An algorithm is a finite set of well-defined instructions for accomplishing some task which, given an initialstate, will result in a corresponding recognisable end-state (contrast with heuristic ).
The concept of an algorithm is often illustrated by the example of a recipe ,although many algorithms are much more complex; algorithms often have steps that repeat ( iterate ) or require decisions (such as logic or comparison ) until the task is completed. Correctly performing an algorithm will notsolve a problem if the algorithm is flawed or not appropriate to the problem. For example, performing the potato salad algorithmwill fail if there are no potatoes present, even if all the motions of preparing the salad are performed as if the potatoes werethere.
Different algorithms may complete the same task with a different set of instructions in more or less time, space, or effortthan others. For example, given two different recipes for making potato salad, one may have peel the potato beforeboil the potato while the other presents the steps in the reverse order, yet they both call for these steps to berepeated for all potatoes and end when the potato salad is ready to be eaten.
In some countries, such as the USA, some algorithms can effectively be patented if aphysical embodiment is possible (for example, a multiplication algorithm may be embodied in the arithmetic unit of amicroprocessor).
Algorithms are essential to the way computers process information, because a computer program is essentially an algorithm that tells thecomputer what specific steps to perform (in what specific order) in order to carry out a specified task, such as calculatingemployees’ paychecks or printing students’ report cards. Thus, an algorithm can be considered to be any sequence ofoperations 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, writtento an output sink or device, and/or stored for further use. Stored data is regarded as part of the internal state of the entityperforming the algorithm.
For any such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possiblecircumstances that could arise. That is, any conditional steps must be systematically dealt with, case-by-case; the criteria foreach 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 thefunctioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting 'from thetop' and going 'down to the bottom', an idea that is described more formally by flow of control .
So far, this discussion of the formalisation of an algorithm has assumed the premises of imperative programming . This is the most common conception, andit 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. Itderives from the intuition of 'memory' as a scratchpad. There is an example below of such an assignment.
Algorithms are not only implemented as computer programs , butoften also 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 onediscipline of computer science , and is often practiced abstractly(without the use of a specific programming language orother implementation). In this sense, it resembles other mathematical disciplines in that the analysis focuses on the underlyingprinciples of the algorithm, and not on any particular implementation. One way to embody (or sometimes codify) analgorithm is the writing of pseudocode .
Some writers restrict the definition of algorithm to procedures that eventually finish. Others include proceduresthat could run forever without stopping, arguing that some entity may be required to carry out such permanent tasks. In thelatter case, success can no longer be defined in terms of halting with a meaningful output. Instead, terms of success that allowfor unbounded output sequences must be defined. For example, an algorithm that verifies if there are more zeros than ones in aninfinite random binary sequence must run forever to be effective. If it is implemented correctly, however, the algorithm's outputwill be useful: for as long as it examines the sequence, the algorithm will give a positive response while the number of examinedzeros outnumber the ones, and a negative response otherwise. Success for this algorithm could then be defined as eventuallyoutputting only positive responses if there are actually more zeros than ones in the sequence, and in any other case outputtingany mixture of positive and negative responses.
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 thinkingabout the solution, you will realise that you must look at every number in the list. Upon further thinking, you will realise thatyou need to look at each number only once. Taking this into account, here is a simple algorithm to accomplish this:
Given: a list "List" largest = List counter = 2 while counter <= length(List): if List[counter] > largest: largest = List[counter] counter = counter + 1 print largest
Notes on notation:
Note also the algorithm assumes that the list contains at least one number. It will fail when presented an empty list. Mostalgorithms have similar assumptions on their inputs, called pre-conditions .
As it happens, most people who implement algorithms want to know how much of a particular resource (such as time or storage) agiven algorithm requires. Methods have been developed for the analysis of algorithms to obtain such quantitative answers; for example, the algorithm above has atime requirement of O(n), using the big O notation withn representing for the length of the list.
The word algorithm comes ultimately from the name of the 9th-century mathematician Abu Abdullah Muhammadbin Musa al-Khwarizmi . The word algorism originally referred only tothe 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 performingtasks.
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 nevercompleted his analytical engine the algorithm was neverimplemented on it.
The lack of mathematical rigor in the "well-definedprocedure" 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 thatevery method yet found for describing "well-defined procedures" advanced by other mathematicians could be emulated on a Turingmachine (a statement known as the Church-Turingthesis ).
Nowadays, a formal criterion for an algorithm is that it is a procedure that can be implemented on a completely-specifiedTuring machine or one of the equivalent formalisms . Turing's initial interest wasin the halting problem : deciding when an algorithm describes aterminating procedure. In practical terms computational complexity theory matters more: it includes the puzzling problem of thealgorithms called NP-complete , which are generally presumed to take more thanpolynomial 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, eachdifferent from the other. Furthermore, each of these categories will include many different types of algorithms. Some commonlyfound paradigms include:
Another way to classify algorithms is by implementation. A recursive algorithm is one that invokes (makes reference to) itself repeatedly until a certaincondition matches, which is a method common to functionalprogramming . Algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at atime. Those computers are sometimes called serial computers. An algorithm designed for such an environment is called a serialalgorithm, as opposed to parallel algorithms , which takeadvantage of computer architectures where several processors can work on a problem at the same time. The various heuristicalgorithm would probably also fall into this category, as their name (e.g. a genetic algorithm) describes its implementation.
A list of algorithms discussed in Wikipedia isavailable.
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