Algorithm

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"Algorithms" redirects here. For the subfield of computer science, see Analysis of algorithms. For other uses, see Algorithm (disambiguation).

In a loop, subtract the larger number against the smaller number. Halt the loop when the subtraction will make a number negative. Assess two numbers, whether one of them is equal to zero or not. If yes, take the other number as the greatest common divisor. If no, put the two numbers in the subtraction loop again.
Flowchart of using successive subtractions to find the greatest common divisor of number r and s

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).

In contrast, a heuristic is an approach to solving problems that do not have well-defined correct or optimal results.[2] For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation.

As an

deterministic; some algorithms, known as randomized algorithms, incorporate random input.[9]

Etymology

Around 825 AD, Persian scientist and polymath Muḥammad ibn Mūsā al-Khwārizmī wrote kitāb al-ḥisāb al-hindī ("Book of Indian computation") and kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī ("Addition and subtraction in Indian arithmetic").[1] In the early 12th century, Latin translations of said al-Khwarizmi texts involving the Hindu–Arabic numeral system and arithmetic appeared, for example Liber Alghoarismi de practica arismetrice, attributed to John of Seville, and Liber Algorismi de numero Indorum, attributed to Adelard of Bath.[10] Hereby, alghoarismi or algorismi is the Latinization of Al-Khwarizmi's name; the text starts with the phrase Dixit Algorismi, or "Thus spoke Al-Khwarizmi".[2] Around 1230, the English word algorism is attested and then by Chaucer in 1391, English adopted the French term.[3][4][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

For a detailed presentation of the various points of view on the definition of "algorithm", see Algorithm characterizations.

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

bureaucratic procedure[12]
or cook-book recipe.[13] In general, a program is an algorithm only if it stops eventually[14]—even though infinite loops may sometimes prove desirable. Boolos, Jeffrey & 1974, 1999 define an algorithm to be an explicit set of instructions for determining an output, that can be followed by a computing machine or a human who could only carry out specific elementary operations on symbols.[15]

Most algorithms are intended to be

electrical circuit
, or a mechanical device.

History

Ancient algorithms

Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in

Arabic mathematics (around 800 AD).[22]

The earliest evidence of algorithms is found in ancient

Hammurabi dynasty c. 1800 – c. 1600 BC, Babylonian clay tablets described algorithms for computing formulas.[23] Algorithms were also used in Babylonian astronomy. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.[24]

Algorithms for arithmetic are also found in ancient

Introduction to Arithmetic by Nicomachus,[25][20]: Ch 9.2  and the Euclidean algorithm, which was first described in Euclid's Elements (c. 300 BC).[20]: Ch 9.1 Examples of ancient Indian mathematics included the Shulba Sutras, the Kerala School, and the Brāhmasphuṭasiddhānta.[17]

The first cryptographic algorithm for deciphering encrypted code was developed by Al-Kindi, a 9th-century Arab mathematician, in A Manuscript On Deciphering Cryptographic Messages. He gave the first description of cryptanalysis by frequency analysis, the earliest codebreaking algorithm.[22]

Computers

Weight-driven clocks

Bolter credits the invention of the weight-driven clock as "the key invention [of

Turing-complete computer instead of just a calculator
. Although a full implementation of Babbage's second device was not realized for decades after her lifetime, Lovelace has been called "history's first programmer".

Electromechanical relay

Bell and Newell (1971) write that the

telegraph, the precursor of the telephone, was in use throughout the world. By the late 19th century, the ticker tape (c. 1870s) was in use, as were Hollerith cards (c. 1890). Then came the teleprinter (c. 1910) with its punched-paper use of Baudot code
on tape.

Telephone-switching networks of

electromechanical relays were invented in 1835. These led to the invention of the digital adding device by George Stibitz in 1937. While working 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".[30][31]

Formalization

Ada Lovelace's diagram from "Note G", the first published computer algorithm

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

Turing machines
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 expressions of algorithms that avoid common ambiguities of natural language. Programming languages are primarily for expressing algorithms in a computer-executable form, but are also used to define or document algorithms.

Turing machines

There are many possible representations and

assembly code called "sets of quadruples", and more. Algorithm representations can also be classified into three accepted levels of Turing machine description: high-level description, implementation description, and formal description.[34] A high-level description describes qualities of the algorithm itself, ignoring how it is implemented on the Turing machine.[34] An implementation description describes the general manner in which the machine moves its head and stores data in order to carry out the algorithm, but does not give exact states.[34] In the most detail, a formal description gives the exact state table and list of transitions of the Turing machine.[34]

Flowchart representation

The graphical aid called a flowchart offers a way to describe and document an algorithm (and a computer program corresponding to it). It has four primary symbols: arrows showing program flow, rectangles (SEQUENCE, GOTO), diamonds (IF-THEN-ELSE), and dots (OR-tie). Sub-structures can "nest" in rectangles, but only if a single exit occurs from the superstructure.

