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Polyhedral Model
The polyhedral model (also called the polytope method) is a mathematical framework for programs that perform large numbers of operations -- too large to be explicitly enumerated -- thereby requiring a ''compact'' representation. Nested loop programs are the typical, but not the only example, and the most common use of the model is for loop nest optimization in program optimization. The polyhedral method treats each loop iteration within nested loops as lattice points inside mathematical objects called polyhedra, performs affine transformations or more general non-affine transformations such as tiling on the polytopes, and then converts the transformed polytopes into equivalent, but optimized (depending on targeted optimization goal), loop nests through polyhedra scanning. Simple example Consider the following example written in C: const int n = 100; int i, j, a n]; for (i = 1; i < n; i++) The essential problem with this code is that each iteration of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Nest Optimization
In computer science and particularly in compiler design, loop nest optimization (LNO) is an optimization technique that applies a set of loop transformations for the purpose of locality optimization or parallelization or another loop overhead reduction of the loop nests. (Nested loops occur when one loop is inside of another loop.) One classical usage is to reduce memory access latency or the cache bandwidth necessary due to cache reuse for some common linear algebra algorithms. The technique used to produce this optimization is called loop tiling, also known as loop blocking or strip mine and interchange. Overview Loop tiling partitions a loop's iteration space into smaller chunks or blocks, so as to help ensure data used in a loop stays in the cache until it is reused. The partitioning of loop iteration space leads to partitioning of a large array into smaller blocks, thus fitting accessed array elements into cache size, enhancing cache reuse and eliminating cache size requir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grayscale
In digital photography, computer-generated imagery, and colorimetry, a grayscale image is one in which the value of each pixel is a single sample representing only an ''amount'' of light; that is, it carries only intensity information. Grayscale images, a kind of black-and-white or gray monochrome, are composed exclusively of shades of gray. The contrast ranges from black at the weakest intensity to white at the strongest. Grayscale images are distinct from one-bit bi-tonal black-and-white images, which, in the context of computer imaging, are images with only two colors: black and white (also called ''bilevel'' or '' binary images''). Grayscale images have many shades of gray in between. Grayscale images can be the result of measuring the intensity of light at each pixel according to a particular weighted combination of frequencies (or wavelengths), and in such cases they are monochromatic proper when only a single frequency (in practice, a narrow band of frequencies) is ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compiler Optimizations
In computing, an optimizing compiler is a compiler that tries to minimize or maximize some attributes of an executable computer program. Common requirements are to minimize a program's execution time, memory footprint, storage size, and power consumption (the last three being popular for portable computers). Compiler optimization is generally implemented using a sequence of ''optimizing transformations'', algorithms which take a program and transform it to produce a semantically equivalent output program that uses fewer resources or executes faster. It has been shown that some code optimization problems are NP-complete, or even undecidable. In practice, factors such as the programmer's willingness to wait for the compiler to complete its task place upper limits on the optimizations that a compiler might provide. Optimization is generally a very CPU- and memory-intensive process. In the past, computer memory limitations were also a major factor in limiting which optimizations co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Unrolling
Loop unrolling, also known as loop unwinding, is a loop transformation technique that attempts to optimize a program's execution speed at the expense of its binary size, which is an approach known as space–time tradeoff. The transformation can be undertaken manually by the programmer or by an optimizing compiler. On modern processors, loop unrolling is often counterproductive, as the increased code size can cause more cache misses; ''cf.'' Duff's device. The goal of loop unwinding is to increase a program's speed by reducing or eliminating instructions that control the loop, such as pointer arithmetic and "end of loop" tests on each iteration; reducing branch penalties; as well as hiding latencies, including the delay in reading data from memory. To eliminate this computational overhead, loops can be re-written as a repeated sequence of similar independent statements. Loop unrolling is also part of certain formal verification techniques, in particular bounded model checking. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Nest Optimization
In computer science and particularly in compiler design, loop nest optimization (LNO) is an optimization technique that applies a set of loop transformations for the purpose of locality optimization or parallelization or another loop overhead reduction of the loop nests. (Nested loops occur when one loop is inside of another loop.) One classical usage is to reduce memory access latency or the cache bandwidth necessary due to cache reuse for some common linear algebra algorithms. The technique used to produce this optimization is called loop tiling, also known as loop blocking or strip mine and interchange. Overview Loop tiling partitions a loop's iteration space into smaller chunks or blocks, so as to help ensure data used in a loop stays in the cache until it is reused. The partitioning of loop iteration space leads to partitioning of a large array into smaller blocks, thus fitting accessed array elements into cache size, enhancing cache reuse and eliminating cache size requir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frameworks Supporting The Polyhedral Model
