Algorithmic efficiency

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In computer science, algorithmic efficiency is a property of an algorithm which relates to the amount of computational resources used by the algorithm. An algorithm must be analyzed to determine its resource usage, and the efficiency of an algorithm can be measured based on the usage of different resources. Algorithmic efficiency can be thought of as analogous to engineering productivity for a repeating or continuous process.

For maximum efficiency it is desirable to minimize resource usage. However, different resources such as time and space complexity cannot be compared directly, so which of two algorithms is considered to be more efficient often depends on which measure of efficiency is considered most important.

For example, bubble sort and timsort are both algorithms to sort a list of items from smallest to largest. Bubble sort sorts the list in time proportional to the number of elements squared ($O(n^2)$, see Big O notation), but only requires a small amount of extra memory which is constant with respect to the length of the list ($O(1)$). Timsort sorts the list in time linearithmic (proportional to a quantity times its logarithm) in the list's length ($O(n\log n)$), but has a space requirement linear in the length of the list ($O(n)$). If large lists must be sorted at high speed for a given application, timsort is a better choice; however, if minimizing the memory footprint of the sorting is more important, bubble sort is a better choice.

Background

The importance of efficiency with respect to time was emphasised by Ada Lovelace in 1843 as applied to Charles Babbage's mechanical analytical engine:

"In almost every computation a great variety of arrangements for the succession of the processes is possible, and various considerations must influence the selections amongst them for the purposes of a calculating engine. One essential object is to choose that arrangement which shall tend to reduce to a minimum the time necessary for completing the calculation"

Early electronic computers had both limited speed and limited random access memory. Therefore, a space–time trade-off occurred. A task could use a fast algorithm using a lot of memory, or it could use a slow algorithm using little memory. The engineering trade-off was then to use the fastest algorithm that could fit in the available memory.

Modern computers are significantly faster than the early computers, and have a much larger amount of memory available (Gigabytes instead of Kilobytes). Nevertheless, Donald Knuth emphasised that efficiency is still an important consideration:

"In established engineering disciplines a 12% improvement, easily obtained, is never considered marginal and I believe the same viewpoint should prevail in software engineering"

Overview

An algorithm is considered efficient if its resource consumption, also known as computational cost, is at or below some acceptable level. Roughly speaking, 'acceptable' means: it will run in a reasonable amount of time or space on an available computer, typically as a function of the size of the input. Since the 1950s computers have seen dramatic increases in both the available computational power and in the available amount of memory, so current acceptable levels would have been unacceptable even 10 years ago. In fact, thanks to the approximate doubling of computer power every 2 years, tasks that are acceptably efficient on modern smartphones and embedded systems may have been unacceptably inefficient for industrial servers 10 years ago.

Computer manufacturers frequently bring out new models, often with higher performance. Software costs can be quite high, so in some cases the simplest and cheapest way of getting higher performance might be to just buy a faster computer, provided it is compatible with an existing computer.

There are many ways in which the resources used by an algorithm can be measured: the two most common measures are speed and memory usage; other measures could include transmission speed, temporary disk usage, long-term disk usage, power consumption, total cost of ownership, response time to external stimuli, etc. Many of these measures depend on the size of the input to the algorithm, i.e. the amount of data to be processed. They might also depend on the way in which the data is arranged; for example, some sorting algorithms perform poorly on data which is already sorted, or which is sorted in reverse order.

In practice, there are other factors which can affect the efficiency of an algorithm, such as requirements for accuracy and/or reliability. As detailed below, the way in which an algorithm is implemented can also have a significant effect on actual efficiency, though many aspects of this relate to optimization issues.

Theoretical analysis

In the theoretical analysis of algorithms, the normal practice is to estimate their complexity in the asymptotic sense. The most commonly used notation to describe resource consumption or "complexity" is Donald Knuth's Big O notation, representing the complexity of an algorithm as a function of the size of the input $n$. Big O notation is an asymptotic measure of function complexity, where $f(n)=O(g(n))$ roughly means the time requirement for an algorithm is proportional to ${\displaystyle g(n)}$, omitting lower-order terms that contribute less than ${\displaystyle g(n)}$ to the growth of the function as $n$ grows arbitrarily large. This estimate may be misleading when $n$ is small, but is generally sufficiently accurate when $n$ is large as the notation is asymptotic. For example, bubble sort may be faster than merge sort when only a few items are to be sorted; however either implementation is likely to meet performance requirements for a small list. Typically, programmers are interested in algorithms that scale efficiently to large input sizes, and merge sort is preferred over bubble sort for lists of length encountered in most data-intensive programs.

Some examples of Big O notation applied to algorithms' asymptotic time complexity include:

Notation	Name	Examples
${\displaystyle O(1)}$	constant	Finding the median from a sorted list of measurements; Using a constant-size lookup table; Using a suitable hash function for looking up an item.
${\displaystyle O(\log n)}$	logarithmic	Finding an item in a sorted array with a binary search or a balanced search tree as well as all operations in a Binomial heap.
${\displaystyle O(n)}$	linear	Finding an item in an unsorted list or a malformed tree (worst case) or in an unsorted array; Adding two n-bit integers by ripple carry.
${\displaystyle O(n\log n)}$	linearithmic, loglinear, or quasilinear	Performing a Fast Fourier transform; heapsort, quicksort (best and average case), or merge sort
${\displaystyle O(n^2)}$	quadratic	Multiplying two n-digit numbers by a simple algorithm; bubble sort (worst case or naive implementation), Shell sort, quicksort (worst case), selection sort or insertion sort
${\displaystyle O(c^n),c>1}$	exponential	Finding the optimal (non-approximate) solution to the travelling salesman problem using dynamic programming; determining if two logical statements are equivalent using brute-force search

Benchmarking: measuring performance

For new versions of software or to provide comparisons with competitive systems, benchmarks are sometimes used, which assist with gauging an algorithms relative performance. If a new sort algorithm is produced, for example, it can be compared with its predecessors to ensure that at least it is efficient as before with known data, taking into consideration any functional improvements. Benchmarks can be used by customers when comparing various products from alternative suppliers to estimate which product will best suit their specific requirements in terms of functionality and performance. For example, in the mainframe world certain proprietary sort products from independent software companies such as Syncsort compete with products from the major suppliers such as IBM for speed.

