In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, computer science Computer science or computing science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe, and transform information. Computer science, and related subjects, an 'algorithm' is an effective method Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions that later were shown to be equivalent; the notion captured by these definitions is known as (recursive) computability for solving a problem expressed as a finite sequence of instructions. Algorithms are used for calculation The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm to the vague heuristics of calculating a strategy in a competition or calculating the chance of a successful relationship between two people, data processing Computer data processing is any process that uses a computer program to enter data and summarise, analyse or otherwise convert data into usable information. The process may be automated and run on a computer. It involves recording, analysing, sorting, summarising, calculating, disseminating and storing data. Because data is most useful when well-, and many other fields. (In more advanced or abstract settings, the instructions do not necessarily constitute a finite sequence, and even not necessarily a sequence; see, e.g., "nondeterministic algorithm In computer science, a nondeterministic algorithm is an algorithm with one or more choice points where multiple different continuations are possible, without any specification of which one will be taken".)

Each algorithm is a list of well-defined instructions for completing a task. Starting from an initial state, the instructions describe a computation that proceeds through a well-defined series of successive states, eventually terminating in a final ending state. The transition from one state to the next is not necessarily deterministic Determinism is the philosophical view that every event, including human cognition, behaviour, decision, and action, is causally determined by the environment. It is, in essence, the view that one's life is predetermined before one is even born. Determinism proposes there is a predetermined unbroken chain of prior occurrences back to the origin of; some algorithms, known as randomized algorithms A randomized algorithm or probabilistic algorithm is an algorithm which employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random bits. Formally,, incorporate randomness.

A partial formalization of the concept began with attempts to solve the Entscheidungsproblem In mathematics, the Entscheidungsproblem is a challenge posed by David Hilbert in 1928. The Entscheidungsproblem asks for an algorithm that will take as input a description of a formal language and a mathematical statement in the language and produce as output either "True" or "False" according to whether the statement is true (the "decision problem") posed by David Hilbert David Hilbert /ˈdaːfɪt ˈhɪlbʌt/ was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of in 1928. Subsequent formalizations were framed as attempts to define "effective calculability In computability theory, the Church–Turing thesis is a combined hypothesis ("thesis") about the nature of effectively calculable (computable) functions by recursion (Church's Thesis), by mechanical device equivalent to a Turing machine (Turing's Thesis) or by use of Church's λ-calculus. The three computational processes (recursion, λ-"[1] or "effective method";[2] those formalizations included the Gödel Kurt Gödel (German pronunciation: [kʊʁt ˈɡøːdl̩] ; April 28, 1906, Brno, Moravia – January 14, 1978, Princeton, New Jersey, USA) was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century,–Herbrand Jacques Herbrand was a French mathematician who was born in Paris, France and died in La Bérarde, Isère, France–Kleene Stephen Cole Kleene was an American mathematician who helped lay the foundations for theoretical computer science. One of many distinguished students of Alonzo Church, Kleene, along with Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory. Kleene's work grounds the study of recursive functions Recursion in computer science is a method where the solution to a problem depends on solutions to smaller instances of the same problem. The approach can be applied to many types of problems, and is one of the central ideas of computer science of 1930, 1934 and 1935, Alonzo Church Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem's lambda calculus In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. It was introduced by Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics. After the original system was shown to be logically inconsistent , of 1936, Emil Post Emil Leon Post, Ph.D., was a mathematician and logician's "Formulation 1 A Post–Turing machine is a "program formulation" of an especially simple type of Turing machine, comprising a variant of Emil Post's Turing-equivalent model of computation described below. A Post-Turing machine uses a binary alphabet, an infinite sequence of binary storage locations, and a primitive programming language with" of 1936, and Alan Turing Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst and computer scientist. He was influential in the development of computer science and providing a formalisation of the concept of the algorithm and computation with the Turing machine, playing a significant role in the creation of the modern computer's Turing machines A Turing machine is a theoretical device that manipulates symbols contained on a strip of tape. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside of a computer. The "Turing" machine was described by Alan Turing in of 1936–7 and 1939.

The adjective "continuous" when applied to the word "algorithm" can mean: 1) An algorithm operating on data that represents continuous quantities, even though this data is represented by discrete approximations – such algorithms are studied in numerical analysis Numerical analysis is the study of algorithms that use numerical approximation for the problems of continuous mathematics (as distinguished from discrete mathematics); or 2) An algorithm in the form of a differential equation A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines that operates continuously on the data, running on an analog computer An analog computer is a form of computer that uses the continuously-changeable aspects of physical phenomena such as electrical, mechanical, or hydraulic quantities to model the problem being solved. In contrast, digital computers represent varying quantities incrementally, as their numerical values change.[3]

