624 research outputs found
Interleaving schemes for multidimensional cluster errors
We present two-dimensional and three-dimensional interleaving techniques for correcting two- and three-dimensional bursts (or clusters) of errors, where a cluster of errors is characterized by its area or volume. Correction of multidimensional error clusters is required in holographic storage, an emerging application of considerable importance. Our main contribution is the construction of efficient two-dimensional and three-dimensional interleaving schemes. The proposed schemes are based on t-interleaved arrays of integers, defined by the property that every connected component of area or volume t consists of distinct integers. In the two-dimensional case, our constructions are optimal: they have the lowest possible interleaving degree. That is, the resulting t-interleaved arrays contain the smallest possible number of distinct integers, hence minimizing the number of codewords required in an interleaving scheme. In general, we observe that the interleaving problem can be interpreted as a graph-coloring problem, and introduce the useful special class of lattice interleavers. We employ a result of Minkowski, dating back to 1904, to establish both upper and lower bounds on the interleaving degree of lattice interleavers in three dimensions. For the case t≡0 mod 6, the upper and lower bounds coincide, and the Minkowski lattice directly yields an optimal lattice interleaver. For t≠0 mod 6, we construct efficient lattice interleavers using approximations of the Minkowski lattice
Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices
Step by step completion of a left-to-right tiling of a rectangular floor with
tiles of a single shape starts from one edge of the floor, considers the
possible ways of inserting a tile at the leftmost uncovered square, passes
through a sequence of rugged shapes of the front line between covered and
uncovered regions of the floor, and finishes with a straight front line at the
opposite edge. We count the tilings by mapping the front shapes to nodes in a
digraph, then counting closed walks on that digraph with the transfer matrix
method.
Generating functions are detailed for tiles of shape 1 x 3, 1 x 4 and 2 x 3
and modestly wide floors. Equivalent results are shown for the 3-dimensional
analog of filling bricks of shape 1x 1 x 2, 1 x 1 x 3, 1 x 1 x 4, 1 x 2 x 2 or
1 x 2 x 3 into rectangular containers of small cross sections.Comment: 21 pages, 21 figure
Tiramisu: A Polyhedral Compiler for Expressing Fast and Portable Code
This paper introduces Tiramisu, a polyhedral framework designed to generate
high performance code for multiple platforms including multicores, GPUs, and
distributed machines. Tiramisu introduces a scheduling language with novel
extensions to explicitly manage the complexities that arise when targeting
these systems. The framework is designed for the areas of image processing,
stencils, linear algebra and deep learning. Tiramisu has two main features: it
relies on a flexible representation based on the polyhedral model and it has a
rich scheduling language allowing fine-grained control of optimizations.
Tiramisu uses a four-level intermediate representation that allows full
separation between the algorithms, loop transformations, data layouts, and
communication. This separation simplifies targeting multiple hardware
architectures with the same algorithm. We evaluate Tiramisu by writing a set of
image processing, deep learning, and linear algebra benchmarks and compare them
with state-of-the-art compilers and hand-tuned libraries. We show that Tiramisu
matches or outperforms existing compilers and libraries on different hardware
architectures, including multicore CPUs, GPUs, and distributed machines.Comment: arXiv admin note: substantial text overlap with arXiv:1803.0041
Storage and Querying of Large Persistent Arrays
The scientic and analytical applications today are increasingly becoming data in-
tensive. Many such applications deal with data that is multidimensional in nature.
Traditionally, relational database systems have been used by many data intensive
application, and relational paradigm has proved to be both natural and ecient.
However, for multidimensional data, when the number of dimensions becomes large,
relational databases are inecient both in terms of storage and query response time.
In this thesis, we explore linearised storage, and indexed and skiplist based retrieval
on persistent arrays. The application programs are provided with a logical view of
multidimensional array. The techniques have been implemented in a home-grown
database management system called MuBase
A Survey on Array Storage, Query Languages, and Systems
Since scientific investigation is one of the most important providers of
massive amounts of ordered data, there is a renewed interest in array data
processing in the context of Big Data. To the best of our knowledge, a unified
resource that summarizes and analyzes array processing research over its long
existence is currently missing. In this survey, we provide a guide for past,
present, and future research in array processing. The survey is organized along
three main topics. Array storage discusses all the aspects related to array
partitioning into chunks. The identification of a reduced set of array
operators to form the foundation for an array query language is analyzed across
multiple such proposals. Lastly, we survey real systems for array processing.
The result is a thorough survey on array data storage and processing that
should be consulted by anyone interested in this research topic, independent of
experience level. The survey is not complete though. We greatly appreciate
pointers towards any work we might have forgotten to mention.Comment: 44 page
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