Some benchmarks provide opportunities for producing an analysis comparing the relative speed of various compiled and interpreted languages for example and The Computer Language Benchmarks Game compares the performance of implementations of typical programming problems in several programming languages.

Even creating "do it yourself" benchmarks can demonstrate the relative performance of different programming languages, using a variety of user specified criteria. This is quite simple, as a "Nine language performance roundup" by Christopher W. Cowell-Shah demonstrates by example.

Implementation concerns

Implementation issues can also have an effect on efficiency, such as the choice of programming language, or the way in which the algorithm is actually coded, or the choice of a compiler for a particular language, or the compilation options used, or even the operating system being used. In many cases a language implemented by an interpreter may be much slower than a language implemented by a compiler. See the articles on just-in-time compilation and interpreted languages.

There are other factors which may affect time or space issues, but which may be outside of a programmer's control; these include data alignment, data granularity, cache locality, cache coherency, garbage collection, instruction-level parallelism, multi-threading (at either a hardware or software level), simultaneous multitasking, and subroutine calls.

Some processors have capabilities for vector processing, which allow a single instruction to operate on multiple operands; it may or may not be easy for a programmer or compiler to use these capabilities. Algorithms designed for sequential processing may need to be completely redesigned to make use of parallel processing, or they could be easily reconfigured. As parallel and distributed computing grow in importance in the late 2010s, more investments are being made into efficient high-level APIs for parallel and distributed computing systems such as CUDA, TensorFlow, Hadoop, OpenMP and MPI.

Another problem which can arise in programming is that processors compatible with the same instruction set (such as x86-64 or ARM) may implement an instruction in different ways, so that instructions which are relatively fast on some models may be relatively slow on other models. This often presents challenges to optimizing compilers, which must have a great amount of knowledge of the specific CPU and other hardware available on the compilation target to best optimize a program for performance. In the extreme case, a compiler may be forced to emulate instructions not supported on a compilation target platform, forcing it to generate code or link an external library call to produce a result that is otherwise incomputable on that platform, even if it is natively supported and more efficient in hardware on other platforms. This is often the case in embedded systems with respect to floating-point arithmetic, where small and low-power microcontrollers often lack hardware support for floating-point arithmetic and thus require computationally expensive software routines to produce floating point calculations.

Measures of resource usage

Measures are normally expressed as a function of the size of the input ${\displaystyle n}$.

The two most common measures are:

• Time: how long does the algorithm take to complete?
• Space: how much working memory (typically RAM) is needed by the algorithm? This has two aspects: the amount of memory needed by the code (auxiliary space usage), and the amount of memory needed for the data on which the code operates (intrinsic space usage).

For computers whose power is supplied by a battery (e.g. laptops and smartphones), or for very long/large calculations (e.g. supercomputers), other measures of interest are:

• Direct power consumption: power needed directly to operate the computer.
• Indirect power consumption: power needed for cooling, lighting, etc.

As of 2018, power consumption is growing as an important metric for computational tasks of all types and at all scales ranging from embedded Internet of things devices to system-on-chip devices to server farms. This trend is often referred to as green computing.

Less common measures of computational efficiency may also be relevant in some cases:

• Transmission size: bandwidth could be a limiting factor. Data compression can be used to reduce the amount of data to be transmitted. Displaying a picture or image (e.g. Google logo) can result in transmitting tens of thousands of bytes (48K in this case) compared with transmitting six bytes for the text "Google". This is important for I/O bound computing tasks.
• External space: space needed on a disk or other external memory device; this could be for temporary storage while the algorithm is being carried out, or it could be long-term storage needed to be carried forward for future reference.
• Response time (latency): this is particularly relevant in a real-time application when the computer system must respond quickly to some external event.
• Total cost of ownership: particularly if a computer is dedicated to one particular algorithm.

Time

Theory

Analyze the algorithm, typically using time complexity analysis to get an estimate of the running time as a function of the size of the input data. The result is normally expressed using Big O notation. This is useful for comparing algorithms, especially when a large amount of data is to be processed. More detailed estimates are needed to compare algorithm performance when the amount of data is small, although this is likely to be of less importance. Algorithms which include parallel processing may be more difficult to analyze.

Practice

Use a benchmark to time the use of an algorithm. Many programming languages have an available function which provides CPU time usage. For long-running algorithms the elapsed time could also be of interest. Results should generally be averaged over several tests.

Run-based profiling can be very sensitive to hardware configuration and the possibility of other programs or tasks running at the same time in a multi-processing and multi-programming environment.

This sort of test also depends heavily on the selection of a particular programming language, compiler, and compiler options, so algorithms being compared must all be implemented under the same conditions.

Space

This section is concerned with use of memory resources (registers, cache, RAM, virtual memory, secondary memory) while the algorithm is being executed. As for time analysis above, analyze the algorithm, typically using space complexity analysis to get an estimate of the run-time memory needed as a function as the size of the input data. The result is normally expressed using Big O notation.

There are up to four aspects of memory usage to consider:

• The amount of memory needed to hold the code for the algorithm.
• The amount of memory needed for the input data.
• The amount of memory needed for any output data.
  • Some algorithms, such as sorting, often rearrange the input data and don't need any additional space for output data. This property is referred to as "in-place" operation.
• The amount of memory needed as working space during the calculation.
  • This includes local variables and any stack space needed by routines called during a calculation; this stack space can be significant for algorithms which use recursive techniques.