Contents

Etymology

Al-Khwārizmī Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī (c. 780, Khwārizm – c. 850) was a Persian mathematician, astronomer and geographer, a scholar in the House of Wisdom in Baghdad (in Persian: خوارزمی, arabic الخوارزمی), Persian The Persian people are defined by the use of the Persian language as their mother tongue. However, the term Persian has also a supra-ethnic significance and has been historically referred to a part of Iranian peoples. The origin of the Persian people is traced to the ancient Indo-Iranians , who arrived in parts of Greater Iran circa 2000-1500 BCE astronomer An astronomer is a scientist who studies celestial bodies such as planets, stars, and galaxies and mathematician A mathematician is a person whose primary area of study or research, or both, is the field of mathematics. Mathematicians are concerned with particular problems related to logic, space, transformations, numbers and more general ideas which encompass these concepts. Some notable mathematicians include Sir Isaac Newton, Muhammad ibn Mūsā al-Khwā, wrote a treatise A treatise is a formal and systematic written discourse on some subject, generally longer and treating it in greater depth than an essay, and more concerned with investigating or exposing the principles of the subject in the Arabic language in 825 AD, On Calculation with Hindu–Arabic numeral system. (See algorism Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist. This system largely superseded earlier calculation systems that used a different set of symbols for each numerical magnitude and in some). It was translated from Arabic into Latin Latin or sometimes Roman is an Italic language originally spoken in Latium and Ancient Rome. Although often considered a dead language, in view of the fact that it has no native, fluent speakers, Latin continues to be taught in schools and has been, and currently is, used in the process of new word production in modern languages from many in the 12th century as Algoritmi de numero Indorum (al-Daffa 1977), whose title is likely intended to mean "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's rendition of the author's name; but people misunderstanding the title treated Algoritmi as a Latin plural and this led to the word "algorithm" (Latin algorismus) coming to mean "calculation method". The intrusive "th" is most likely due to a false cognate False cognates are pairs of words in the same or different languages that are similar in form and meaning but have different roots. That is, they appear to be, or are sometimes considered, cognates, when in fact they are not with the Greek ἀριθμός (arithmos) meaning "numbers".

Why algorithms are necessary: an informal definition

For a detailed presentation of the various points of view around the definition of "algorithm" see Algorithm characterizations The word algorithm does not have a generally accepted definition. Researchers are actively working in formalizing this term. This article will present some of the "characterizations" of the notion of "algorithm" in more detail. For examples of simple addition algorithms specified in the detailed manner described in Algorithm characterizations The word algorithm does not have a generally accepted definition. Researchers are actively working in formalizing this term. This article will present some of the "characterizations" of the notion of "algorithm" in more detail, see Algorithm examples The Turing machine model is primitive, but not as primitive as it can be. As noted in the above quotes this is a source of concern when studying complexity and equivalence of algorithms. Although the observations quoted below concern the Random access machine model – a Turing-machine equivalent – the problem remains for any Turing-equivalent.

While there is no generally accepted formal definition of "algorithm," an informal definition could be "a process that performs some sequence of operations." For some people, a program is only an algorithm if it stops eventually. For others, a program is only an algorithm if it stops before a given number of calculation steps.

A prototypical example of an algorithm is Euclid's algorithm In mathematics, the Euclidean algorithm[a] is an efficient method for computing the greatest common divisor (GCD), also known as the greatest common factor (GCF) or highest common factor (HCF). It is named after the Greek mathematician Euclid, who described it in Books VII and X of his Elements to determine the maximum common divisor of two integers.

We can derive clues to the issues involved and an informal meaning of the word from the following quotation from Boolos & Jeffrey (1974, 1999) (boldface added):

No human being can write fast enough, or long enough, or small enough† ( †"smaller and smaller without limit ...you'd be trying to write on molecules, on atoms, on electrons") to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols[4]

The term "enumerably infinite" means "countable using integers perhaps extending to infinity." Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be chosen from 0 to infinity. Thus we might expect an algorithm to be an algebraic equation such as y = m + n — two arbitrary "input variables" m and n that produce an output y. As we see in Algorithm characterizations The word algorithm does not have a generally accepted definition. Researchers are actively working in formalizing this term. This article will present some of the "characterizations" of the notion of "algorithm" in more detail — the word algorithm implies much more than this, something on the order of (for our addition example):

Precise instructions (in language understood by "the computer") for a "fast, efficient, good" process that specifies the "moves" of "the computer" (machine or human, equipped with the necessary internally contained information and capabilities) to find, decode, and then munch arbitrary input integers/symbols m and n, symbols + and = ... and (reliably, correctly, "effectively") produce, in a "reasonable" time, output-integer y at a specified place and in a specified format.

The concept of algorithm is also used to define the notion of decidability In logic, the term decidable refers to the existence of an effective method for determining membership in a set of formulas. Logical systems such as propositional logic are decidable if membership in their set of logically valid formulas can be effectively determined. A theory (set of formulas closed under logical consequence) in a fixed logical. That notion is central for explaining how formal systems In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive (to conclude) one expression from one or more other expressions (premises) antecedently supposed (axioms) or derived (theorems). The axioms and rules may be called a deductive apparatus. A formal system may be formulated and studied for its come into being starting from a small set of axioms In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other truths and rules. In logic Logic is the study of reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension. From such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete (in some sense) and abstract usage of the term.

