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d formal language for calculating a function. Starting from an initial state and initial input perhaps empty, the instructions describe a computation that, when executed, proceeds through a finite number of welldefined 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.
History
The concept of algorithm has existed since antiquity. Arithmetic algorithms, such as a division algorithm, were used by ancient Babylonian mathematicians c. 2500 BC and Egyptian mathematicians c. 1550 BC. Greek mathematicians later used algorithms in 240 BC in the sieve of Eratosthenes for finding prime numbers, and the Euclidean algorithm for finding the greatest common divisor of two numbers. Arabic mathematicians such as alKindi in the 9th century used cryptographic algorithms for codebreaking, based on frequency analysis.
The word algorithm is |
derived from the name of the 9thcentury Persian mathematician Muammad ibn Ms alKhwrizm, whose nisba identifying him as from Khwarazm was Latinized as Algoritmi Arabized Persian c. 780850.
Muammad ibn Ms alKhwrizm was a mathematician, astronomer, geographer, and scholar in the House of Wisdom in Baghdad, whose name means 'the native of Khwarazm', a region that was part of Greater Iran and is now in Uzbekistan. About 825, alKhwarizmi wrote an Arabic language treatise on the HinduArabic numeral system, which was translated into Latin during the 12th century. The manuscript starts with the phrase Dixit Algorizmi 'Thus spake AlKhwarizmi', where "Algorizmi" was the translator's Latinization of AlKhwarizmi's name. AlKhwarizmi was the most widely read mathematician in Europe in the late Middle Ages, primarily through another of his books, the Algebra. In late medieval Latin, algorismus, English 'algorism', the corruption of his name, simply meant the "decimal number system". In the 15th century, under the influence |
of the Greek word arithmos, 'number' cf. 'arithmetic', the Latin word was altered to algorithmus, and the corresponding English term 'algorithm' is first attested in the 17th century; the modern sense was introduced in the 19th century.
Indian mathematics was predominantly algorithmic.
Algorithms that are representative of the Indian mathematical tradition range from the ancient ulbastrs to the medieval texts of the Kerala School.
In English, the word algorithm was first used in about 1230 and then by Chaucer in 1391. English adopted the French term, but it was not until the late 19th century that "algorithm" took on the meaning that it has in modern English.
Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu. It begins with
which translates to
The poem is a few hundred lines long and summarizes the art of calculating with the new styled Indian dice Tali Indorum, or Hindu numerals.
A partial formalization of the modern concept of algor |
ithm began with attempts to solve the Entscheidungsproblem decision problem posed by David Hilbert in 1928. Later formalizations were framed as attempts to define "effective calculability" or "effective method". Those formalizations included the GdelHerbrandKleene 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 193637 and 1939.
Informal definition
An informal definition could be "a set of rules that precisely defines a sequence of operations", which would include all computer programs including programs that do not perform numeric calculations, and for example any prescribed bureaucratic procedure
or cookbook recipe.
In general, a program is only an algorithm if it stops eventuallyeven though infinite loops may sometimes prove desirable.
A prototypical example of an algorithm is the Euclidean algorithm, which is used to determine the maximum common divisor of two integers; an example there are oth |
ers is described by the flowchart above and as an example in a later section.
offer an informal meaning of the word "algorithm" in the following quotation
No human being can write fast enough, or long enough, or small enough "smaller and smaller without limit ... you'd be trying to write on molecules, on atoms, on electrons" to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols.
An "enumerably infinite set" is one whose elements can be put into onetoone correspondence with the integers. Thus Boolos and Jeffrey are saying that an algorithm implies inst |
ructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be arbitrarily large. For example, an algorithm can be an algebraic equation such as y m n i.e., two arbitrary "input variables" m and n that produce an output y, but various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of for the addition example
Precise instructions in a language understood by "the computer" for a fast, efficient, "good" process that specifies the "moves" of "the computer" machine or human, equipped with the necessary internally contained information and capabilities to find, decode, and then process arbitrary input integerssymbols m and n, symbols and ... and "effectively" produce, in a "reasonable" time, outputinteger y at a specified place and in a specified format.