Early electronic computers, and early home computers, had relatively small amounts of working memory. For example, the 1949 Electronic Delay Storage Automatic Calculator (EDSAC) had a maximum working memory of 1024 17-bit words, while the 1980 Sinclair ZX80 came initially with 1024 8-bit bytes of working memory. In the late 2010s, it is typical for personal computers to have between 4 and 32 GB of RAM, an increase of over 300 million times as much memory.

Caching and memory hierarchy

Further information: Memory hierarchy

Current computers can have relatively large amounts of memory (possibly Gigabytes), so having to squeeze an algorithm into a confined amount of memory is much less of a problem than it used to be. But the presence of four different categories of memory can be significant:

• Processor registers, the fastest of computer memory technologies with the least amount of storage space. Most direct computation on modern computers occurs with source and destination operands in registers before being updated to the cache, main memory and virtual memory if needed. On a processor core, there are typically on the order of hundreds of bytes or fewer of register availability, although a register file may contain more physical registers than architectural registers defined in the instruction set architecture.
• Cache memory is the second fastest and second smallest memory available in the memory hierarchy. Caches are present in CPUs, GPUs, hard disk drives and external peripherals, and are typically implemented in static RAM. Memory caches are multi-leveled; lower levels are larger, slower and typically shared between processor cores in multi-core processors. In order to process operands in cache memory, a processing unit must fetch the data from the cache, perform the operation in registers and write the data back to the cache. This operates at speeds comparable (about 2-10 times slower) with the CPU or GPU's arithmetic logic unit or floating-point unit if in the L1 cache. It is about 10 times slower if there is an L1 cache miss and it must be retrieved from and written to the L2 cache, and a further 10 times slower if there is an L2 cache miss and it must be retrieved from an L3 cache, if present.
• Main physical memory is most often implemented in dynamic RAM (DRAM). The main memory is much larger (typically gigabytes compared to ≈8 megabytes) than an L3 CPU cache, with read and write latencies typically 10-100 times slower. As of 2018, RAM is increasingly implemented on-chip of processors, as CPU or GPU memory.
• Virtual memory is most often implemented in terms of secondary storage such as a hard disk, and is an extension to the memory hierarchy that has much larger storage space but much larger latency, typically around 1000 times slower than a cache miss for a value in RAM. While originally motivated to create the impression of higher amounts of memory being available than were truly available, virtual memory is more important in contemporary usage for its time-space tradeoff and enabling the usage of virtual machines. Cache misses from main memory are called page faults, and incur huge performance penalties on programs.

An algorithm whose memory needs will fit in cache memory will be much faster than an algorithm which fits in main memory, which in turn will be very much faster than an algorithm which has to resort to virtual memory. Because of this, cache replacement policies are extremely important to high-performance computing, as are cache-aware programming and data alignment. To further complicate the issue, some systems have up to three levels of cache memory, with varying effective speeds. Different systems will have different amounts of these various types of memory, so the effect of algorithm memory needs can vary greatly from one system to another.

In the early days of electronic computing, if an algorithm and its data wouldn't fit in main memory then the algorithm couldn't be used. Nowadays the use of virtual memory appears to provide much memory, but at the cost of performance. If an algorithm and its data will fit in cache memory, then very high speed can be obtained; in this case minimizing space will also help minimize time. This is called the principle of locality, and can be subdivided into locality of reference, spatial locality and temporal locality. An algorithm which will not fit completely in cache memory but which exhibits locality of reference may perform reasonably well.

Criticism of the current state of programming

• David May FRS a British computer scientist and currently Professor of Computer Science at University of Bristol and founder and CTO of XMOS Semiconductor, believes one of the problems is that there is a reliance on Moore's law to solve inefficiencies. He has advanced an 'alternative' to Moore's law (May's law) stated as follows:

Software efficiency halves every 18 months, compensating Moore's Law

May goes on to state:

In ubiquitous systems, halving the instructions executed can double the battery life and big data sets bring big opportunities for better software and algorithms: Reducing the number of operations from N × N to N × log(N) has a dramatic effect when N is large ... for N = 30 billion, this change is as good as 50 years of technology improvements.

• Software author Adam N. Rosenburg in his blog "The failure of the Digital computer", has described the current state of programming as nearing the "Software event horizon", (alluding to the fictitious "shoe event horizon" described by Douglas Adams in his Hitchhiker's Guide to the Galaxy book). He estimates there has been a 70 dB factor loss of productivity or "99.99999 percent, of its ability to deliver the goods", since the 1980s—"When Arthur C. Clarke compared the reality of computing in 2001 to the computer HAL 9000 in his book 2001: A Space Odyssey, he pointed out how wonderfully small and powerful computers were but how disappointing computer programming had become".

Competitions for the best algorithms

The following competitions invite entries for the best algorithms based on some arbitrary criteria decided by the judges:

• Wired magazine

Data compression

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In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compression reduces bits by identifying and eliminating statistical redundancy. No information is lost in lossless compression. Lossy compression reduces bits by removing unnecessary or less important information. Typically, a device that performs data compression is referred to as an encoder, and one that performs the reversal of the process (decompression) as a decoder.

The process of reducing the size of a data file is often referred to as data compression. In the context of data transmission, it is called source coding; encoding done at the source of the data before it is stored or transmitted. Source coding should not be confused with channel coding, for error detection and correction or line coding, the means for mapping data onto a signal.

Compression is useful because it reduces the resources required to store and transmit data. Computational resources are consumed in the compression and decompression processes. Data compression is subject to a space–time complexity trade-off. For instance, a compression scheme for video may require expensive hardware for the video to be decompressed fast enough to be viewed as it is being decompressed, and the option to decompress the video in full before watching it may be inconvenient or require additional storage. The design of data compression schemes involves trade-offs among various factors, including the degree of compression, the amount of distortion introduced (when using lossy data compression), and the computational resources required to compress and decompress the data.