Formalization

Algorithms are essential to the way computers process information. Many computer programs A computer program is a sequence of instructions written to perform a specified task for a computer. A computer requires programs to function, typically executing the program's instructions in a central processor. The program has an executable form that the computer can use directly to execute the instructions. The same program in its human- contain algorithms that specify the specific instructions a computer should perform (in a specific order) to carry out a specified task, such as calculating employees' paychecks or printing students' report cards. Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete Turing completeness, named after Alan Turing, is significant in that every plausible design for a computing device so far advanced can be emulated by a universal Turing machine — an observation that has become known as the Church-Turing thesis. Thus, a machine that can act as a universal Turing machine can, in principle, perform any calculation system. Authors who assert this thesis include Minsky (1967), Savage (1987) and Gurevich (2000):

Minsky: "But we will also maintain, with Turing . . . that any procedure which could "naturally" be called effective, can in fact be realized by a (simple) machine. Although this may seem extreme, the arguments . . . in its favor are hard to refute".[5]

Gurevich: "...Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine ... according to Savage [1987], an algorithm is a computational process defined by a Turing machine".[6]

Typically, when an algorithm is associated with processing information, data is read from an input source, written to an output device, and/or stored for further processing. Stored data is regarded as part of the internal state of the entity performing the algorithm. In practice, the state is stored in one or more data structures In computer science, a data structure is a particular way of storing and organizing data in a computer so that it can be used efficiently.

For any such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be systematically dealt with, case-by-case; the criteria for each case must be clear (and computable).

Because an algorithm is a precise list of precise steps, the order of computation will always be critical to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting "from the top" and going "down to the bottom", an idea that is described more formally by flow of control In computer science, control flow refers to the order in which the individual statements, instructions, or function calls of an imperative or a declarative program are executed or evaluated.

So far, this discussion of the formalization of an algorithm has assumed the premises of imperative programming In computer science, imperative programming is a programming paradigm that describes computation in terms of statements that change a program state. In much the same way that imperative mood in natural languages expresses commands to take action, imperative programs define sequences of commands for the computer to perform. This is the most common conception, and it attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the assignment operation In computer science the assignment statement sets or re-sets the value stored in the storage location denoted by a variable name. In most imperative computer programming languages the assignment statement is one of the basic statements, setting the value of a variable. It derives from the intuition of "memory In psychology, memory is an organism's ability to store, retain, and recall information. Traditional studies of memory began in the fields of philosophy, including techniques of artificially enhancing the memory. The late nineteenth and early twentieth century put memory within the paradigms of cognitive psychology. In recent decades, it has" as a scratchpad. There is an example below of such an assignment.

For some alternate conceptions of what constitutes an algorithm see functional programming In computer science, functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. It emphasizes the application of functions, in contrast to the imperative programming style, which emphasizes changes in state. Functional programming has its roots in the and logic programming Logic programming is, in its broadest sense, the use of mathematical logic for computer programming. In this view of logic programming, which can be traced at least as far back as John McCarthy's advice-taker proposal, logic is used as a purely declarative representation language, and a theorem-prover or model-generator is used as the problem- .

Termination

Some writers restrict the definition of algorithm to procedures that eventually finish. In such a category Kleene places the "decision procedure or decision method or algorithm for the question".[7] Others, including Kleene, include procedures that could run forever without stopping; such a procedure has been called a "computational method"[8] or "calculation procedure or algorithm (and hence a calculation problem) in relation to a general question which requires for an answer, not yes or no, but the exhibiting of some object".[9]

Minsky makes the pertinent observation, in regards to determining whether an algorithm will eventually terminate (from a particular starting state):

But if the length of the process isn't known in advance, then "trying" it may not be decisive, because if the process does go on forever—then at no time will we ever be sure of the answer.[5]

As it happens, no other method can do any better, as was shown by Alan Turing with his celebrated result on the undecidability of the so-called halting problem. There is no algorithmic procedure for determining of arbitrary algorithms whether or not they terminate from given starting states. The analysis of algorithms for their likelihood of termination is called termination analysis.

See the examples of (im-)"proper" subtraction at partial function for more about what can happen when an algorithm fails for certain of its input numbers—e.g., (i) non-termination, (ii) production of "junk" (output in the wrong format to be considered a number) or no number(s) at all (halt ends the computation with no output), (iii) wrong number(s), or (iv) a combination of these. Kleene proposed that the production of "junk" or failure to produce a number is solved by having the algorithm detect these instances and produce e.g., an error message (he suggested "0"), or preferably, force the algorithm into an endless loop.[10] Davis (1958) does this to his subtraction algorithm—he fixes his algorithm in a second example so that it is proper subtraction and it terminates.[11] Along with the logical outcomes "true" and "false" Kleene (1952) also proposes the use of a third logical symbol "u" — undecided[12] — thus an algorithm will always produce something when confronted with a "proposition". The problem of wrong answers must be solved with an independent "proof" of the algorithm e.g., using induction:

We normally require auxiliary evidence for this [that the algorithm correctly defines a mu recursive function], e.g., in the form of an inductive proof that, for each argument value, the computation terminates with a unique value.[13]

Expressing algorithms

Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, programming languages or control tables (processed by interpreters). Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode, flowcharts and control tables are structured ways to express algorithms that avoid many of the ambiguities common in natural language statements, while remaining independent of a particular implementation language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are often used as a way to define or document algorithms.

There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables (see more at finite state machine and state transition table), as flowcharts (see more at state diagram), or as a form of rudimentary machine code or assembly code called "sets of quadruples" (see more at Turing machine).

Sometimes it is helpful in the description of an algorithm to supplement small "flow charts" (state diagrams) with natural-language and/or arithmetic expressions written inside "block diagrams" to summarize what the "flow charts" are accomplishing.