The concept of algorithm is also used to define the notion of decidabilitya 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 implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network for example, the human brain implementing arithmetic or an insect looking for food, in an electrical circuit, or in a mechanical device.
Formalization
Algorithms are essential to the way computers process data. Many computer programs contain algorithms that detail the specific instructions a computer should performin a specific orderto carry out a specified task, such as calculating employees' paychecks or printing students' report cards. Thus, an algorithm c |
an be considered to be any sequence of operations that can be simulated by a Turingcomplete system. Authors who assert this thesis include Minsky 1967, Savage 1987 and Gurevich 2000
Minsky "But we will also maintain, with Turing ... that any procedure which could "naturally" be called effective, can, in fact, be realized by a simple machine. Although this may seem extreme, the arguments ... in its favor are hard to refute".
Gurevich " Turing's informal argument in favor of his thesis justifies a stronger thesis every algorithm can be simulated by a Turing machine according to Savage 1987, an algorithm is a computational process defined by a Turing machine".Turing machines can define computational processes that do not terminate. The informal definitions of algorithms generally require that the algorithm always terminates. This requirement renders the task of deciding whether a formal procedure is an algorithm impossible in the general casedue to a major theorem of computability theory known as the halting |
problem.
Typically, when an algorithm is associated with processing information, data can be read from an input source, written to an output device and stored for further processing. Stored data are regarded as part of the internal state of the entity performing the algorithm. In practice, the state is stored in one or more data structures.
For some of these computational processes, the algorithm must be rigorously defined specified in the way it applies in all possible circumstances that could arise. This means that any conditional steps must be systematically dealt with, casebycase; the criteria for each case must be clear and computable.
Because an algorithm is a precise list of precise steps, the order of computation is always crucial 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, the discussion on the formalizat |
ion of an algorithm has assumed the premises of imperative programming. This is the most common conceptionone which attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the assignment operation, which sets the value of a variable. It derives from the intuition of "memory" as a scratchpad. An example of such an assignment can be found below.
For some alternate conceptions of what constitutes an algorithm, see functional programming and logic programming.
Expressing algorithms
Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, drakoncharts, 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, drakoncharts and control tables are structured ways to express algorithms that avoid many of the ambiguities common in the statements |
based on natural language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are also often used as a way to define or document algorithms.
There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables see finitestate machine, state transition table and control table for more, as flowcharts and drakoncharts see state diagram for more, or as a form of rudimentary machine code or assembly code called "sets of quadruples" see Turing machine for more.
Representations of algorithms can be classed into three accepted levels of Turing machine description, as follows
1 Highlevel description
"...prose to describe an algorithm, ignoring the implementation details. At this level, we do not need to mention how the machine manages its tape or head."
2 Implementation description
"...prose used to define the way the Turing machine uses its head and the way that it stores data |
on its tape. At this level, we do not give details of states or transition function."
3 Formal description
Most detailed, "lowest level", gives the Turing machine's "state table".
For an example of the simple algorithm "Add mn" described in all three levels, see Examples.
Design
Algorithm design refers to a method or a mathematical process for problemsolving and engineering algorithms. The design of algorithms is part of many solution theories of operation research, such as dynamic programming and divideandconquer. 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 runtime, memory usage efficiency; the big O notation is used to describe e.g. an algorithm's runtime growth as the size of its input increases.
Typical steps in the development of algorithms
Problem definition
Development of a model
Specificati |
on of the algorithm
Designing an algorithm
Checking the correctness of the algorithm
Analysis of algorithm
Implementation of algorithm
Program testing
Documentation preparation
Computer algorithms
"Elegant" compact programs, "good" fast programs The notion of "simplicity and elegance" appears informally in Knuth and precisely in Chaitin
Knuth " ... we want good algorithms in some loosely defined aesthetic sense. One criterion ... is the length of time taken to perform the algorithm .... Other criteria are adaptability of the algorithm to computers, its simplicity and elegance, etc."
Chaitin " ... a program is 'elegant,' by which I mean that it's the smallest possible program for producing the output that it does"
Chaitin prefaces his definition with "I'll show you can't prove that a program is 'elegantsuch a proof would solve the Halting problem ibid.