Lossless

Main article: Lossless compression

Lossless data compression algorithms usually exploit statistical redundancy to represent data without losing any information, so that the process is reversible. Lossless compression is possible because most real-world data exhibits statistical redundancy. For example, an image may have areas of color that do not change over several pixels; instead of coding "red pixel, red pixel, ..." the data may be encoded as "279 red pixels". This is a basic example of run-length encoding; there are many schemes to reduce file size by eliminating redundancy.

The Lempel–Ziv (LZ) compression methods are among the most popular algorithms for lossless storage. DEFLATE is a variation on LZ optimized for decompression speed and compression ratio, but compression can be slow. In the mid-1980s, following work by Terry Welch, the Lempel–Ziv–Welch (LZW) algorithm rapidly became the method of choice for most general-purpose compression systems. LZW is used in GIF images, programs such as PKZIP, and hardware devices such as modems. LZ methods use a table-based compression model where table entries are substituted for repeated strings of data. For most LZ methods, this table is generated dynamically from earlier data in the input. The table itself is often Huffman encoded. Grammar-based codes like this can compress highly repetitive input extremely effectively, for instance, a biological data collection of the same or closely related species, a huge versioned document collection, internet archival, etc. The basic task of grammar-based codes is constructing a context-free grammar deriving a single string. Other practical grammar compression algorithms include Sequitur and Re-Pair.

The strongest modern lossless compressors use probabilistic models, such as prediction by partial matching. The Burrows–Wheeler transform can also be viewed as an indirect form of statistical modelling. In a further refinement of the direct use of probabilistic modelling, statistical estimates can be coupled to an algorithm called arithmetic coding. Arithmetic coding is a more modern coding technique that uses the mathematical calculations of a finite-state machine to produce a string of encoded bits from a series of input data symbols. It can achieve superior compression compared to other techniques such as the better-known Huffman algorithm. It uses an internal memory state to avoid the need to perform a one-to-one mapping of individual input symbols to distinct representations that use an integer number of bits, and it clears out the internal memory only after encoding the entire string of data symbols. Arithmetic coding applies especially well to adaptive data compression tasks where the statistics vary and are context-dependent, as it can be easily coupled with an adaptive model of the probability distribution of the input data. An early example of the use of arithmetic coding was in an optional (but not widely used) feature of the JPEG image coding standard. It has since been applied in various other designs including H.263, H.264/MPEG-4 AVC and HEVC for video coding.

Archive software typically has the ability to adjust the "dictionary size", where a larger size demands more random access memory during compression and decompression, but compresses stronger, especially on repeating patterns in files' content.

Lossy

Main article: Lossy compression

In the late 1980s, digital images became more common, and standards for lossless image compression emerged. In the early 1990s, lossy compression methods began to be widely used. In these schemes, some loss of information is accepted as dropping nonessential detail can save storage space. There is a corresponding trade-off between preserving information and reducing size. Lossy data compression schemes are designed by research on how people perceive the data in question. For example, the human eye is more sensitive to subtle variations in luminance than it is to the variations in color. JPEG image compression works in part by rounding off nonessential bits of information. A number of popular compression formats exploit these perceptual differences, including psychoacoustics for sound, and psychovisuals for images and video.

Most forms of lossy compression are based on transform coding, especially the discrete cosine transform (DCT). It was first proposed in 1972 by Nasir Ahmed, who then developed a working algorithm with T. Natarajan and K. R. Rao in 1973, before introducing it in January 1974. DCT is the most widely used lossy compression method, and is used in multimedia formats for images (such as JPEG and HEIF), video (such as MPEG, AVC and HEVC) and audio (such as MP3, AAC and Vorbis).

Lossy image compression is used in digital cameras, to increase storage capacities. Similarly, DVDs, Blu-ray and streaming video use lossy video coding formats. Lossy compression is extensively used in video.

In lossy audio compression, methods of psychoacoustics are used to remove non-audible (or less audible) components of the audio signal. Compression of human speech is often performed with even more specialized techniques; speech coding is distinguished as a separate discipline from general-purpose audio compression. Speech coding is used in internet telephony, for example, audio compression is used for CD ripping and is decoded by the audio players.

Lossy compression can cause generation loss.

Theory

Main article: Shannon's source coding theorem

The theoretical basis for compression is provided by information theory and, more specifically, algorithmic information theory for lossless compression and rate–distortion theory for lossy compression. These areas of study were essentially created by Claude Shannon, who published fundamental papers on the topic in the late 1940s and early 1950s. Other topics associated with compression include coding theory and statistical inference.

Machine learning

There is a close connection between machine learning and compression. A system that predicts the posterior probabilities of a sequence given its entire history can be used for optimal data compression (by using arithmetic coding on the output distribution). Conversely, an optimal compressor can be used for prediction (by finding the symbol that compresses best, given the previous history). This equivalence has been used as a justification for using data compression as a benchmark for "general intelligence".

An alternative view can show compression algorithms implicitly map strings into implicit feature space vectors, and compression-based similarity measures compute similarity within these feature spaces. For each compressor C(.) we define an associated vector space ℵ, such that C(.) maps an input string x, corresponding to the vector norm ||~x||. An exhaustive examination of the feature spaces underlying all compression algorithms is precluded by space; instead, feature vectors chooses to examine three representative lossless compression methods, LZW, LZ77, and PPM.

According to AIXI theory, a connection more directly explained in Hutter Prize, the best possible compression of x is the smallest possible software that generates x. For example, in that model, a zip file's compressed size includes both the zip file and the unzipping software, since you can't unzip it without both, but there may be an even smaller combined form.