Algorithmic analysis

Main article: Analysis of algorithms

It is often important to know how much time, storage, or other cost an algorithm may require. Methods have been developed for the analysis of algorithms to obtain such quantitative answers (estimates); for example, an algorithm that adds up the elements of a list of n numbers would have a time requirement of ⁠⁠, using big O notation. The algorithm only needs to remember two values: the sum of all the elements so far, and its current position in the input list. If the space required to store the input numbers is not counted, it has a space requirement of ⁠⁠, otherwise ⁠⁠ is required.

Different algorithms may complete the same task with a different set of instructions in less or more time, space, or 'effort' than others. For example, a binary search algorithm (with cost ⁠⁠) outperforms a sequential search (cost ⁠⁠ ) 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. Algorithms are often studied abstractly, without referencing any specific programming language or implementation. Algorithm analysis resembles other mathematical disciplines as it focuses on the algorithm's properties, not implementation. Pseudocode is typical for analysis as it is a simple and general representation. Most algorithms are implemented on particular hardware/software platforms and their algorithmic efficiency is tested using real code. The efficiency of a particular algorithm may be insignificant for many "one-off" problems but it may be critical for algorithms designed for fast interactive, commercial or long life scientific usage. Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign.

Empirical testing is useful for uncovering 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 non-trivial to perform fairly.[35]

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.[36] In general, speed improvements depend on special properties of the problem, which are very common in practical applications.[37] 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 is a method or 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,[38] 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.[39]

Structured programming

Per the

proofs of correctness using mathematical induction.[43]

See also: Software patent

By themselves, algorithms 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

LZW patent. Additionally, some cryptographic algorithms have export restrictions (see export of cryptography
).

Classification

By implementation

Recursion
A
stacks to solve problems. Problems may be suited for one implementation or the other. The Tower of Hanoi
is a puzzle commonly solved 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 on serial computers. Serial algorithms are designed for these environments, unlike parallel or distributed algorithms. Parallel algorithms take advantage of computer architectures where multiple processors can work on a problem at the same time. Distributed algorithms use multiple machines connected via a computer network. Parallel and distributed algorithms divide the problem into subproblems and collect the results back together. Resource consumption in these 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 close to the true solution. Such algorithms have practical value for many hard problems. For example, the Knapsack problem, where there is a set of items and the goal is to pack the knapsack to get the maximum total value. Each item has some weight and some value. The 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.[45]
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. Some common paradigms are:

Brute-force or exhaustive search
Brute force is a problem-solving method of systematically trying every possible option until the optimal solution is found. This approach can be very time-consuming, testing every possible combination of variables. It is often used when other methods are unavailable or too complex. Brute force can solve a variety of problems, including finding the shortest path between two points and cracking passwords.
Divide and conquer
A
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 find approximate solutions when finding exact solutions may be impractical (see heuristic method below). For some problems the fastest approximations must involve some randomness.[46] Whether randomized algorithms with polynomial time complexity can be the fastest algorithm 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 transforms difficult problems into better-known problems solvable with (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 by linear equality and inequality constraints, the constraints can be used directly to produce optimal solutions. There are algorithms that can solve any problem in this category, such as the popular simplex algorithm.[47] Problems that can be solved with linear programming include the maximum flow problem for directed graphs. If a problem also requires that any of the unknowns be integers, 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 optimal substructures—meaning the optimal solution 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. For example, Floyd–Warshall algorithm, the shortest path between a start and goal vertex in a weighted graph can be found using the shortest path to the goal from all adjacent vertices. Dynamic programming and memoization
go together. Unlike divide and conquer, dynamic programming subproblems often overlap. The difference between dynamic programming and simple recursion is the caching or memoization of recursive calls. When subproblems are independent and do not repeat, memoization does not help; hence dynamic programming is not applicable to all complex problems. Using memoization dynamic programming reduces the complexity of many problems from exponential to polynomial.
The greedy method
Sollin
are greedy algorithms that can solve this optimization problem.
The heuristic method
In
heuristic algorithms find solutions close to the optimal solution when finding the optimal solution is impractical. These algorithms get 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. They can ideally find a solution very close to the optimal solution in a relatively short time. These algorithms include local search, tabu search, simulated annealing, and genetic algorithms. Some, 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
.

Examples

Further information: List of algorithms

One of the simplest algorithms finds 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 described in plain English as:

High-level description:

  1. If a set of numbers is empty, then there is no highest number.
  2. Assume the first number in the set is the largest.
  3. For each remaining number in the set: if this number is greater than the current largest, it becomes the new largest.
  4. When there are no unchecked numbers left in the set, consider the current largest number to be the largest in 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
  • "←" denotes assignment. For instance, "largest ← item" means that the value of largest changes to the value of item.
  • "return" terminates the algorithm and outputs the following value.