Algorithm versus function computable by an algorithm For a given function multiple algorithms may exist. This is true, even without expanding the av |
ailable instruction set available to the programmer. Rogers observes that "It is ... important to distinguish between the notion of algorithm, i.e. procedure and the notion of function computable by algorithm, i.e. mapping yielded by procedure. The same function may have several different algorithms".
Unfortunately, there may be a tradeoff between goodness speed and elegance compactnessan elegant program may take more steps to complete a computation than one less elegant. An example that uses Euclid's algorithm appears below.
Computers and computors, models of computation A computer or human "computor" is a restricted type of machine, a "discrete deterministic mechanical device" that blindly follows its instructions. Melzak's and Lambek's primitive models reduced this notion to four elements i discrete, distinguishable locations, ii discrete, indistinguishable counters iii an agent, and iv a list of instructions that are effective relative to the capability of the agent.
Minsky describes a more congenial v |
ariation of Lambek's "abacus" model in his "Very Simple Bases for Computability". Minsky's machine proceeds sequentially through its five or six, depending on how one counts instructions unless either a conditional IFTHEN GOTO or an unconditional GOTO changes program flow out of sequence. Besides HALT, Minsky's machine includes three assignment replacement, substitution operations ZERO e.g. the contents of location replaced by 0 L 0, SUCCESSOR e.g. L L1, and DECREMENT e.g. L L 1. Rarely must a programmer write "code" with such a limited instruction set. But Minsky shows as do Melzak and Lambek that his machine is Turing complete with only four general types of instructions conditional GOTO, unconditional GOTO, assignmentreplacementsubstitution, and HALT. However, a few different assignment instructions e.g. DECREMENT, INCREMENT, and ZEROCLEAREMPTY for a Minsky machine are also required for Turingcompleteness; their exact specification is somewhat up to the designer. The unconditional GOTO is a convenienc |
e; it can be constructed by initializing a dedicated location to zero e.g. the instruction " Z 0 "; thereafter the instruction IF Z0 THEN GOTO xxx is unconditional.
Simulation of an algorithm computer computor language Knuth advises the reader that "the best way to learn an algorithm is to try it . . . immediately take pen and paper and work through an example". But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a language that the simulatorcomputercomputor can effectively execute. Stone gives an example of this when computing the roots of a quadratic equation the computor must know how to take a square root. If they don't, then the algorithm, to be effective, must provide a set of rules for extracting a square root.
This means that the programmer must know a "language" that is effective relative to the target computing agent computercomputor.
But what model should be used for the simulation? Van Emde Boas observes "even if we base complexity theor |
y on abstract instead of concrete machines, arbitrariness of the choice of a model remains. It is at this point that the notion of simulation enters". When speed is being measured, the instruction set matters. For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a "modulus" instruction available rather than just subtraction or worse just Minsky's "decrement".
Structured programming, canonical structures Per the ChurchTuring thesis, any algorithm can be computed by a model known to be Turing complete, and per Minsky's demonstrations, Turing completeness requires only four instruction typesconditional GOTO, unconditional GOTO, assignment, HALT. Kemeny and Kurtz observe that, while "undisciplined" use of unconditional GOTOs and conditional IFTHEN 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 pro |
grams in a structured language". Tausworthe augments the three BhmJacopini canonical structures SEQUENCE, IFTHENELSE, and WHILEDO, with two more DOWHILE and CASE. An additional benefit of a structured program is that it lends itself to proofs of correctness using mathematical induction.
Canonical flowchart symbols The graphical aide called a flowchart, offers a way to describe and document an algorithm and a computer program of one. 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 IFTHENELSE, and the dot ORtie. The BhmJacopini canonical structures are made of these primitive shapes. Substructures 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.
Examples
Algorithm example
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 highlevel description in English prose, as
Highlevel description
If there are no numbers in the set then there is no highest number.
Assume the first number in the set is the largest number in the set.
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.
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.
