and pruning
which illustrates the close connection with foldr/build list-fusion, aka ``short-cut deforestation''.

Predecessor and lists are not representable in simply typed lambda-calculus
Therefore, higher-rank or recursive/inductive types are necessary for lists.

Parallel composition of streams: several sources to one sink
Folding over multiple streams using monad transformers

Last updated January 1, 2017

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