Algorithm 2

• Serial or parallel or distributed: Algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at a time. Those computers are sometimes called serial computers. An algorithm designed for such an environment is called a serial algorithm, as opposed to parallel algorithms or distributed algorithms. Parallel algorithms take advantage of computer architectures where several processors can work on a problem at the same time, whereas distributed algorithms utilize multiple machines connected with a network. Parallel or distributed algorithms divide the problem into more symmetrical or asymmetrical subproblems and collect the results back together. The resource consumption in such algorithms is not only processor cycles on each processor but also the communication overhead between the processors. Sorting algorithms can be parallelized efficiently, but their communication overhead is expensive. Iterative algorithms are generally parallelizable. Some problems have no parallel algorithms, and are called inherently serial problems.

• Deterministic or non-deterministic: Deterministic algorithms solve the problem with exact decision at every step of the algorithm whereas non-deterministic algorithms solve problems via guessing although typical guesses are made more accurate through the use of heuristics.

• Exact or approximate: While many algorithms reach an exact solution, approximation algorithms seek an approximation that is close to the true solution. Approximation may use either a deterministic or a random strategy. Such algorithms have practical value for many hard problems.

• Quantum algorithm: Quantum algorithm run on a realistic model of quantum computation. The term is usually used for those algorithms which seem inherently quantum, or use some essential feature of quantum computation such as quantum superposition or quantum entanglement. 

Another 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:

• Brute-force or exhaustive search. This is the naïve method of trying every possible solution to see which is best.

• Divide and conquer. A divide and conquer algorithm repeatedly reduces an instance of a problem to one or more smaller instances of the same problem (usually recursively) until the instances are small enough to solve easily. One such example of divide and conquer is merge sorting. Sorting can be done on each segment of data after dividing data into segments and sorting of entire data can be obtained in the conquer phase by merging the segments. A simpler variant of divide and conquer is called a decrease and conquer algorithm, that solves an identical subproblem and uses the solution of this subproblem to solve the bigger problem. Divide and conquer divides the problem into multiple subproblems and so the conquer stage will be more complex than decrease and conquer algorithms. An example of decrease and conquer algorithm is the binary search algorithm.

• Dynamic programming. When a problem shows optimal substructure, meaning the optimal solution to a problem can be constructed from optimal solutions to subproblems, and overlapping subproblems, meaning the same subproblems are used to solve many different problem instances, a quicker approach called dynamic programming avoids recomputing solutions that have already been computed. For example, Floyd–Warshall algorithm, the shortest path to a goal from a vertex in a weighted graph can be found by using the shortest path to the goal from all adjacent vertices. Dynamic programming and memoization go together. The main difference between dynamic programming and divide and conquer is that subproblems are more or less independent in divide and conquer, whereas subproblems overlap in dynamic programming. The difference between dynamic programming and straightforward recursion is in caching or memoization of recursive calls. When subproblems are independent and there is no repetition, memoization does not help; hence dynamic programming is not a solution for all complex problems. By using memoization or maintaining atable of subproblems already solved, dynamic programming reduces the exponential nature of many problems to polynomial complexity.

• The greedy method. A greedy algorithm is similar to a dynamic programming algorithm, but the difference is that solutions to the subproblems do not have to be known at each stage; instead a "greedy" choice can be made of what looks best for the moment. The greedy method extends the solution with the best possible decision (not all feasible decisions) at an algorithmic stage based on the current local optimum and the best decision (not all possible decisions) made in a previous stage. It is not exhaustive, and does not give an accurate answer to many problems. But when it works, it will be the fastest method. The most popular greedy algorithm is finding the minimal spanning tree as given by Huffman Tree, Kruskal, Prim, Sollin.

• Linear programming. When solving a problem using linear programming, specific inequalities involving the inputs are found and then an attempt is made to maximize (or minimize) some linear function of the inputs. Many problems (such as the maximum flow for directed graphs) can be stated in a linear programming way, and then be solved by a 'generic' algorithm such as the simplex algorithm. A more complex variant of linear programming is called integer programming, where the solution space is restricted to the integers.

• Reduction. This technique involves solving a difficult problem by transforming it into a better known problem for which we have (hopefully) asymptotically optimal algorithms. The goal is to find a reducing algorithm whose complexity is not dominated by the resulting reduced algorithm's. For example, one selection algorithm for finding the median in an unsorted list involves first sorting the list (the expensive portion) and then pulling out the middle element in the sorted list (the cheap portion). This technique is also known as transform and conquer.

• Search and enumeration. Many problems (such as playing chess) can be modeled as problems on graphs. A graph exploration algorithm specifies rules for moving around a graph and is useful for such problems. This category also includes search algorithms, branch and bound enumeration and backtracking.

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 evolutionaryprocesses, 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.

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, medical 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.

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.

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.

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; or

2. An algorithm in the form of a differential equation that operates continuously on the data, running on an analog computer.

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).

The word "Algorithm" or "Algorism" in some other writing versions, comes from the name Al-Khwārizmī (c. 780-850), a Persian mathematician, astronomer, geographer and a scholar in the House of Wisdom in Baghdad, whose name means "the native of Khwarezm", a city that was part of the Greater Iran during his era and now is in modern day Uzbekistan He wrote a treatise in the Arabic language during the 9th century, which was translated into Latin in the 12th century under the title Algoritmi de numero Indorum. This title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name.[58] Al-Khwarizmi was the most widely read mathematician in Europe in the late Middle Ages, primarily through his other book, theAlgebra. In late medieval Latin, algorismus, the corruption of his name, simply meant the "decimal number system" that is still the meaning of modern English algorism. In 17th century French the word's form, but not its meaning, changed to algorithme. English adopted the French very soon afterwards, but it wasn't until the late 19th century that "Algorithm" took on the meaning that it has in modern English.

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 systemarithmetic used in Turing machine and Post–Turing machine computations.

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.

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 that provides us with the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical automata" beginning in the 13th century and finally to "computational machines"—the difference engine andanalytical engines of Charles Babbage and Countess Ada Lovelace.

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 aLogical 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".

This machine he displayed in 1870 before the Fellows of the Royal Society. 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".

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.[68] By the mid-19th century 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 19th century the ticker tape (ca 1870s) was in use, as was the use of Hollerith cards in the 1890 U.S. census. Then came the teleprinter (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".

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."

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".

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". 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. 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 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. Church's proof 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. Alan Turing's proof of that the Entscheidungsproblem was unsolvable by use of his "a- [automatic-] machine"—in effect almost identical to Post's "formulation", J. Barkley Rosser's definition of "effective method" in terms of "a machine". S. C. Kleene's proposal of a precursor to "Church thesis" that he called "Thesis I", and a few years later Kleene's renaming his Thesis "Church's Thesis"[81] and proposing "Turing's Thesis".

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]". See more at Post–Turing machine

Alan Turing's work 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'". Given the prevalence of Morse code and telegraphy, ticker tape machines, and teletypewriters 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.

"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".

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.

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