Quasiformal description
Written in prose but much closer to the highlevel 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 L0
for each item in L, do
if |
item largest, then
largest item
return largest
Euclid's algorithm
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor GCD of two integers numbers, the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements c. 300 BC. It is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other numbertheoretic and cryptographic calculations.
Euclid poses the problem thus "Given two numbers not prime to one another, to find their greatest common measure". He defines "A number to be a multitude composed of units" a counting number, a positive integer not including zero. To "measure" is to place a shorter measuring length s successively q times along longer length l until the remaining portion r is less than the shorter length s. In modern words, remainder r l qs, |
q being the quotient, or remainder r is the "modulus", the integerfractional part left over after the division.
For Euclid's method to succeed, the starting lengths must satisfy two requirements i the lengths must not be zero, AND ii the subtraction must be "proper"; i.e., a test must guarantee that the smaller of the two numbers is subtracted from the larger or the two can be equal so their subtraction yields zero.
Euclid's original proof adds a third requirement the two lengths must not be prime to one another. Euclid stipulated this so that he could construct a reductio ad absurdum proof that the two numbers' common measure is in fact the greatest. While Nicomachus' algorithm is the same as Euclid's, when the numbers are prime to one another, it yields the number "1" for their common measure. So, to be precise, the following is really Nicomachus' algorithm.
Computer language for Euclid's algorithm
Only a few instruction types are required to execute Euclid's algorithmsome logical tests conditional GOTO |
, unconditional GOTO, assignment replacement, and subtraction.
A location is symbolized by upper case letters, e.g. S, A, etc.
The varying quantity number in a location is written in lower case letters and usually associated with the location's name. For example, location L at the start might contain the number l 3009.
An inelegant program for Euclid's algorithm
The following algorithm is framed as Knuth's fourstep version of Euclid's and Nicomachus', but, rather than using division to find the remainder, it uses successive subtractions of the shorter length s from the remaining length r until r is less than s. The highlevel description, shown in boldface, is adapted from Knuth 197324
INPUT
Into two locations L and S put the numbers l and s that represent the two lengths
INPUT L, S
Initialize R make the remaining length r equal to the startinginitialinput length l
R L
E0 Ensure r s.
Ensure the smaller of the two numbers is in S and the larger in R
IF R S THEN
the contents of L is |
the larger number so skip over the exchangesteps 4, 5 and 6
GOTO step 7
ELSE
swap the contents of R and S.
L R this first step is redundant, but is useful for later discussion.
R S
S L
E1 Find remainder Until the remaining length r in R is less than the shorter length s in S, repeatedly subtract the measuring number s in S from the remaining length r in R.
IF S R THEN
done measuring so
GOTO 10
ELSE
measure again,
R R S
Remainderloop
GOTO 7.
E2 Is the remainder zero? EITHER i the last measure was exact, the remainder in R is zero, and the program can halt, OR ii the algorithm must continue the last measure left a remainder in R less than measuring number in S.
IF R 0 THEN
done so
GOTO step 15
ELSE
CONTINUE TO step 11,
E3 Interchange s and r The nut of Euclid's algorithm. Use remainder r to measure what was previously smaller number s; L serves as a temporary location.
L R
R S
S L
Repeat the measuring pro |
cess
GOTO 7
OUTPUT
Done. S contains the greatest common divisor
PRINT S
DONE
HALT, END, STOP.
An elegant program for Euclid's algorithm
The flowchart of "Elegant" can be found at the top of this article. In the unstructured Basic language, the steps are numbered, and the instruction LET is the assignment instruction symbolized by .
5 REM Euclid's algorithm for greatest common divisor
6 PRINT "Type two integers greater than 0"
10 INPUT A,B
20 IF B0 THEN GOTO 80
30 IF A B THEN GOTO 60
40 LET BBA
50 GOTO 20
60 LET AAB
70 GOTO 20
80 PRINT A
90 END
How "Elegant" works In place of an outer "Euclid loop", "Elegant" shifts back and forth between two "coloops", an A B loop that computes A A B, and a B A loop that computes B B A. This works because, when at last the minuend M is less than or equal to the subtrahend S Difference Minuend Subtrahend, the minuend can become s the new measuring length and the subtrahend can become the new r the length to be measured; in ot |
her words the "sense" of the subtraction reverses.