Representations of algorithms are generally classed into three accepted levels of Turing machine description:[14]

"...prose to describe an algorithm, ignoring the implementation details. At this level we do not need to mention how the machine manages its tape or head."
"...prose used to define the way the Turing machine uses its head and the way that it stores data on its tape. At this level we do not give details of states or transition function."
Most detailed, "lowest level", gives the Turing machine's "state table".
For an example of the simple algorithm "Add m+n" described in all three levels see Algorithm examples.

Computer algorithms

In computer systems, an algorithm is basically an instance of logic written in software by software developers to be effective for the intended "target" computer(s), in order for the software on the target machines to do something. For instance, if a person is writing software that is supposed to print out a PDF document located at the operating system folder "/My Documents" at computer drive "D:" every Friday at 10 pm, they will write an algorithm that specifies the following actions: "If today's date (computer time) is 'Friday,' open the document at 'D:/My Documents' and call the 'print' function". While this simple algorithm does not look into whether the printer has enough paper or whether the document has been moved into a different location, one can make this algorithm more robust and anticipate these problems by rewriting it as a formal CASE statement[15] or as a (carefully crafted) sequence of IF-THEN-ELSE statements.[16] For example the CASE statement might appear as follows (there are other possibilities):

CASE 1: IF today's date is NOT Friday THEN exit this CASE instruction ELSE
CASE 2: IF today's date is Friday AND the document is located at 'D:/My Documents' AND there is paper in the printer THEN print the document (and exit this CASE instruction) ELSE
CASE 3: IF today's date is Friday AND the document is NOT located at 'D:/My Documents' THEN display 'document not found' error message (and exit this CASE instruction) ELSE
CASE 4: IF today's date is Friday AND the document is located at 'D:/My Documents' AND there is NO paper in the printer THEN (i) display 'out of paper' error message and (ii) exit.

Note that CASE 3 includes two possibilities: (i) the document is NOT located at 'D:/My Documents' AND there's paper in the printer OR (ii) the document is NOT located at 'D:/My Documents' AND there's NO paper in the printer.

The sequence of IF-THEN-ELSE tests might look like this:

TEST 1: IF today's date is NOT Friday THEN done ELSE TEST 2:
TEST 2: IF the document is NOT located at 'D:/My Documents' THEN display 'document not found' error message ELSE TEST 3:
TEST 3: IF there is NO paper in the printer THEN display 'out of paper' error message ELSE print the document.

These examples' logic grants precedence to the instance of "NO document at 'D:/My Documents' ". Also observe that in a well-crafted CASE statement or sequence of IF-THEN-ELSE statements the number of distinct actions—4 in these examples: do nothing, print the document, display 'document not found', display 'out of paper' – equals the number of cases.

Given unlimited memory, a computational machine with the ability to execute either a set of CASE statements or a sequence of IF-THEN-ELSE statements is Turing complete. Therefore, anything that is computable can be computed by this machine. This form of algorithm is fundamental to computer programming in all its forms (see more at McCarthy formalism).

Implementation

Most algorithms are intended to be implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain implementing arithmetic or an insect looking for food), in an electrical circuit, or in a mechanical device.

Example

Further information: Algorithm examples An animation of the quicksort algorithm sorting an array of randomized values. The red bars mark the pivot element; at the start of the animation, the element farthest to the right hand side is chosen as the pivot.

One of the simplest algorithms is to find the largest number in an (unsorted) list of numbers. The solution necessarily requires looking at every number in the list, but only once at each. From this follows a simple algorithm, which can be stated in a high-level description English prose, as:

High-level description:

  1. Assume the first item is largest.
  2. Look at each of the remaining items in the list and if it is larger than the largest item so far, make a note of it.
  3. The last noted item is the largest in the list when the process is complete.

(Quasi-)formal description: Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm in pseudocode or pidgin code:

Algorithm LargestNumber
Input: A non-empty list of numbers L.
Output: The largest number in the list L.
largest ← L0
for each item in the list (Length(L)≥1), do
if the item > largest, then
largest ← the item
return largest

For a more complex example of an algorithm, see Euclid's algorithm for the greatest common divisor, one of the earliest algorithms known.

Algorithmic analysis

Main article: Analysis of algorithms

It is frequently important to know how much of a particular resource (such as time or storage) is theoretically required for a given algorithm. Methods have been developed for the analysis of algorithms to obtain such quantitative answers (estimates); for example, the algorithm above has a time requirement of O(n), using the big O notation with n as the length of the list. At all times the algorithm only needs to remember two values: the largest number found so far, and its current position in the input list. Therefore it is said to have a space requirement of O(1), if the space required to store the input numbers is not counted, or O(n) if it is counted.

Different algorithms may complete the same task with a different set of instructions in less or more time, space, or 'effort' than others. For example, a binary search algorithm will usually outperform a brute force sequential search when used for table lookups on sorted lists.

Formal versus empirical

Main articles: Empirical algorithmics, Profiling (computer programming), and Program optimization
This section requires expansion.

The analysis and study of algorithms is a discipline of computer science, and is often practiced abstractly without the use of a specific programming language or implementation. In this sense, algorithm analysis resembles other mathematical disciplines in that it focuses on the underlying properties of the algorithm and not on the specifics of any particular implementation. Usually pseudocode is used for analysis as it is the simplest and most general representation. However, ultimately, most algorithms are usually implemented on particular hardware / software platforms and their algorithmic efficiency is eventually put to the test using real code.