Data differencing

Main article: Data differencing

Data compression can be viewed as a special case of data differencing. Data differencing consists of producing a difference given a source and a target, with patching reproducing the target given a source and a difference. Since there is no separate source and target in data compression, one can consider data compression as data differencing with empty source data, the compressed file corresponding to a difference from nothing. This is the same as considering absolute entropy (corresponding to data compression) as a special case of relative entropy (corresponding to data differencing) with no initial data.

The term differential compression is used to emphasize the data differencing connection.

Uses

Image

Main article: Image compression

Entropy coding originated in the 1940s with the introduction of Shannon–Fano coding, the basis for Huffman coding which was developed in 1950. Transform coding dates back to the late 1960s, with the introduction of fast Fourier transform (FFT) coding in 1968 and the Hadamard transform in 1969.

An important image compression technique is the discrete cosine transform (DCT), a technique developed in the early 1970s. DCT is the basis for JPEG, a lossy compression format which was introduced by the Joint Photographic Experts Group (JPEG) in 1992. JPEG greatly reduces the amount of data required to represent an image at the cost of a relatively small reduction in image quality and has become the most widely used image file format. Its highly efficient DCT-based compression algorithm was largely responsible for the wide proliferation of digital images and digital photos.

Lempel–Ziv–Welch (LZW) is a lossless compression algorithm developed in 1984. It is used in the GIF format, introduced in 1987. DEFLATE, a lossless compression algorithm specified in 1996, is used in the Portable Network Graphics (PNG) format.

Wavelet compression, the use of wavelets in image compression, began after the development of DCT coding. The JPEG 2000 standard was introduced in 2000. In contrast to the DCT algorithm used by the original JPEG format, JPEG 2000 instead uses discrete wavelet transform (DWT) algorithms. JPEG 2000 technology, which includes the Motion JPEG 2000 extension, was selected as the video coding standard for digital cinema in 2004.

Audio

See also: Audio coding format and Audio codec

Audio data compression, not to be confused with dynamic range compression, has the potential to reduce the transmission bandwidth and storage requirements of audio data. Audio compression algorithms are implemented in software as audio codecs. In both lossy and lossless compression, information redundancy is reduced, using methods such as coding, quantization, discrete cosine transform and linear prediction to reduce the amount of information used to represent the uncompressed data.

Lossy audio compression algorithms provide higher compression and are used in numerous audio applications including Vorbis and MP3. These algorithms almost all rely on psychoacoustics to eliminate or reduce fidelity of less audible sounds, thereby reducing the space required to store or transmit them.

The acceptable trade-off between loss of audio quality and transmission or storage size depends upon the application. For example, one 640 MB compact disc (CD) holds approximately one hour of uncompressed high fidelity music, less than 2 hours of music compressed losslessly, or 7 hours of music compressed in the MP3 format at a medium bit rate. A digital sound recorder can typically store around 200 hours of clearly intelligible speech in 640 MB.

Lossless audio compression produces a representation of digital data that can be decoded to an exact digital duplicate of the original. Compression ratios are around 50–60% of the original size, which is similar to those for generic lossless data compression. Lossless codecs use curve fitting or linear prediction as a basis for estimating the signal. Parameters describing the estimation and the difference between the estimation and the actual signal are coded separately.

A number of lossless audio compression formats exist. See list of lossless codecs for a listing. Some formats are associated with a distinct system, such as Direct Stream Transfer, used in Super Audio CD and Meridian Lossless Packing, used in DVD-Audio, Dolby TrueHD, Blu-ray and HD DVD.

Some audio file formats feature a combination of a lossy format and a lossless correction; this allows stripping the correction to easily obtain a lossy file. Such formats include MPEG-4 SLS (Scalable to Lossless), WavPack, and OptimFROG DualStream.

When audio files are to be processed, either by further compression or for editing, it is desirable to work from an unchanged original (uncompressed or losslessly compressed). Processing of a lossily compressed file for some purpose usually produces a final result inferior to the creation of the same compressed file from an uncompressed original. In addition to sound editing or mixing, lossless audio compression is often used for archival storage, or as master copies.

Lossy audio compression

Comparison of spectrograms of audio in an uncompressed format and several lossy formats. The lossy spectrograms show bandlimiting of higher frequencies, a common technique associated with lossy audio compression.

Lossy audio compression is used in a wide range of applications. In addition to standalone audio-only applications of file playback in MP3 players or computers, digitally compressed audio streams are used in most video DVDs, digital television, streaming media on the Internet, satellite and cable radio, and increasingly in terrestrial radio broadcasts. Lossy compression typically achieves far greater compression than lossless compression, by discarding less-critical data based on psychoacoustic optimizations.

Psychoacoustics recognizes that not all data in an audio stream can be perceived by the human auditory system. Most lossy compression reduces redundancy by first identifying perceptually irrelevant sounds, that is, sounds that are very hard to hear. Typical examples include high frequencies or sounds that occur at the same time as louder sounds. Those irrelevant sounds are coded with decreased accuracy or not at all.

Due to the nature of lossy algorithms, audio quality suffers a digital generation loss when a file is decompressed and recompressed. This makes lossy compression unsuitable for storing the intermediate results in professional audio engineering applications, such as sound editing and multitrack recording. However, lossy formats such as MP3 are very popular with end-users as the file size is reduced to 5-20% of the original size and a megabyte can store about a minute's worth of music at adequate quality.

Coding methods

To determine what information in an audio signal is perceptually irrelevant, most lossy compression algorithms use transforms such as the modified discrete cosine transform (MDCT) to convert time domain sampled waveforms into a transform domain, typically the frequency domain. Once transformed, component frequencies can be prioritized according to how audible they are. Audibility of spectral components is assessed using the absolute threshold of hearing and the principles of simultaneous masking—the phenomenon wherein a signal is masked by another signal separated by frequency—and, in some cases, temporal masking—where a signal is masked by another signal separated by time. Equal-loudness contours may also be used to weigh the perceptual importance of components. Models of the human ear-brain combination incorporating such effects are often called psychoacoustic models.

