Algorithm

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In mathematics and computer science, an algorithm (/ˈælɡərɪðəm/ ^ⓘ) is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation.^[1] Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning), achieving automation eventually. Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus".^[2]

In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result.^[3] For example, social media recommender systems rely on heuristics in such a way that, although widely characterized as "algorithms" in 21st century popular media, cannot deliver correct results due to the nature of the problem.

As an

In the early 12th century, Latin translations of said al-Khwarizmi texts involving the Hindu–Arabic numeral system and arithmetic appeared: Liber Alghoarismi de practica arismetrice (attributed to John of Seville) and Liber Algorismi de numero Indorum (attributed to Adelard of Bath).^[2] Hereby, alghoarismi or algorismi is the Latinization of Al-Khwarizmi's name; the text starts with the phrase Dixit Algorismi ("Thus spoke Al-Khwarizmi").^[3]

Around 1230, the English word algorism is attested and then by Chaucer in 1391, English adopted the French term.^{[4][5][clarification needed]} In the 15th century, under the influence of the Greek word ἀριθμός (arithmos, "number"; cf. "arithmetic"), the Latin word was altered to algorithmus.^{[citation needed]}

Definition

One informal definition is "a set of rules that precisely defines a sequence of operations",

The concept of algorithm is also used to define the notion of decidability—a notion that is central for explaining how formal systems come into being starting from a small set of axioms and rules. In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related to the 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.

Most algorithms are intended to be

History

This section is missing information about 20th and 21st century development of computer algorithms. Please expand the section to include this information. Further details may exist on the talk page. (October 2023)

Ancient algorithms

Since antiquity, step-by-step procedures for solving mathematical problems have been attested. This includes

In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve the

of 1936–37 and 1939.

Representations

Algorithms can be expressed in many kinds of notation, including

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, drakon-charts and control tables are structured ways to express algorithms that avoid many of the ambiguities common in statements based on natural language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but they are also often used as a way to define or document algorithms.

Turing machines

There is a wide variety of representations possible and one can express a given

assembly code called "sets of quadruples" (see Turing machine for more). Representations of algorithms can also be classified into three accepted levels of Turing machine description: high level description, implementation description, and formal description.^[32] A high level description describes qualities of the algorithm itself, ignoring how it is implemented on the turing machine.^[32] An implementation description describes the general manner in which the turing machine moves its head and stores data in order to carry out the algorithm, but doesn't give exact states.^[32] In the most detail, a formal description gives the exact state table and list of transitions of the turing machine.^[32]

Flowchart representation

The graphical aid called a flowchart offers a way to describe and document an algorithm (and a computer program corresponding to it). Like the program flow of a Minsky machine, a flowchart always starts at the top of a page and proceeds down. Its primary symbols are only four: the directed arrow showing program flow, the rectangle (SEQUENCE, GOTO), the diamond (IF-THEN-ELSE), and the dot (OR-tie). The Böhm–Jacopini canonical structures are made of these primitive shapes. Sub-structures can "nest" in rectangles, but only if a single exit occurs from the superstructure. The symbols and their use to build the canonical structures are shown in the diagram.^[33]

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, an algorithm which adds up the elements of a list of n numbers would have a time requirement of ${\displaystyle O(n)}$, using big O notation. At all times the algorithm only needs to remember two values: the sum of all the elements so far, and its current position in the input list. Therefore, it is said to have a space requirement of ${\displaystyle O(1)}$, if the space required to store the input numbers is not counted, or ${\displaystyle O(n)}$ if it is counted.

Different algorithms may complete the same task with a different set of instructions in less or more time, space, or '

binary search

algorithm (with cost ${\displaystyle O(\log n)}$) outperforms a sequential search (cost ${\displaystyle O(n)}$ ) when used for table lookups on sorted lists or arrays.

Formal versus empirical

Main articles: Empirical algorithmics, Profiling (computer programming), and Program optimization

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. For the solution of a "one off" problem, the efficiency of a particular algorithm may not have significant consequences (unless n is extremely large) but for algorithms designed for fast interactive, commercial or long life scientific usage it may be critical. Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign.

Empirical testing is useful because it may uncover unexpected interactions that affect performance. Benchmarks may be used to compare before/after potential improvements to an algorithm after program optimization. Empirical tests cannot replace formal analysis, though, and are not trivial to perform in a fair manner.^[34]

Execution efficiency

Main article: Algorithmic efficiency

To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to FFT algorithms (used heavily in the field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging.^[35] In general, speed improvements depend on special properties of the problem, which are very common in practical applications.^[36] Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.