Use of the polyhedral model (also called the polytope model) within a compiler requires software to represent the objects of this framework (sets of integer-valued points in regions of various spaces) and perform operations upon them (e.g., testing whether the set is empty). For more detail about the objects and operations in this model, and an example relating the model to the programs being compiled, see the polyhedral model page. There are many frameworks supporting the polyhedral model. Some of these frameworks use one or more libraries for performing polyhedral operations. Others, notably Omega, combine everything in a single package. Some commonly used libraries are the Omega Library (and a more recent fork), piplib,Paul Feautrier. ''Parametric Integer Programming.'' 1988 PolyLib, PPL, isl, the Cloog polyhedral code generator,Cedric Bastoul. ''Code Generation in the Polyhedral Model Is Easier Than You Think.'' PACT'13 IEEE International Conference on Parallel Architecture a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope Model Skewed
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Optimization
In compiler theory, loop optimization is the process of increasing execution speed and reducing the overheads associated with loops. It plays an important role in improving cache performance and making effective use of parallel processing capabilities. Most execution time of a scientific program is spent on loops; as such, many compiler optimization techniques have been developed to make them faster. Representation of computation and transformations Since instructions inside loops can be executed repeatedly, it is frequently not possible to give a bound on the number of instruction executions that will be impacted by a loop optimization. This presents challenges when reasoning about the correctness and benefits of a loop optimization, specifically the representations of the computation being optimized and the optimization(s) being performed.In the book Reasoning About Program Transformations', Jean-Francois Collard discusses in depth the general question of representing execu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Loop , United States
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Inner loop may refer to: *Inner loop in computer programs *Inner Loop (Phoenix), a section of Interstate 10 in downtown Phoenix, Arizona, United States *Inner Loop (Rochester), an expressway around downtown Rochester, New York, United States * Inner Loop (Washington, D.C.), a previously proposed freeway loop in Washington, D.C., United States *Inner–outer directions, where "inner loop" is used to describe the clockwise traveling lanes of a roadway **This usage of "inner loop" is commonly applied to the clockwise roadway of Interstate 495 (Capital Beltway) *Interstate 610 (Texas), the innermost highway loop around the central area of Houston, Texas Houston (; ) is the most populous city in Texas, the most populous city in the Southern United States, the fourth-most populous city in the United States, and the sixth-most populous city in North America, with a population of 2,304,580 in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floyd–Steinberg Dithering
Floyd–Steinberg dithering is an image dithering algorithm first published in 1976 by Robert W. Floyd and Louis Steinberg. It is commonly used by image manipulation software, for example when an image is converted into GIF format that is restricted to a maximum of 256 colors. Implementation The algorithm achieves dithering using error diffusion, meaning it pushes (adds) the residual quantization error of a pixel onto its neighboring pixels, to be dealt with later. It spreads the debt out according to the distribution (shown as a map of the neighboring pixels): : \begin & & * & \frac & \ldots \\ \ldots & \frac & \frac & \frac & \ldots \\ \end The pixel indicated with a star (*) indicates the pixel currently being scanned, and the blank pixels are the previously-scanned pixels. The algorithm scans the image from left to right, top to bottom, quantizing pixel values one by one. Each time ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Program Optimization
In computer science, program optimization, code optimization, or software optimization, is the process of modifying a software system to make some aspect of it work more efficiently or use fewer resources. In general, a computer program may be optimized so that it executes more rapidly, or to make it capable of operating with less memory storage or other resources, or draw less power. General Although the word "optimization" shares the same root as "optimal", it is rare for the process of optimization to produce a truly optimal system. A system can generally be made optimal not in absolute terms, but only with respect to a given quality metric, which may be in contrast with other possible metrics. As a result, the optimized system will typically only be optimal in one application or for one audience. One might reduce the amount of time that a program takes to perform some task at the price of making it consume more memory. In an application where memory space is at a premium, on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dither
Dither is an intentionally applied form of image noise, noise used to randomize quantization error, preventing large-scale patterns such as color banding in images. Dither is routinely used in processing of both digital audio and digital video, video data, and is often one of the last stages of Audio mastering, mastering audio to a compact disc, CD. A common use of dither is converting a grayscale image to Binary image, black and white, such that the density of black dots in the new image approximates the average gray level in the original. Etymology The term ''dither'' was published in books on analog computation and hydraulically controlled guns shortly after World War II. Though he did not use the term ''dither'', the concept of dithering to reduce quantization patterns was first applied by Lawrence G. Roberts in his 1961 MIT master's thesis and 1962 article. By 1964 dither was being used in the modern sense described in this article. The technique was in use at least ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