Empirical testing is useful because it may uncover unexpected interactions that affect performance. For instance an algorithm that has no locality of reference may have much poorer performance than predicted because it 'thrashes the cache'. Benchmarks may be used to compare before/after potential improvements to an algorithm after program optimization.

Classification

There are various ways to classify algorithms, each with its own merits.

By implementation

One way to classify algorithms is by implementation means.[original research?]

By design paradigm

Another[original research?] way of classifying algorithms is by their design methodology or paradigm. There is a certain number of paradigms, each different from the other. Furthermore, each of these categories will include many different types of algorithms. Some commonly found paradigms include:

  1. Randomized algorithms are those that make some choices randomly (or pseudo-randomly); for some problems, it can in fact be proven that the fastest solutions must involve some randomness. There are two large classes of such algorithms:
    1. Monte Carlo algorithms return a correct answer with high-probability. E.g. RP is the subclass of these that run in polynomial time)
    2. Las Vegas algorithms always return the correct answer, but their running time is only probabilistically bound, e.g. ZPP.
  2. In optimization problems, heuristic algorithms do not try to find an optimal solution, but an approximate solution where the time or resources are limited. They are not practical to find perfect solutions. An example of this would be local search, tabu search, or simulated annealing algorithms, a class of heuristic probabilistic algorithms that vary the solution of a problem by a random amount. The name "simulated annealing" alludes to the metallurgic term meaning the heating and cooling of metal to achieve freedom from defects. The purpose of the random variance is to find close to globally optimal solutions rather than simply locally optimal ones, the idea being that the random element will be decreased as the algorithm settles down to a solution. Approximation algorithms are those heuristic algorithms that additionally provide some bounds on the error. Genetic algorithms attempt to find solutions to problems by mimicking biological evolutionary processes, with a cycle of random mutations yielding successive generations of "solutions". Thus, they emulate reproduction and "survival of the fittest". In genetic programming, this approach is extended to algorithms, by regarding the algorithm itself as a "solution" to a problem.

By field of study

See also: List of algorithms

Every field of science has its own problems and needs efficient algorithms. Related problems in one field are often studied together. Some example classes are search algorithms, sorting algorithms, merge algorithms, numerical algorithms, graph algorithms, string algorithms, computational geometric algorithms, combinatorial algorithms, machine learning, cryptography, data compression algorithms and parsing techniques.

Fields tend to overlap with each other, and algorithm advances in one field may improve those of other, sometimes completely unrelated, fields. For example, dynamic programming was invented for optimization of resource consumption in industry, but is now used in solving a broad range of problems in many fields.

By complexity

See also: Complexity class and Parameterized complexity

Algorithms can be classified by the amount of time they need to complete compared to their input size. There is a wide variety: some algorithms complete in linear time relative to input size, some do so in an exponential amount of time or even worse, and some never halt. Additionally, some problems may have multiple algorithms of differing complexity, while other problems might have no algorithms or no known efficient algorithms. There are also mappings from some problems to other problems. Owing to this, it was found to be more suitable to classify the problems themselves instead of the algorithms into equivalence classes based on the complexity of the best possible algorithms for them.

By computing power

Another[dubious – discuss] way to classify algorithms is by computing power. This is typically done by considering some collection (class) of algorithms. A recursive class of algorithms is one that includes algorithms for all Turing computable functions. Looking at classes of algorithms allows for the possibility of restricting the available computational resources (time and memory) used in a computation. A subrecursive class of algorithms is one in which not all Turing computable functions can be obtained. For example, the algorithms that run in polynomial time suffice for many important types of computation but do not exhaust all Turing computable functions. The class of algorithms implemented by primitive recursive functions is another subrecursive class.

Burgin (2005, p. 24) uses a generalized definition of algorithms that relaxes the common requirement that the output of the algorithm that computes a function must be determined after a finite number of steps. He defines a super-recursive class of algorithms as "a class of algorithms in which it is possible to compute functions not computable by any Turing machine" (Burgin 2005, p. 107). This is closely related to the study of methods of hypercomputation.

Legal issues

See also: Software patents for a general overview of the patentability of software, including computer-implemented algorithms.

Algorithms, by themselves, are not usually patentable. In the United States, a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals does not constitute "processes" (USPTO 2006), and hence algorithms are not patentable (as in Gottschalk v. Benson). However, practical applications of algorithms are sometimes patentable. For example, in Diamond v. Diehr, the application of a simple feedback algorithm to aid in the curing of synthetic rubber was deemed patentable. The patenting of software is highly controversial, and there are highly criticized patents involving algorithms, especially data compression algorithms, such as Unisys' LZW patent.

Additionally, some cryptographic algorithms have export restrictions (see export of cryptography).

History: Development of the notion of "algorithm"

Discrete and distinguishable symbols

Tally-marks: To keep track of their flocks, their sacks of grain and their money the ancients used tallying: accumulating stones or marks scratched on sticks, or making discrete symbols in clay. Through the Babylonian and Egyptian use of marks and symbols, eventually Roman numerals and the abacus evolved (Dilson, p. 16–41). Tally marks appear prominently in unary numeral system arithmetic used in Turing machine and Post–Turing machine computations.