The following version can be used with programming languages from the Cfamily
Euclid's algorithm for greatest common divisor
int euclidAlgorithm int A, int B
AabsA;
BabsB;
while B!0
while AB AAB;
BBA;
return A;
Testing the Euclid algorithms
Does an algorithm do what its author wants it to do? A few test cases usually give some confidence in the core functionality. But tests are not enough. For test cases, one source uses 3009 and 884. Knuth suggested 40902, 24140. Another interesting case is the two relatively prime numbers 14157 and 5950.
But "exceptional cases" must be identified and tested. Will "Inelegant" perform properly when R S, S R, R S? Ditto for "Elegant" B A, A B, A B? Yes to all. What happens when one number is zero, both numbers are zero? "Inelegant" computes forever in all cases; "Elegant" computes forever when A 0. What happens if negative numbers are entered? Fractional numbers? If the |
input numbers, i.e. the domain of the function computed by the algorithmprogram, is to include only positive integers including zero, then the failures at zero indicate that the algorithm and the program that instantiates it is a partial function rather than a total function. A notable failure due to exceptions is the Ariane 5 Flight 501 rocket failure June 4, 1996.
Proof of program correctness by use of mathematical induction Knuth demonstrates the application of mathematical induction to an "extended" version of Euclid's algorithm, and he proposes "a general method applicable to proving the validity of any algorithm". Tausworthe proposes that a measure of the complexity of a program be the length of its correctness proof.
Measuring and improving the Euclid algorithms
Elegance compactness versus goodness speed With only six core instructions, "Elegant" is the clear winner, compared to "Inelegant" at thirteen instructions. However, "Inelegant" is faster it arrives at HALT in fewer steps. Algorithm analysis |
indicates why this is the case "Elegant" does two conditional tests in every subtraction loop, whereas "Inelegant" only does one. As the algorithm usually requires many loopthroughs, on average much time is wasted doing a "B 0?" test that is needed only after the remainder is computed.
Can the algorithms be improved? Once the programmer judges a program "fit" and "effective"that is, it computes the function intended by its authorthen the question becomes, can it be improved?
The compactness of "Inelegant" can be improved by the elimination of five steps. But Chaitin proved that compacting an algorithm cannot be automated by a generalized algorithm; rather, it can only be done heuristically; i.e., by exhaustive search examples to be found at Busy beaver, trial and error, cleverness, insight, application of inductive reasoning, etc. Observe that steps 4, 5 and 6 are repeated in steps 11, 12 and 13. Comparison with "Elegant" provides a hint that these steps, together with steps 2 and 3, can be eliminated. Th |
is reduces the number of core instructions from thirteen to eight, which makes it "more elegant" than "Elegant", at nine steps.
The speed of "Elegant" can be improved by moving the "B0?" test outside of the two subtraction loops. This change calls for the addition of three instructions B 0?, A 0?, GOTO. Now "Elegant" computes the examplenumbers faster; whether this is always the case for any given A, B, and R, S would require a detailed analysis.
Algorithmic analysis
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 On, 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 t |
o have a space requirement of O1, if the space required to store the input numbers is not counted, or On if it is counted.
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 Olog n outperforms a sequential search cost On 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, 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 hardwaresoftware platforms and thei |
r 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 beforeafter 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.
Execution efficiency
To illustrate the potential improvements possible even in wellestablished 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 appli |
cations 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 like digital cameras and medical equipment to consume less power.
Classification
There are various ways to classify algorithms, each with its own merits.
By implementation
One way to classify algorithms is by implementation means.
Recursion
A recursive algorithm is one that invokes makes reference to itself repeatedly until a certain condition also known as termination condition matches, which is a method common to functional programming. Iterative algorithms use repetitive constructs like loops and sometimes additional data structures like stacks to solve the given problems. Some problems are naturally suited for one implementation or the other. For example, 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.
