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# Algorithm

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An algorithm, broadly defined, is an understandable and finite set of instructions for accomplishing some task which, given a defined set of inputs, will result in some recognisable end-state (contrast with heuristic). Algorithms often have steps that repeat (iterate) or require decisions (such as logic or comparison) until the task is completed.

Different algorithms may complete the same task with a different set of instructions in more or less time, space, or effort than others. A cooking recipe is an example of an algorithm. Given two different recipes for making potato salad, one may have peel the potato before boil the potato while the other presents the steps in the reverse order, yet they both call for these steps to be repeated for all potatoes and end when the potato salad is ready to be eaten.

Correctly performing an algorithm will not solve a problem if the algorithm is flawed or not appropriate to the problem. For example, performing the potato salad algorithm will fail if there are no potatoes present, even if all the motions of preparing the salad are performed as if the potatoes were there.

 Table of contents 1 Formalized algorithms 2 Implementing algorithms 3 Example 4 History 5 Classes of algorithms 6 See also 7 References

## Formalized algorithms

Algorithms are essential to the way computers process information, because a computer program is essentially an algorithm that tells the computer what specific steps to perform (in what specific order) in 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 which can be performed by a Turing-complete system.

Typically, when an algorithm is associated with processing information, data is read from an input source or device, written to an output sink or device, and/or stored for further use. Stored data is regarded as part of the internal state of the entity performing the algorithm.

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 almost 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.

So far, this discussion of the formalization of an algorithm has assumed the premises of imperative programming. 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, setting the value of a variable. It derives from the intuition of 'memor' as a scratchpad. There is an example below of such an assignment.

See functional programming and logic programming for alternate conceptions of what constitutes an algorithm.

## Implementing algorithms

An algorithm is a method or procedure for carrying out a task (such as solving a problem in mathematics, finding the freshest produce in a supermarket, or manipulating information in general).

Algorithms are sometimes implemented as computer programs but are more often implemented by other means, such as in a biological neural network (for example, the human brain implementing arithmetic or an insect relocating food), or in electric circuits or in a mechanical device.

The analysis and study of algorithms is one discipline of computer science, and is often practiced abstractly (without the use of a specific programming language or other implementation). In this sense, it resembles other mathematical disciplines in that the analysis focuses on the underlying principles of the algorithm, and not on any particular implementation. One way to embody (or sometimes codify) an algorithm is the writing of pseudocode.

Some writers restrict the definition of algorithm to procedures that eventually finish. Others include procedures that could run forever without stopping, arguing that some entity may be required to carry out such permanent tasks. In the latter case, it can be difficult to determine whether a given algorithm successfully completes its tasks.

## Example

Here is a simple example of an algorithm.

Imagine you have an unsorted list of random numbers. Our goal is to find the highest number in this list. Upon first thinking about the solution, you will realize that you must look at every number in the list. Upon further thinking, you will realize that you need to look at each number only once. Taking this into account, here is a simple algorithm to accomplish this:

And here is a more formal coding of the algorithm in a pseudocode that is similar to most programming languages:
```Given: a list "List"

counter = 1
largest = List[counter]
while counter <= length(List):
if List[counter] > largest:
largest = List[counter]
counter = counter + 1
print largest

```
Notes on notation:
• = as used here indicates assignment. That is, the value on the right-hand side of the expression is assigned to the container (or variable) on the left-hand side of the expression.
• List[counter] as used here indicates the counterth element of the list. For example: if the value of counter is 5, then List[counter] refers to the 5th element of the list.
• <= as used here indicates 'less than or equal to'

As it happens, most people who implement algorithms want to know how much of a particular resource (such as time or storage) a given algorithm requires. Methods have been developed for the analysis of algorithms to obtain such quantitative answers; for example, the algorithm above has a time requirement of O(n), using the big O notation with n representing for the length of the list.

## History

The word algorithm is a corruption of early English algorisme, which came from Latin algorismus, which came from the name of the Persian mathematician Abu Ja'far Mohammed ibn Musa al-Khwarizmi (ca. 780 - ca. 845). He was the author of the book Kitab al-jabr w'al-muqabala (Rules of Restoration and Reduction) which introduced algebra to people in the West. The word algebra itself originates from al-Jabr in the title of the book. The word algorism originally referred only to the rules of performing arithmetic using Arabic numerals but evolved into algorithm by the 18th century. The word has now evolved to include all definite procedures for solving problems or performing tasks.