Other types of lossy compressors, such as the linear predictive coding (LPC) used with speech, are source-based coders. LPC uses a model of the human vocal tract to analyze speech sounds and infer the parameters used by the model to produce them moment to moment. These changing parameters are transmitted or stored and used to drive another model in the decoder which reproduces the sound.

Lossy formats are often used for the distribution of streaming audio or interactive communication (such as in cell phone networks). In such applications, the data must be decompressed as the data flows, rather than after the entire data stream has been transmitted. Not all audio codecs can be used for streaming applications.

Latency is introduced by the methods used to encode and decode the data. Some codecs will analyze a longer segment, called a frame, of the data to optimize efficiency, and then code it in a manner that requires a larger segment of data at one time to decode. The inherent latency of the coding algorithm can be critical; for example, when there is a two-way transmission of data, such as with a telephone conversation, significant delays may seriously degrade the perceived quality.

In contrast to the speed of compression, which is proportional to the number of operations required by the algorithm, here latency refers to the number of samples that must be analyzed before a block of audio is processed. In the minimum case, latency is zero samples (e.g., if the coder/decoder simply reduces the number of bits used to quantize the signal). Time domain algorithms such as LPC also often have low latencies, hence their popularity in speech coding for telephony. In algorithms such as MP3, however, a large number of samples have to be analyzed to implement a psychoacoustic model in the frequency domain, and latency is on the order of 23 ms.

Speech encoding

Speech encoding is an important category of audio data compression. The perceptual models used to estimate what aspects of speech a human ear can hear are generally somewhat different from those used for music. The range of frequencies needed to convey the sounds of a human voice is normally far narrower than that needed for music, and the sound is normally less complex. As a result, speech can be encoded at high quality using a relatively low bit rate.

This is accomplished, in general, by some combination of two approaches:

• Only encoding sounds that could be made by a single human voice.
• Throwing away more of the data in the signal—keeping just enough to reconstruct an "intelligible" voice rather than the full frequency range of human hearing.

The earliest algorithms used in speech encoding (and audio data compression in general) were the A-law algorithm and the μ-law algorithm.

History

Solidyne 922: The world's first commercial audio bit compression sound card for PC, 1990

Early audio research was conducted at Bell Labs. There, in 1950, C. Chapin Cutler filed the patent on differential pulse-code modulation (DPCM). In 1973, Adaptive DPCM (ADPCM) was introduced by P. Cummiskey, Nikil S. Jayant and James L. Flanagan.

Perceptual coding was first used for speech coding compression, with linear predictive coding (LPC). Initial concepts for LPC date back to the work of Fumitada Itakura (Nagoya University) and Shuzo Saito (Nippon Telegraph and Telephone) in 1966. During the 1970s, Bishnu S. Atal and Manfred R. Schroeder at Bell Labs developed a form of LPC called adaptive predictive coding (APC), a perceptual coding algorithm that exploited the masking properties of the human ear, followed in the early 1980s with the code-excited linear prediction (CELP) algorithm which achieved a significant compression ratio for its time. Perceptual coding is used by modern audio compression formats such as MP3 and AAC.

Discrete cosine transform (DCT), developed by Nasir Ahmed, T. Natarajan and K. R. Rao in 1974, provided the basis for the modified discrete cosine transform (MDCT) used by modern audio compression formats such as MP3, Dolby Digital, and AAC. MDCT was proposed by J. P. Princen, A. W. Johnson and A. B. Bradley in 1987, following earlier work by Princen and Bradley in 1986.

The world's first commercial broadcast automation audio compression system was developed by Oscar Bonello, an engineering professor at the University of Buenos Aires. In 1983, using the psychoacoustic principle of the masking of critical bands first published in 1967, he started developing a practical application based on the recently developed IBM PC computer, and the broadcast automation system was launched in 1987 under the name Audicom. Twenty years later, almost all the radio stations in the world were using similar technology manufactured by a number of companies.

A literature compendium for a large variety of audio coding systems was published in the IEEE's Journal on Selected Areas in Communications (JSAC), in February 1988. While there were some papers from before that time, this collection documented an entire variety of finished, working audio coders, nearly all of them using perceptual techniques and some kind of frequency analysis and back-end noiseless coding.

Video

See also: Video coding format and Video codec

Uncompressed video requires a very high data rate. Although lossless video compression codecs perform at a compression factor of 5 to 12, a typical H.264 lossy compression video has a compression factor between 20 and 200.

The two key video compression techniques used in video coding standards are the discrete cosine transform (DCT) and motion compensation (MC). Most video coding standards, such as the H.26x and MPEG formats, typically use motion-compensated DCT video coding (block motion compensation).

Most video codecs are used alongside audio compression techniques to store the separate but complementary data streams as one combined package using so-called container formats.

Encoding theory

Video data may be represented as a series of still image frames. Such data usually contains abundant amounts of spatial and temporal redundancy. Video compression algorithms attempt to reduce redundancy and store information more compactly.

Most video compression formats and codecs exploit both spatial and temporal redundancy (e.g. through difference coding with motion compensation). Similarities can be encoded by only storing differences between e.g. temporally adjacent frames (inter-frame coding) or spatially adjacent pixels (intra-frame coding). Inter-frame compression (a temporal delta encoding) (re)uses data from one or more earlier or later frames in a sequence to describe the current frame. Intra-frame coding, on the other hand, uses only data from within the current frame, effectively being still-image compression.

The intra-frame video coding formats used in camcorders and video editing employ simpler compression that uses only intra-frame prediction. This simplifies video editing software, as it prevents a situation in which a compressed frame refers to data that the editor has deleted.