See also

Notes

  1. ^ a b "Definition of ALGORITHM". Merriam-Webster Online Dictionary. Archived from the original on February 14, 2020. Retrieved November 14, 2019.
  2. ^
    ISBN 1402030045
  3. ^ a b "Any classical mathematical algorithm, for example, can be described in a finite number of English words" (Rogers 1987:2).
  4. ^ a b Well defined concerning the agent that executes the algorithm: "There is a computing agent, usually human, which can react to the instructions and carry out the computations" (Rogers 1987:2).
  5. ^ "an algorithm is a procedure for computing a function (concerning some chosen notation for integers) ... this limitation (to numerical functions) results in no loss of generality", (Rogers 1987:1).
  6. zero or more inputs, i.e., quantities
    which are given to it initially before the algorithm begins" (Knuth 1973:5).
  7. ^ "A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a 'computational method'" (Knuth 1973:5).
  8. ^ "An algorithm has one or more outputs, i.e., quantities which have a specified relation to the inputs" (Knuth 1973:5).
  9. ^ Whether or not a process with random interior processes (not including the input) is an algorithm is debatable. Rogers opines that: "a computation is carried out in a discrete stepwise fashion, without the use of continuous methods or analog devices ... carried forward deterministically, without resort to random methods or devices, e.g., dice" (Rogers 1987:2).
  10. ^ Blair, Ann, Duguid, Paul, Goeing, Anja-Silvia and Grafton, Anthony. Information: A Historical Companion, Princeton: Princeton University Press, 2021. p. 247
  11. ^ Stone 1973:4
  12. ^
    ISBN 9780262536370. Archived
    from the original on December 22, 2019. Retrieved May 27, 2019. [...] the next level of abstraction of central bureaucracy: globally operating algorithms.
  13. ^ Dietrich, Eric (1999). "Algorithm". In Wilson, Robert Andrew; Keil, Frank C. (eds.). The MIT Encyclopedia of the Cognitive Sciences. MIT Cognet library. Cambridge, Massachusetts: MIT Press (published 2001). p. 11.
    ISBN 9780262731447
    . Retrieved July 22, 2020. An algorithm is a recipe, method, or technique for doing something.
  14. ^ Stone requires that "it must terminate in a finite number of steps" (Stone 1973:7–8).
  15. ^ Boolos and Jeffrey 1974,1999:19
  16. ^
    ISBN 9783642181924
    .
  17. ^
    ISBN 978-93-86279-25-5
    .
  18. ^ Hayashi, T. (2023, January 1). Brahmagupta. Encyclopedia Britannica.
  19. JSTOR 3027363
    .
  20. ^
    ISBN 978-1-118-46029-0
    .
  21. ISBN 978-3-540-63369-3
    .
  22. ^
    ISBN 9783319016283
    .
  23. S2CID 7829945. Archived from the original
    (PDF) on December 24, 2012.
  24. ISBN 978-0-387-95136-2
    .
  25. ^ Ast, Courtney. "Eratosthenes". Wichita State University: Department of Mathematics and Statistics. Archived from the original on February 27, 2015. Retrieved February 27, 2015.
  26. ^ Bolter 1984:24
  27. ^ Bolter 1984:26
  28. ^ Bolter 1984:33–34, 204–206.
  29. ^ Bell and Newell diagram 1971:39, cf. Davis 2000
  30. ^ Melina Hill, Valley News Correspondent, A Tinkerer Gets a Place in History, Valley News West Lebanon NH, Thursday, March 31, 1983, p. 13.
  31. ^ Davis 2000:14
  32. ^ Kleene 1943 in Davis 1965:274
  33. ^ Rosser 1939 in Davis 1965:225
  34. ^ a b c d Sipser 2006:157
  35. S2CID 40772241
    .
  36. ^ Gillian Conahan (January 2013). "Better Math Makes Faster Data Networks". discovermagazine.com. Archived from the original on May 13, 2014. Retrieved May 13, 2014.
  37. ^ Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price, "ACM-SIAM Symposium On Discrete Algorithms (SODA) Archived July 4, 2013, at the Wayback Machine, Kyoto, January 2012. See also the sFFT Web Page Archived February 21, 2012, at the Wayback Machine.
  38. ISBN 978-0-471-38365-9. Archived
    from the original on April 28, 2015. Retrieved June 14, 2018.
  39. ^ "Big-O notation (article) | Algorithms". Khan Academy. Retrieved June 3, 2024.
  40. ISBN 0-201-13433-0
    .
  41. ^ Tausworthe 1977:101
  42. ^ Tausworthe 1977:142
  43. ^ Knuth 1973 section 1.2.1, expanded by Tausworthe 1977 at pages 100ff and Chapter 9.1
  44. ISSN 0099-9660
    . Retrieved March 29, 2017.
  45. S2CID 28836720. Archived
    from the original on October 18, 2017. Retrieved September 19, 2017.
  46. S2CID 13268711
    .
  47. ^
    George B. Dantzig
    and Mukund N. Thapa. 2003. Linear Programming 2: Theory and Extensions. Springer-Verlag.

Bibliography

  • Zaslavsky, C. (1970). Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria. The Two-Year College Mathematics Journal, 1(2), 76–99. https://doi.org/10.2307/3027363

Further reading

Look up algorithm in Wiktionary, the free dictionary.
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