The first case of an algorithm written for a computer was Ada Byron's notes on the analytical engine written in 1842, for which she is considered by many to be the world's first programmer. However, since Charles Babbage never completed his analytical engine the algorithm was never implemented on it.

The lack of mathematical rigor in the "well-defined procedure" definition of algorithms posed some difficulties for mathematicians and logicians of the 19th and early 20th centuries. This problem was largely solved with the description of the Turing machine, an abstract model of a computer formulated by Alan Turing, and the demonstration that every method yet found for describing "well-defined procedures" advanced by other mathematicians could be emulated on a Turing machine (a statement known as the Church-Turing thesis).

Nowadays, a formal criterion for an algorithm is that it is a procedure implementable on a completely-specified Turing machine or one of the equivalent formalisms. Turing's initial interest was in the halting problem: deciding when an algorithm describes a terminating procedure. In practical terms computational complexity theory matters more: it includes the puzzling problem of the algorithms called NP-complete, which are generally presumed to take more than polynomial time.

## Classes of algorithms

There are many ways to classify algorithms, and the merits of each classification have been the subject of ongoing debate.

One 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 algorithm. Some commonly found paradigms include:

• The greedy method. A greedy algorithm works by making a series of simple decisions that are never reconsidered.
• Divide and conquer. A divide-and-conquer algorithm 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 be directly expressible in the programming language employed (what is 'direct' is often discretionary).
• Dynamic programming. A dynamic programming algorithm works bottom-up by building progressively larger solutions to subproblems arising from the original problem, and then uses those solutions to obtain the final result.
• 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 the search algorithms and backtracking.
• The probabilistic and heuristic paradigm. Algorithms belonging to this class fit the definition of an algorithm more loosely. Probabilistic algorithms are those that make some choices randomly (or pseudo-randomly). 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. Also there are heuristic algorithms, whose general purpose is not to find a final solution, but an approximate solution where the time or resources to find a perfect solution are not practical. An example of this would be simulated annealing algorithms, a class of heuristic probabilistic algorithms that vary the solution of a problem a by 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.

Another way to classify algorithms is by implementation. A recursive algorithm is one that invokes (makes reference to) itself repeatedly until a certain condition matches, which is a method common to functional programming. Algorithms are usually discussed with the assumption that computers execute each 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, which take advantage of computer architectures where several processors can work on a problem at the same time. The various heuristic algorithm would probably also fall into this category, as their name (eg. a genetic algorithm) describes its implementation.

A list of algorithms discussed in Wikipedia is available.

## References

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Stony Brook Algorithm Repository
This is a collection of implementations for 75 fundamental algorithms problems, including data structures, numerical and combinatorial algorithms,graph algorithms, and computational geometry. Implementations are available in C++, Java, Fortran, and other languages.
http://www.cs.sunysb.edu/~algorith/

The Grail Project
A symbolic computation environment for finite-state machines, regular expressions, and finite languages.
http://www.csd.uwo.ca/research/grail/

The Algorithm Base
Database of algorithms. May be scanned through or can be questioned through a knowledge based assistant. Links to originating web sites.
http://www.intelligenceunited.com/index2.html

Priority Queues
Electronic bibliography on priority queues (heaps). Links to downloadable reports, researchers' home pages, and software.
http://www.leekillough.com/heaps/

ACM SIGACT
Special Interest Group on Algorithms and Computation Theory, the ACM special interest group for Theoretical Computer Science. Site has membership information, meetings, reports and a newsletter for members.
http://sigact.acm.org/

Algorithms in the Real World
Notes for a course at Carnegie Mellon University.
http://www-2.cs.cmu.edu/~guyb/realworld.html

Design and Analysis of Computer Algorithms
Lecture notes; applets and code in C, C++, and Java; links regarding books, journals, computability, quantum computing, societies and organizations.
http://www.personal.kent.edu/~rmuhamma/Algorithms/algorithm.html

Fundamental Algorithms
Data structures and code for some important algorithms.
http://www.greatsnakes.com/savannah/algorithms.asp

Self-stabilizing Algorithms
A project to create tools for developing and testing self-stabilizing algorithms.
http://www.eti.pg.gda.pl/~kuszner/self-stab/