Usually, video compression additionally employs lossy compression techniques like quantization that reduce aspects of the source data that are (more or less) irrelevant to the human visual perception by exploiting perceptual features of human vision. For example, small differences in color are more difficult to perceive than are changes in brightness. Compression algorithms can average a color across these similar areas in a manner similar to those used in JPEG image compression. As in all lossy compression, there is a trade-off between video quality and bit rate, cost of processing the compression and decompression, and system requirements. Highly compressed video may present visible or distracting artifacts.

Other methods other than the prevalent DCT-based transform formats, such as fractal compression, matching pursuit and the use of a discrete wavelet transform (DWT), have been the subject of some research, but are typically not used in practical products. Wavelet compression is used in still-image coders and video coders without motion compensation. Interest in fractal compression seems to be waning, due to recent theoretical analysis showing a comparative lack of effectiveness of such methods.

Inter-frame coding

Main article: Inter frame

Further information: Motion compensation

In inter-frame coding, individual frames of a video sequence are compared from one frame to the next, and the video compression codec records the differences to the reference frame. If the frame contains areas where nothing has moved, the system can simply issue a short command that copies that part of the previous frame into the next one. If sections of the frame move in a simple manner, the compressor can emit a (slightly longer) command that tells the decompressor to shift, rotate, lighten, or darken the copy. This longer command still remains much shorter than data generated by intra-frame compression. Usually, the encoder will also transmit a residue signal which describes the remaining more subtle differences to the reference imagery. Using entropy coding, these residue signals have a more compact representation than the full signal. In areas of video with more motion, the compression must encode more data to keep up with the larger number of pixels that are changing. Commonly during explosions, flames, flocks of animals, and in some panning shots, the high-frequency detail leads to quality decreases or to increases in the variable bitrate.

Hybrid block-based transform formats

Further information: Discrete cosine transform

 

Processing stages of a typical video encoder

Today, nearly all commonly used video compression methods (e.g., those in standards approved by the ITU-T or ISO) share the same basic architecture that dates back to H.261 which was standardized in 1988 by the ITU-T. They mostly rely on the DCT, applied to rectangular blocks of neighboring pixels, and temporal prediction using motion vectors, as well as nowadays also an in-loop filtering step.

In the prediction stage, various deduplication and difference-coding techniques are applied that help decorrelate data and describe new data based on already transmitted data.

Then rectangular blocks of remaining pixel data are transformed to the frequency domain. In the main lossy processing stage, frequency domain data gets quantized in order to reduce information that is irrelevant to human visual perception.

In the last stage statistical redundancy gets largely eliminated by an entropy coder which often applies some form of arithmetic coding.

In an additional in-loop filtering stage various filters can be applied to the reconstructed image signal. By computing these filters also inside the encoding loop they can help compression because they can be applied to reference material before it gets used in the prediction process and they can be guided using the original signal. The most popular example are deblocking filters that blur out blocking artifacts from quantization discontinuities at transform block boundaries.

History

Main article: Video coding format § History

In 1967, A.H. Robinson and C. Cherry proposed a run-length encoding bandwidth compression scheme for the transmission of analog television signals. Discrete cosine transform (DCT), which is fundamental to modern video compression, was introduced by Nasir Ahmed, T. Natarajan and K. R. Rao in 1974.

H.261, which debuted in 1988, commercially introduced the prevalent basic architecture of video compression technology. It was the first video coding format based on DCT compression. H.261 was developed by a number of companies, including Hitachi, PictureTel, NTT, BT and Toshiba.

The most popular video coding standards used for codecs have been the MPEG standards. MPEG-1 was developed by the Motion Picture Experts Group (MPEG) in 1991, and it was designed to compress VHS-quality video. It was succeeded in 1994 by MPEG-2/H.262, which was developed by a number of companies, primarily Sony, Thomson and Mitsubishi Electric. MPEG-2 became the standard video format for DVD and SD digital television. In 1999, it was followed by MPEG-4/H.263. It was also developed by a number of companies, primarily Mitsubishi Electric, Hitachi and Panasonic.

H.264/MPEG-4 AVC was developed in 2003 by a number of organizations, primarily Panasonic, Godo Kaisha IP Bridge and LG Electronics. AVC commercially introduced the modern context-adaptive binary arithmetic coding (CABAC) and context-adaptive variable-length coding (CAVLC) algorithms. AVC is the main video encoding standard for Blu-ray Discs, and is widely used by video sharing websites and streaming internet services such as YouTube, Netflix, Vimeo, and iTunes Store, web software such as Adobe Flash Player and Microsoft Silverlight, and various HDTV broadcasts over terrestrial and satellite television.

Genetics

Genetics compression algorithms are the latest generation of lossless algorithms that compress data (typically sequences of nucleotides) using both conventional compression algorithms and genetic algorithms adapted to the specific datatype. In 2012, a team of scientists from Johns Hopkins University published a genetic compression algorithm that does not use a reference genome for compression. HAPZIPPER was tailored for HapMap data and achieves over 20-fold compression (95% reduction in file size), providing 2- to 4-fold better compression and is less computationally intensive than the leading general-purpose compression utilities. For this, Chanda, Elhaik, and Bader introduced MAF-based encoding (MAFE), which reduces the heterogeneity of the dataset by sorting SNPs by their minor allele frequency, thus homogenizing the dataset. Other algorithms developed in 2009 and 2013 (DNAZip and GenomeZip) have compression ratios of up to 1200-fold—allowing 6 billion basepair diploid human genomes to be stored in 2.5 megabytes (relative to a reference genome or averaged over many genomes).

Outlook and currently unused potential

It is estimated that the total amount of data that is stored on the world's storage devices could be further compressed with existing compression algorithms by a remaining average factor of 4.5:1. It is estimated that the combined technological capacity of the world to store information provides 1,300 exabytes of hardware digits in 2007, but when the corresponding content is optimally compressed, this only represents 295 exabytes of Shannon information.

