Algorithm


In mathematics and computer science, an algorithm is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation. 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 and deduce valid inferences.
In contrast, a heuristic is an approach to solving problems without well-defined correct or optimal results. 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 effective method, an algorithm can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. Starting from an initial state and initial input, the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.

Etymology

Around 825 AD, Persian scientist and polymath Muḥammad ibn Mūsā al-Khwārizmī wrote kitāb al-ḥisāb al-hindī and kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī. In the early 12th century, Latin translations of these 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. Here, alghoarismi or algorismi is the Latinization of Al-Khwarizmi's name; the text starts with the phrase Dixit Algorismi, or "Thus spoke Al-Khwarizmi".
The word algorism in English came to mean the use of place-value notation in calculations; it occurs in the Ancrene Wisse from circa 1225. By the time Geoffrey Chaucer wrote The Canterbury Tales in the late 14th century, he used a variant of the same word in describing augrym stones, stones used for place-value calculation. In the 15th century, under the influence of the Greek word ἀριθμός, the Latin word was altered to algorithmus. By 1596, this form of the word was used in English, as algorithm, by Thomas Hood.

Definition

One informal definition is "a set of rules that precisely defines a sequence of operations", which would include all computer programs, and any prescribed bureaucratic procedure
or cook-book recipe. In general, a program is an algorithm only if it stops eventually—even though infinite loops may sometimes prove desirable. 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.
Most algorithms are intended to be implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network, in an 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 Babylonian mathematics, Egyptian mathematics, Indian mathematics, the Ifa Oracle, Greek mathematics, Chinese mathematics (around 200 BC and later), and Arabic mathematics.
The earliest evidence of algorithms is found in ancient Mesopotamian mathematics. A Sumerian clay tablet found in Shuruppak near Baghdad and dated to describes the earliest division algorithm. During the Hammurabi dynasty, Babylonian clay tablets described algorithms for computing formulas. 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.
Algorithms for arithmetic are also found in ancient Egyptian mathematics, dating back to the Rhind Mathematical Papyrus. Algorithms were later used in ancient Hellenistic mathematics. Two examples are the Sieve of Eratosthenes, which was described in the Introduction to Arithmetic by Nicomachus, and the Euclidean algorithm, which was first described in Euclid's Elements.Examples of ancient Indian mathematics included the Shulba Sutras, the Kerala School, and the Brāhmasphuṭasiddhānta.
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.

Computers

Weight-driven clocks

David Bolter credits the invention of the weight-driven clock as "the key invention," specifically the verge escapement mechanism producing the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical automata" in the 13th century and "computational machines"—the difference and analytical engines of Charles Babbage and Ada Lovelace in the mid-19th century. Lovelace designed the first algorithm intended for processing on a computer, Babbage's analytical engine, which is the first device considered a real Turing-complete computer instead of just a calculator. Although the 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 write that the Jacquard loom, a precursor to Hollerith cards, and "telephone switching technologies" led to the development of the first computers. By the mid-19th century, the telegraph, the precursor of the telephone, was in use throughout the world. By the late 19th century, the ticker tape was in use, as were Hollerith cards. Then came the teleprinter 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".

Formalization

In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve the Entscheidungsproblem posed by David Hilbert. Later formalizations were framed as attempts to define "effective calculability" or "effective method". Those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, and Alan Turing's Turing machines of 1936–37 and 1939.

Modern Algorithms

Algorithms have evolved and improved in many ways as time goes on. Common uses of algorithms today include social media apps like Instagram and YouTube. Algorithms are used as a way to analyze what people like and push more of those things to the people who interact with them. Quantum computing uses quantum algorithm procedures to solve problems faster. More recently, in 2024, NIST updated their post-quantum encryption standards, which includes new encryption algorithms to enhance defenses against attacks using quantum computing.

Representations

Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, drakon-charts, programming languages or control tables. 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 Turing machine programs can be expressed as a sequence of machine tables, as flowcharts and drakon-charts, as a form of rudimentary machine code or 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. A high-level description describes the qualities of the algorithm itself, ignoring how it is implemented on the Turing machine. An implementation description describes the general manner in which the machine moves its head and stores data to carry out the algorithm, but does not give exact states. In the most detail, a formal description gives the exact state table and list of transitions of the Turing machine.

Flowchart representation

The graphical aid called a flowchart offers a way to describe and document an algorithm. It has four primary symbols: arrows showing program flow, rectangles, diamonds, and dots. Sub-structures can "nest" in rectangles, but only if a single exit occurs from the superstructure.

Algorithmic analysis

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 ; 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 outperforms a sequential search when used for table lookups on sorted lists or arrays.

Formal versus empirical

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.

Execution efficiency

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

Best Case and Worst Case

The best case of an algorithm refers to the scenario or input for which the algorithm or data structure takes the least time and resources to complete its tasks. The worst case of an algorithm is the case that causes the algorithm or data structure to consume the maximum period of time and computational resources.

Design

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 divide-and-conquer or dynamic programming within operation research. Techniques for designing and implementing algorithm designs are also called algorithm design patterns, with examples including the template method pattern and the decorator pattern. One of the most important aspects of algorithm design is resource efficiency; the big O notation is used to describe e.g., an algorithm's run-time growth as the size of its input increases.

Structured programming

Per the Church–Turing thesis, any algorithm can be computed by any Turing complete model. Turing completeness only requires four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. However, Kemeny and Kurtz observe that, while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in "spaghetti code", a programmer can write structured programs using only these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language". Tausworthe augments the three Böhm-Jacopini canonical structures: SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE. An additional benefit of a structured program is that it lends itself to proofs of correctness using mathematical induction.

Legal status

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", so algorithms are not patentable. However practical applications of algorithms are sometimes patentable. For example, in Diamond v. Diehr, the application of a simple feedback algorithm to aid in the curing of synthetic rubber was deemed patentable. The patenting of software is controversial, and there are criticized patents involving algorithms, especially data compression algorithms, such as Unisys's LZW patent. Additionally, some cryptographic algorithms have export restrictions.

Classification

By implementation

; Recursion
; Serial, parallel or distributed
; Deterministic or non-deterministic
; Exact or approximate
; Quantum algorithm

By design paradigm

Another way of classifying algorithms is by their design methodology or paradigm. Some common paradigms are:
; Brute-force or exhaustive search
; Divide and conquer
; Search and enumeration
;Randomized algorithm
  1. Monte Carlo algorithms return a correct answer with high probability. E.g. RP is the subclass of these that run in polynomial time.
  2. Las Vegas algorithms always return the correct answer, but their running time is only probabilistically bound, e.g. ZPP.
; Reduction of complexity
; Back tracking

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
; Dynamic programming
; The greedy method
;The heuristic method

Examples

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.
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:
Input: A list of numbers L.
Output: The largest number in the list L.
if L.size = 0 return null
largest ← L
for each item in L, do
if item > largest, then
largest ← item
return ''largest''