Combinatorial Algorithms
Lecture notes of a course at San Diego State University.
http://www.eli.sdsu.edu/courses/fall95/cs660/notes/

Computer Programming Algorithms Directory
Resources that describe computer programming algorithms.
http://www.algosort.com/

Algorithms, Calculations and Formulae
Resources on algorithms, calculations and formulae, including books, faq, urls, and a forum.
http://www.algorithms.org/

Combinatorial Algorithms
Course material, syllabus and notes for a course by Roger Whitney at SDSU.
http://www.eli.sdsu.edu/courses/fall95/cs660/

Data Structures And Number Systems
Web text by Brian Brown.
http://www.ibilce.unesp.br/courseware/datas/data1.htm

Data Structures and Algorithms
Course Notes, University of Western Australia
http://ciips.ee.uwa.edu.au/~morris/Year2/PLDS210/ds_ToC.html

Introduction to Quantum Algorithms
An introduction to quantum algorithms by Matthew Hayward for those new to the field and who do not have an extensive physics background.
http://www.imsa.edu/~matth/cs299/

Circular Queues
A brief discussion and implementation of circular queues in C.
http://www.cs.utk.edu/~calloway/prg/cq/

CATS: Combinatorial Algorithms Test Sets
Searchable index of problems, links and methodology.
http://www.jea.acm.org/CATS/

Tree Automata Techniques and Applications
An evolving web text in PostScript and PDF, with related software.
http://www.grappa.univ-lille3.fr/tata

Sourcebank - Computer Science - Algorithms
A collection of source code for various topics.
http://www.devx.com/sourcebank/directorybrowse.asp?adType=&dir_id=653&showResType=0&timeSelect=&showFilter=showAll

Algorithms Courses
Links to courses in algorithms maintained at various university computer science departments.
http://www.cs.pitt.edu/~kirk/algorithmcourses/

On the Road to Algorithms
Information on algorithms such as Bubble Sort and Random Number Generation, using HTML, Java and Perl. Collected by Lam Ka Chun (Raymond).
http://www.hlcmklam.com/

Analysis Of Algorithms
An initiative of attendees of the 1997 Dagstuhl seminar, these pages provide research papers, a bulletin board, and links to researchers and other resources in the field. The focus is on average case and probabilistic analysis.
http://pauillac.inria.fr/algo/AofA/

Web Data Structures and Algorithms
Lecture notes and links for a course by Godfried Toussaint.
http://cgm.cs.mcgill.ca/~godfried/teaching/algorithms-web.html

Problems in Analysis of Algorithms
A list of open problems with updates and solutions.
http://pauillac.inria.fr/algo/AofA/Problems/

Abstract State Machines
A formal method for specifying and verifying algorithms. Tools, meetings, researchers in the area.
http://www.eecs.umich.edu/gasm/

Algorithm Design Paradigms
A course by Paul Dunne at the University of Liverpool. Slides and notes in HTML and PS.
http://www.csc.liv.ac.uk/~ped/teachadmin/algor/algor.html

Data Structures
Lecture Notes by Steven Skiena.
http://www.cs.sunysb.edu/~skiena/214/lectures/

Data Structures
Introduction to data structures, with Java code, by Peter M. Williams.
http://www.cogs.susx.ac.uk/local/teach/dats/dats.html

Resources for the Analysis of Algorithms
Links to papers, conferences and other sites, maintained by Helmut Prodinger.
http://www.wits.ac.za/helmut/AofA/Resources/

Softpanorama Virtual Library
Section on Algorithms and Data Structures. A compilation of links.
http://www.softpanorama.org/Algorithms/algorithms.shtml

Hoshen-Kopelman Algorithm for Cluster Identification
An algorithm for identifying connected clusters on a lattice where sites may be occupied or non-occupied. With example C code.
http://splorg.org/~tobin/kb/hoshenkopelman.html

Dictionary of Algorithms, Data Structures, and Problems
A dictionary of algorithms, algorithmic techniques, data structures, and archetypical problems, with related definitions. Many entries have links to implementations, tutorials, and bibliographical references.
http://www.nist.gov/dads/

Pattern Matching Pointers
A collection of links for and to researchers in the subject.
http://www.cs.ucr.edu/~stelo/pattern.html

OOPWeb Algorithms Directory
Algorithms lecture notes, courses, tutorials, references, guides and online books.
http://www.oopweb.com/Algorithms/Files/Algorithms.html

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