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Algorithm

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:

1. Pretend the first number in the list is the largest number.
2. Look at the next number, and compare it with this largest number.
3. Only if this next number is larger, then keep that as the new largest number.
4. Repeat steps 2 and 3 until you have gone through the whole list.

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

• = 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 counter^th element of the list. For example: if the value of counter is 5, then List[counter] refers to the 5^th element of the list.
• <= as used here indicates 'less than or equal to'

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.

A list of algorithms discussed in Wikipedia is available.

See also

• Algorism
• Bulletproof algorithms
• Numerical analysis
• Cryptographic algorithms
• Sort algorithms
• Search algorithms
• Merge algorithms
• String algorithms
• List of algorithms
• Timeline of algorithms
• Data structure
• Genetic Algorithms
• Randomized algorithms
• Computability logic

• Donald E Knuth: The Art of Computer Programming, Vol 1-3, Addison Wesley 1998. Widely held as a definitive reference. ISBN 0201485419.
• Gaston H. Gonnet and Ricardo Baeza-Yates: Example programs from Handbook of Algorithms and Data Structures. Free source code for lots of important algorithms.
• Dictionary of Algorithms and Data Structures. "This is a dictionary of algorithms, algorithmic techniques, data structures, archetypical problems, and related definitions."
• Important algorithm-related publications

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