Design

See also: Algorithm § By design paradigm

Algorithm design refers to a method or a mathematical process for problem-solving and engineering algorithms. The design of algorithms is part of many solution theories, such as

operation research. Techniques for designing and implementing algorithm designs are also called algorithm design patterns,^[37] with examples including the template method pattern and the decorator pattern. One of the most important aspects of algorithm design is resource (run-time, memory usage) efficiency; the big O notation is used to describe e.g., an algorithm's run-time growth as the size of its input increases.^{[citation needed}

]

Structured programming

Per the

proofs of correctness using mathematical induction.^[41]

Classification

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

By implementation

One way to classify algorithms is by implementation means.

int gcd(int A, int B) { if (B == 0) return A; else if (A > B) return gcd(A-B,B); else return gcd(A,B-A); }

Recursive C implementation of Euclid's algorithm from the above flowchart

Recursion

A

towers of Hanoi

is well understood using recursive implementation. Every recursive version has an equivalent (but possibly more or less complex) iterative version, and vice versa.

Serial, 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 are algorithms that take advantage of computer architectures where multiple processors can work on a problem at the same time. Distributed algorithms are algorithms that use multiple machines connected with a computer network. Parallel and distributed algorithms divide the problem into more symmetrical or asymmetrical subproblems and collect the results back together. For example, a CPU would be an example of a parallel algorithm. The resource consumption in such algorithms is not only processor cycles on each processor but also the communication overhead between the processors. Some sorting algorithms can be parallelized efficiently, but their communication overhead is expensive. Iterative algorithms are generally parallelizable, but some problems have no parallel algorithms and are called inherently serial problems.

Deterministic or non-deterministic

heuristics

.

Exact or approximate

While many algorithms reach an exact solution, approximation algorithms seek an approximation that is closer to the true solution. The approximation can be reached by either using a deterministic or a random strategy. Such algorithms have practical value for many hard problems. One of the examples of an approximate algorithm is the Knapsack problem, where there is a set of given items. Its goal is to pack the knapsack to get the maximum total value. Each item has some weight and some value. Total weight that can be carried is no more than some fixed number X. So, the solution must consider weights of items as well as their value.^[42]

Quantum algorithm

quantum computation. The term is usually used for those algorithms which seem inherently quantum, or use some essential feature of Quantum computing such as quantum superposition or quantum entanglement

.

By design paradigm

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 includes many different types of algorithms. Some common paradigms are:

Brute-force or exhaustive search

Brute force is a method of problem-solving that involves systematically trying every possible option until the optimal solution is found. This approach can be very time consuming, as it requires going through every possible combination of variables. However, it is often used when other methods are not available or too complex. Brute force can be used to solve a variety of problems, including finding the shortest path between two points and cracking passwords.

Divide and conquer

A

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, which 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 is more complex than decrease and conquer algorithms. An example of a decrease and conquer algorithm is the binary search algorithm

.

Search and enumeration

Many problems (such as playing

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

.

Randomized algorithm

Such algorithms make some choices randomly (or pseudo-randomly). They can be very useful in finding approximate solutions for problems where finding exact solutions can be impractical (see heuristic method below). For some of these problems, it is known that the fastest approximations must involve some randomness.^[43] Whether randomized algorithms with polynomial time complexity can be the fastest algorithms for some problems is an open question known as the P versus NP problem. There are two large classes of such algorithms:

1. polynomial time
   .
2. ZPP
   .

Reduction of complexity

This technique involves solving a difficult problem by transforming it into a better-known problem for which we have (hopefully)

transform and conquer

.

Back tracking

In this approach, multiple solutions are built incrementally and abandoned when it is determined that they cannot lead to a valid full solution.

Optimization problems

For optimization problems there is a more specific classification of algorithms; an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following:

Linear programming

When searching for optimal solutions to a linear function bound to linear equality and inequality constraints, the constraints of the problem can be used directly in producing the optimal solutions. There are algorithms that can solve any problem in this category, such as the popular simplex algorithm.^[44] Problems that can be solved with linear programming include the maximum flow problem for directed graphs. If a problem additionally requires that one or more of the unknowns must be an integer then it is classified in integer programming. A linear programming algorithm can solve such a problem if it can be proved that all restrictions for integer values are superficial, i.e., the solutions satisfy these restrictions anyway. In the general case, a specialized algorithm or an algorithm that finds approximate solutions is used, depending on the difficulty of the problem.