Manipulation of symbols as "place holders" for numbers: algebra

The work of the ancient Greek geometers (Euclidean algorithm), Persian mathematician Al-Khwarizmi (from whose name the terms "algorism" and "algorithm" are derived), and Western European mathematicians culminated in Leibniz's notion of the calculus ratiocinator (ca 1680):

A good century and a half ahead of his time, Leibniz proposed an algebra of logic, an algebra that would specify the rules for manipulating logical concepts in the manner that ordinary algebra specifies the rules for manipulating numbers.[19]

Mechanical contrivances with discrete states

The clock: Bolter credits the invention of the weight-driven clock as "The key invention [of Europe in the Middle Ages]", in particular the verge escapement[20] that provides us with the tick and tock of a mechanical clock. "The accurate automatic machine"[21] led immediately to "mechanical automata" beginning in the thirteenth century and finally to "computational machines" – the difference engine and analytical engines of Charles Babbage and Countess Ada Lovelace.[22]

Logical machines 1870 – Stanley Jevons' "logical abacus" and "logical machine": The technical problem was to reduce Boolean equations when presented in a form similar to what are now known as Karnaugh maps. Jevons (1880) describes first a simple "abacus" of "slips of wood furnished with pins, contrived so that any part or class of the [logical] combinations can be picked out mechanically . . . More recently however I have reduced the system to a completely mechanical form, and have thus embodied the whole of the indirect process of inference in what may be called a Logical Machine" His machine came equipped with "certain moveable wooden rods" and "at the foot are 21 keys like those of a piano [etc] . . .". With this machine he could analyze a "syllogism or any other simple logical argument".[23]

This machine he displayed in 1870 before the Fellows of the Royal Society.[24] Another logician John Venn, however, in his 1881 Symbolic Logic, turned a jaundiced eye to this effort: "I have no high estimate myself of the interest or importance of what are sometimes called logical machines ... it does not seem to me that any contrivances at present known or likely to be discovered really deserve the name of logical machines"; see more at Algorithm characterizations. But not to be outdone he too presented "a plan somewhat analogous, I apprehend, to Prof. Jevon's abacus ... [And] [a]gain, corresponding to Prof. Jevons's logical machine, the following contrivance may be described. I prefer to call it merely a logical-diagram machine ... but I suppose that it could do very completely all that can be rationally expected of any logical machine".[25]

Jacquard loom, Hollerith punch cards, telegraphy and telephony—the electromechanical relay: Bell and Newell (1971) indicate that the Jacquard loom (1801), precursor to Hollerith cards (punch cards, 1887), and "telephone switching technologies" were the roots of a tree leading to the development of the first computers.[26] By the mid-1800s the telegraph, the precursor of the telephone, was in use throughout the world, its discrete and distinguishable encoding of letters as "dots and dashes" a common sound. By the late 1800s the ticker tape (ca 1870s) was in use, as was the use of Hollerith cards in the 1890 U.S. census. Then came the Teletype (ca. 1910) with its punched-paper use of Baudot code on tape.

Telephone-switching networks of electromechanical relays (invented 1835) was behind the work of George Stibitz (1937), the inventor of the digital adding device. As he worked in Bell Laboratories, he observed the "burdensome' use of mechanical calculators with gears. "He went home one evening in 1937 intending to test his idea... When the tinkering was over, Stibitz had constructed a binary adding device".[27]

Davis (2000) observes the particular importance of the electromechanical relay (with its two "binary states" open and closed):

It was only with the development, beginning in the 1930s, of electromechanical calculators using electrical relays, that machines were built having the scope Babbage had envisioned."[28]

Mathematics during the 1800s up to the mid-1900s

Symbols and rules: In rapid succession the mathematics of George Boole (1847, 1854), Gottlob Frege (1879), and Giuseppe Peano (1888–1889) reduced arithmetic to a sequence of symbols manipulated by rules. Peano's The principles of arithmetic, presented by a new method (1888) was "the first attempt at an axiomatization of mathematics in a symbolic language".[29]

But Heijenoort gives Frege (1879) this kudos: Frege's is "perhaps the most important single work ever written in logic. ... in which we see a " 'formula language', that is a lingua characterica, a language written with special symbols, "for pure thought", that is, free from rhetorical embellishments ... constructed from specific symbols that are manipulated according to definite rules".[30] The work of Frege was further simplified and amplified by Alfred North Whitehead and Bertrand Russell in their Principia Mathematica (1910–1913).

The paradoxes: At the same time a number of disturbing paradoxes appeared in the literature, in particular the Burali-Forti paradox (1897), the Russell paradox (1902–03), and the Richard Paradox.[31] The resultant considerations led to Kurt Gödel's paper (1931) — he specifically cites the paradox of the liar—that completely reduces rules of recursion to numbers.