Moral nihilism

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Moral_nihilism

Moral nihilism (also known as ethical nihilism) is the meta-ethical view that nothing is morally right or wrong.

Moral nihilism is distinct from moral relativism, which allows for actions to be wrong relative to a particular culture or individual. It is also distinct from expressivism, according to which when we make moral claims, "We are not making an effort to describe the way the world is ... we are venting our emotions, commanding others to act in certain ways, or revealing a plan of action".

Moral nihilism today broadly tends to take the form of an Error Theory: The view developed originally by J.L. Mackie in his 1977 book Ethics: Inventing Right and Wrong. Error theory and nihilism broadly take the form of a negative claim about the existence of objective values or properties. Under traditional views there are moral properties or methods which hold objectively in some sense beyond our contingent interests which morally obligate us to act. For Mackie and the Error Theorists, such properties do not exist in the world, and therefore morality conceived of by reference to objective facts must also not exist. Therefore, morality in the traditional sense does not exist.

However, holding nihilism does not necessarily imply that we should give up using moral or ethical language; some nihilists contend that it remains a useful tool. In fact Mackie and other contemporary defenders of Error Theory, such as Richard Joyce, defend the use of moral or ethical talk and action even in knowledge of their fundamental falsity. The legitimacy of this activity however is questionable and is a subject of great debate in philosophy at the moment.

Forms of nihilism

Moral nihilists agree that all claims such as 'murder is morally wrong' are not true. But different nihilistic views differ in two ways.

Some may say that such claims are neither true nor false; others say that they are all false.

Nihilists differ in the scope of their theories. Error theorists typically claim that it is only distinctively moral claims which are false; practical nihilists claim that there are no reasons for action of any kind; some nihilists extend this claim to include reasons for belief.

Ethical language: false versus not truth-apt

J. L. Mackie argues that moral assertions are only true if there are moral properties, but because there are none, all such claims are false. Under such a view moral propositions which express beliefs are then systematically in error. For under Mackie's view, if there are to be moral properties, they must be objective and therefore not amenable to differences in subjective desires and preferences. Moreover any claims that these moral properties, if they did exist would need to be intrinsically motivating by being in some primitive relation to our consciousness. They must be able of guiding us morally just by the fact of being in some clear awareness of their truth. But this is not the case, and such ideas in his views are plainly queer.

Other versions of the theory claim that moral assertions are not true because they are neither true nor false. This form of moral nihilism claims that moral beliefs and assertions presuppose the existence of moral facts that do not exist. Consider, for example, the claim that the present king of France is bald. Some argue that this claim is neither true nor false because it presupposes that there is currently a king of France, but there is not. The claim suffers from "presupposition failure". Richard Joyce argues for this form of moral nihilism under the name "fictionalism".

The scope question

Error theory is built on three principles:

1. There are no moral features in this world; nothing is right or wrong.
2. Therefore, no moral judgments are true.
3. However, our sincere moral judgments try, but always fail, to describe the moral features of things.

Thus, we always lapse into error when thinking in moral terms. We are trying to state the truth when we make moral judgments. But since there is no moral truth, all of our moral claims are mistaken. Hence the error. These three principles lead to the conclusion that there is no moral knowledge. Knowledge requires truth. If there is no moral truth, there can be no moral knowledge. Thus moral values are purely chimerical.

Arguments for nihilism

Argument from queerness

The most prominent argument for nihilism is the argument from queerness.

J. L. Mackie argues that there are no objective ethical values, by arguing that they would be queer (strange):

If there were objective values, then they would be entities or qualities or relations of a very strange sort, utterly different from anything else in the universe.

For all those who also find such entities queer (prima facie implausible), there is reason to doubt the existence of objective values.

In his book Morality without Foundations: A Defense of Ethical Contextualism (1999), Mark Timmons provides a reconstruction of Mackie's views in the form of the two related arguments. These are based on the rejection of properties, facts, and relationships that do not fit within the worldview of philosophical naturalism, the idea "that everything—including any particular events, facts, properties, and so on—is part of the natural physical world that science investigates" (1999, p. 12). Timmons adds, "The undeniable attraction of this outlook in contemporary philosophy no doubt stems from the rise of modern science and the belief that science is our best avenue for discovering the nature of reality".

There are several ways in which moral properties are supposedly queer:

• our ordinary moral discourse purports to refer to intrinsically prescriptive properties and facts "that would somehow motivate us or provide us with reasons for action independent of our desires and aversions"—but such properties and facts do not comport with philosophical naturalism.
• given that objective moral properties supposedly supervene upon natural properties (such as biological or psychological properties), the relation between the moral properties and the natural properties is metaphysically mysterious and does not comport with philosophical naturalism.
• a moral realist who countenances the existence of metaphysically queer properties, facts, and relations must also posit some special faculty by which we have knowledge of them.

Responses and criticisms

Christine Korsgaard responds to Mackie by saying:

Of course there are entities that meet these criteria. It's true that they are queer sorts of entities and that knowing them isn't like anything else. But that doesn't mean that they don't exist. ... For it is the most familiar fact of human life that the world contains entities that can tell us what to do and make us do it. They are people, and the other animals.

Other criticisms of the argument include noting that the very fact that such entities would have to be something fundamentally different from what we normally experience, therefore assumably outside our sphere of experience, we cannot prima facie have reason to either doubt or affirm their existence. Therefore if one had independent grounds for supposing such things to exist (such as a reductio ad absurdum of the contrary) the argument from queerness cannot give one any particular reason to think otherwise. An argument along these lines has been provided by e.g. Akeel Bilgrami.

Argument from explanatory impotence

Gilbert Harman argued that we do not need to posit the existence of objective values in order to explain our 'moral observations'.