Dynamic programming

When a problem shows

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 a table

of subproblems already solved, dynamic programming reduces the exponential nature of many problems to polynomial complexity.

The greedy method

A

Sollin

are greedy algorithms that can solve this optimization problem.

The heuristic method

In

heuristic algorithms can be used to find a solution close to the optimal solution in cases where finding the optimal solution is impractical. These algorithms work by getting closer and closer to the optimal solution as they progress. In principle, if run for an infinite amount of time, they will find the optimal solution. Their merit is that they can find a solution very close to the optimal solution in a relatively short time. Such algorithms include local search, tabu search, simulated annealing, and genetic algorithms. Some of them, like simulated annealing, are non-deterministic algorithms while others, like tabu search, are deterministic. When a bound on the error of the non-optimal solution is known, the algorithm is further categorized as an approximation algorithm

.

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), so algorithms are not patentable (as in

One of the simplest algorithms is to find the largest number in a list of numbers of random order. Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be stated in a high-level description in English prose, as:

High-level description:

1. If there are no numbers in the set, then there is no highest number.
2. Assume the first number in the set is the largest number in the set.
3. For each remaining number in the set: if this number is larger than the current largest number, consider this number to be the largest number in the set.
4. When there are no numbers left in the set to iterate over, consider the current largest number to be the largest number of the set.

(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 list of numbers L.
Output: The largest number in the list L.

if L.size = 0 return null
largest ← L[0]
for each item in L, do
    if item > largest, then
        largest ← item
return largest

See also

Notes

Bibliography

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  on September 3, 2014. Retrieved September 30, 2013. Presented to the American Mathematical Society, September 1935. Reprinted in The Undecidable, p. 237ff. Kleene's definition of "general recursion" (known now as mu-recursion) was used by Church in his 1935 paper An Unsolvable Problem of Elementary Number Theory that proved the "decision problem" to be "undecidable" (i.e., a negative result).
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Further reading

• ISBN 978-0-520-25419-0
  .
• Berlinski, David (2001). The Advent of the Algorithm: The 300-Year Journey from an Idea to the Computer. Harvest Books.
  ISBN 978-0-15-601391-8
  .
• Chabert, Jean-Luc (1999). A History of Algorithms: From the Pebble to the Microchip. Springer Verlag.
  ISBN 978-3-540-63369-3
  .
• Thomas H. Cormen; Charles E. Leiserson; Ronald L. Rivest; Clifford Stein (2009). Introduction To Algorithms (3rd ed.). MIT Press.
  ISBN 978-0-262-03384-8
  .
• Harel, David; Feldman, Yishai (2004). Algorithmics: The Spirit of Computing. Addison-Wesley.
  ISBN 978-0-321-11784-7
  .
• Hertzke, Allen D.; McRorie, Chris (1998). "The Concept of Moral Ecology". In Lawler, Peter Augustine; McConkey, Dale (eds.). Community and Political Thought Today. Westport, CT:
  Praeger
  .
• Knuth, Donald E. (2000). Selected Papers on Analysis of Algorithms Archived July 1, 2017, at the Wayback Machine. Stanford, California: Center for the Study of Language and Information.
• Knuth, Donald E. (2010). Selected Papers on Design of Algorithms Archived July 16, 2017, at the Wayback Machine. Stanford, California: Center for the Study of Language and Information.
• Wallach, Wendell; Allen, Colin (November 2008). Moral Machines: Teaching Robots Right from Wrong. US: Oxford University Press.
  ISBN 978-0-19-537404-9
  .
• Bleakley, Chris (2020). Poems that Solve Puzzles: The History and Science of Algorithms. Oxford University Press.
  ISBN 978-0-19-885373-2
  .

External links

Look up algorithm in Wiktionary, the free dictionary.

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

Wikimedia Commons has media related to Algorithms.

• "Algorithm". Encyclopedia of Mathematics. EMS Press. 2001 [1994].
• Algorithms at
  Curlie
• Weisstein, Eric W. "Algorithm". MathWorld.
• Dictionary of Algorithms and Data Structures – National Institute of Standards and Technology

Algorithm repositories

• The Stony Brook Algorithm Repository –
  State University of New York at Stony Brook
• Collected Algorithms of the ACM – Associations for Computing Machinery
• The Stanford GraphBase Archived December 6, 2015, at the Wayback Machine – Stanford University