Effective calculability: In an effort to solve the Entscheidungsproblem defined precisely by Hilbert in 1928, mathematicians first set about to define what was meant by an "effective method" or "effective calculation" or "effective calculability" (i.e., a calculation that would succeed). In rapid succession the following appeared: Alonzo Church, Stephen Kleene and J.B. Rosser's λ-calculus[32] a finely honed definition of "general recursion" from the work of Gödel acting on suggestions of Jacques Herbrand (cf. Gödel's Princeton lectures of 1934) and subsequent simplifications by Kleene.[33] Church's proof[34] that the Entscheidungsproblem was unsolvable, Emil Post's definition of effective calculability as a worker mindlessly following a list of instructions to move left or right through a sequence of rooms and while there either mark or erase a paper or observe the paper and make a yes-no decision about the next instruction.[35] Alan Turing's proof of that the Entscheidungsproblem was unsolvable by use of his "a- [automatic-] machine"[36] – in effect almost identical to Post's "formulation", J. Barkley Rosser's definition of "effective method" in terms of "a machine".[37] S. C. Kleene's proposal of a precursor to "Church thesis" that he called "Thesis I",[38] and a few years later Kleene's renaming his Thesis "Church's Thesis"[39] and proposing "Turing's Thesis".[40]

Emil Post (1936) and Alan Turing (1936–7, 1939)

Here is a remarkable coincidence of two men not knowing each other but describing a process of men-as-computers working on computations—and they yield virtually identical definitions.

Emil Post (1936) described the actions of a "computer" (human being) as follows:

"...two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions.

His symbol space would be

"a two way infinite sequence of spaces or boxes... The problem solver or worker is to move and work in this symbol space, being capable of being in, and operating in but one box at a time.... a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke.
"One box is to be singled out and called the starting point. ...a specific problem is to be given in symbolic form by a finite number of boxes [i.e., INPUT] being marked with a stroke. Likewise the answer [i.e., OUTPUT] is to be given in symbolic form by such a configuration of marked boxes....
"A set of directions applicable to a general problem sets up a deterministic process when applied to each specific problem. This process will terminate only when it comes to the direction of type (C ) [i.e., STOP]".[41] See more at Post–Turing machine

Alan Turing's work[42] preceded that of Stibitz (1937); it is unknown whether Stibitz knew of the work of Turing. Turing's biographer believed that Turing's use of a typewriter-like model derived from a youthful interest: "Alan had dreamt of inventing typewriters as a boy; Mrs. Turing had a typewriter; and he could well have begun by asking himself what was meant by calling a typewriter 'mechanical'".[43] Given the prevalence of Morse code and telegraphy, ticker tape machines, and Teletypes we might conjecture that all were influences.

Turing—his model of computation is now called a Turing machine — begins, as did Post, with an analysis of a human computer that he whittles down to a simple set of basic motions and "states of mind". But he continues a step further and creates a machine as a model of computation of numbers.[44]

"Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book....I assume then that the computation is carried out on one-dimensional paper, i.e., on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite....
"The behavior of the computer at any moment is determined by the symbols which he is observing, and his "state of mind" at that moment. We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite...
"Let us imagine that the operations performed by the computer to be split up into 'simple operations' which are so elementary that it is not easy to imagine them further divided".[45]

Turing's reduction yields the following:

"The simple operations must therefore include:
"(a) Changes of the symbol on one of the observed squares
"(b) Changes of one of the squares observed to another square within L squares of one of the previously observed squares.

"It may be that some of these change necessarily invoke a change of state of mind. The most general single operation must therefore be taken to be one of the following:

"(A) A possible change (a) of symbol together with a possible change of state of mind.
"(B) A possible change (b) of observed squares, together with a possible change of state of mind"
"We may now construct a machine to do the work of this computer"[45].

A few years later, Turing expanded his analysis (thesis, definition) with this forceful expression of it:

"A function is said to be "effectively calculable" if its values can be found by some purely mechanical process. Although it is fairly easy to get an intuitive grasp of this idea, it is nevertheless desirable to have some more definite, mathematical expressible definition . . . [he discusses the history of the definition pretty much as presented above with respect to Gödel, Herbrand, Kleene, Church, Turing and Post] . . . We may take this statement literally, understanding by a purely mechanical process one which could be carried out by a machine. It is possible to give a mathematical description, in a certain normal form, of the structures of these machines. The development of these ideas leads to the author's definition of a computable function, and to an identification of computability † with effective calculability . . . .
"† We shall use the expression "computable function" to mean a function calculable by a machine, and we let "effectively calculable" refer to the intuitive idea without particular identification with any one of these definitions".[46]

J. B. Rosser (1939) and S. C. Kleene (1943)

J. Barkley Rosser boldly defined an 'effective [mathematical] method' in the following manner (boldface added):

"'Effective method' is used here in the rather special sense of a method each step of which is precisely determined and which is certain to produce the answer in a finite number of steps. With this special meaning, three different precise definitions have been given to date. [his footnote #5; see discussion immediately below]. The simplest of these to state (due to Post and Turing) says essentially that an effective method of solving certain sets of problems exists if one can build a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer. All three definitions are equivalent, so it doesn't matter which one is used. Moreover, the fact that all three are equivalent is a very strong argument for the correctness of any one." (Rosser 1939:225–6)

Rosser's footnote #5 references the work of (1) Church and Kleene and their definition of λ-definability, in particular Church's use of it in his An Unsolvable Problem of Elementary Number Theory (1936); (2) Herbrand and Gödel and their use of recursion in particular Gödel's use in his famous paper On Formally Undecidable Propositions of Principia Mathematica and Related Systems I (1931); and (3) Post (1936) and Turing (1936–7) in their mechanism-models of computation.

Stephen C. Kleene defined as his now-famous "Thesis I" known as the Church–Turing thesis. But he did this in the following context (boldface in original):

"12. Algorithmic theories... In setting up a complete algorithmic theory, what we do is to describe a procedure, performable for each set of values of the independent variables, which procedure necessarily terminates and in such manner that from the outcome we can read a definite answer, "yes" or "no," to the question, "is the predicate value true?"" (Kleene 1943:273)

History after 1950

A number of efforts have been directed toward further refinement of the definition of "algorithm", and activity is on-going because of issues surrounding, in particular, foundations of mathematics (especially the Church–Turing Thesis) and philosophy of mind (especially arguments around artificial intelligence). For more, see Algorithm characterizations.

See also

Wikibooks has a book on the topic of Algorithms
At Wikiversity you can learn more and teach others about Algorithm at: The Department of Algorithm

Notes

  1. ^ Kleene 1943 in Davis 1965:274
  2. ^ Rosser 1939 in Davis 1965:225
  3. ^ Adaptation and learning in automatic systems, page 54, Ya. Z. Tsypkin, Z. J. Nikolic, Academic Press, 1971, ISBN 978-0-12-702050-1
  4. ^ Boolos and Jeffrey 1974,1999:19
  5. ^ a b Minsky 1967:105
  6. ^ Gurevich 2000:1, 3
  7. ^ Kleene 1952:136
  8. ^ Knuth 1997:5
  9. ^ Boldface added, Kleene 1952:137
  10. ^ Kleene 1952:325
  11. ^ Davis 1958:12–15
  12. ^ Kleene 1952:332
  13. ^ Minsky 1967:186
  14. ^ Sipser 2006:157
  15. ^ Kleene 1952:229 shows that "Definition by cases" is primitive recursive. CASES requires that the list of testable instances within the CASE definition to be (i) mutually exclusive and (ii) collectively exhaustive i.e. it must include or "cover" all possibility. The CASE statement proceeds in numerical order and exits at the first successful test; see more at Boolos–Burgess–Jeffrey Fourth edition 2002:74
  16. ^ An IF-THEN-ELSE or "logical test with branching" is just a CASE instruction reduced to two outcomes: (i) test is successful, (ii) test is unsuccessful. The IF-THEN-ELSE is closely related to the AND-OR-INVERT logic function from which all 16 logical "operators" of one or two variables can be derived; see more at Propositional formula. Like definition by cases, a sequence of IF-THEN-ELSE logical tests must be mutually exclusive and collectively exhaustive over the variables tested.
  17. ^ Kowalski 1979
  18. ^ Sue Carroll, Taz Daughtrey (2007-07-04). Fundamental Concepts for the Software Quality Engineer. pp. 282 et seq.. ISBN 9780873897204. http://books.google.com/?id=bz_cl3B05IcC&pg=PA282.
  19. ^ Davis 2000:18
  20. ^ Bolter 1984:24
  21. ^ Bolter 1984:26
  22. ^ Bolter 1984:33–34, 204–206)
  23. ^ All quotes from W. Stanley Jevons 1880 Elementary Lessons in Logic: Deductive and Inductive, Macmillan and Co., London and New York. Republished as a googlebook; cf Jevons 1880:199–201. Louis Couturat 1914 the Algebra of Logic, The Open Court Publishing Company, Chicago and London. Republished as a googlebook; cf Couturat 1914:75–76 gives a few more details; interestingly he compares this to a typewriter as well as a piano. Jevons states that the account is to be found at Jan . 20, 1870 The Proceedings of the Royal Society.
  24. ^ Jevons 1880:199–200
  25. ^ All quotes from John Venn 1881 Symbolic Logic, Macmillan and Co., London. Republished as a googlebook. cf Venn 1881:120–125. The interested reader can find a deeper explanation in those pages.
  26. ^ Bell and Newell diagram 1971:39, cf. Davis 2000
  27. ^ * Melina Hill, Valley News Correspondent, A Tinkerer Gets a Place in History, Valley News West Lebanon NH, Thursday March 31, 1983, page 13.
  28. ^ Davis 2000:14
  29. ^ van Heijenoort 1967:81ff
  30. ^ van Heijenoort's commentary on Frege's Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought in van Heijenoort 1967:1
  31. ^ Dixon 1906, cf. Kleene 1952:36–40
  32. ^ cf. footnote in Alonzo Church 1936a in Davis 1965:90 and 1936b in Davis 1965:110
  33. ^ Kleene 1935–6 in Davis 1965:237ff, Kleene 1943 in Davis 1965:255ff
  34. ^ Church 1936 in Davis 1965:88ff
  35. ^ cf. "Formulation I", Post 1936 in Davis 1965:289–290
  36. ^ Turing 1936–7 in Davis 1965:116ff
  37. ^ Rosser 1939 in Davis 1965:226
  38. ^ Kleene 1943 in Davis 1965:273–274
  39. ^ Kleene 1952:300, 317
  40. ^ Kleene 1952:376
  41. ^ Turing 1936–7 in Davis 1965:289–290
  42. ^ Turing 1936 in Davis 1965, Turing 1939 in Davis 1965:160
  43. ^ Hodges, p. 96
  44. ^ Turing 1936–7:116)
  45. ^ a b Turing 1936–7 in Davis 1965:136
  46. ^ Turing 1939 in Davis 1965:160

References

Secondary references

Further reading

External links

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