Logical
An algorithm may be viewed as controlled logical deduction. This notion may be expressed as Algorithm logic control. The logic component expresses the axioms that may be used in the computation and the control component determines the way in which deduction is applied to the axioms. This is the basis for the logic programming paradigm. In pure logic programming languages, the control component is fixed and algorithms are specified by supplying only the logic component. The appeal of this approach is the elegant semantics a change in the axioms produces a welldefined change in the algorithm.
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 a |
lgorithms. Parallel algorithms take advantage of computer architectures where several processors can work on a problem at the same time, whereas distributed algorithms utilize multiple machines connected with a computer network. Parallel or distributed algorithms divide the problem into more symmetrical or asymmetrical subproblems and collect the results back together. 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. Some problems have no parallel algorithms and are called inherently serial problems.
Deterministic or nondeterministic
Deterministic algorithms solve the problem with exact decision at every step of the algorithm whereas nondeterministic algorithms solve problems via guessing although typical guesses are made more accurate through the use of |
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.
Quantum algorithm
They run on a realistic model of 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 p |
aradigm. 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
Bruteforce or exhaustive search
This is the naive method of trying every possible solution to see which is best.
Divide and conquer
A divide and conquer algorithm repeatedly 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 solve easily. One such example of divide and conquer is 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 conqu |
er 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 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 search algorithms, branch and bound enumeration and backtracking.
Randomized algorithm
Such algorithms make some choices randomly or pseudorandomly. 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. 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
Monte Carlo algorithms return a correct |
answer with highprobability. E.g. RP is the subclass of these that run in polynomial time.
Las Vegas algorithms always return the correct answer, but their running time is only probabilistically bound, e.g. ZPP.
Reduction of complexity
This technique involves solving a difficult problem by transforming it into a betterknown problem for which we have hopefully asymptotically optimal algorithms. The goal is to find a reducing algorithm whose complexity is not dominated by the resulting reduced algorithm's. For example, one selection algorithm for finding the median in an unsorted list involves first sorting the list the expensive portion and then pulling out the middle element in the sorted list the cheap portion. This technique is also known as 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 classi |
fication 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. 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, dependi |
ng on the difficulty of the problem.
Dynamic programming
When a problem shows optimal substructuresmeaning the optimal solution to a problem can be constructed from optimal solutions to subproblemsand 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, FloydWarshall 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 greedy algorithm is similar to a dynamic programming algorithm in that it works by examining substructures, in this case not of the problem but of a given solution. Such algorithms start with some solution, which may be given or have been constructed in some way, and improve it by making small modifications. For some problems they can find the optimal solution while for others they stop at local optima, that is, at solutions that cannot be improved by the algorithm but are not optimum. The most popular use of greedy algorithms is for finding the minimal spanning tree where finding the optimal solution is possible with this method. Huffman Tree, Kruskal, Prim, Sollin are greedy algorithms that can solve this optimization proble |
m.
The heuristic method
In optimization problems, 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 nondeterministic algorithms while others, like tabu search, are deterministic. When a bound on the error of the nonoptimal solution is known, the algorithm is further categorized as an approximation algorithm.
By field of study
Every field of science has its own problems and needs efficient algorithms. Related problems in one field are often studied together. Some example classes are search algo |
rithms, sorting algorithms, merge algorithms, numerical algorithms, graph algorithms, string algorithms, computational geometric algorithms, combinatorial algorithms, medical algorithms, machine learning, cryptography, data compression algorithms and parsing techniques.
Fields tend to overlap with each other, and algorithm advances in one field may improve those of other, sometimes completely unrelated, fields. For example, dynamic programming was invented for optimization of resource consumption in industry but is now used in solving a broad range of problems in many fields.
By complexity
Algorithms can be classified by the amount of time they need to complete compared to their input size
Constant time if the time needed by the algorithm is the same, regardless of the input size. E.g. an access to an array element.
Logarithmic time if the time is a logarithmic function of the input size. E.g. binary search algorithm.
Linear time if the time is proportional to the input size. E.g. the traverse of a lis |
t.
Polynomial time if the time is a power of the input size. E.g. the bubble sort algorithm has quadratic time complexity.
Exponential time if the time is an exponential function of the input size. E.g. Bruteforce search.
Some problems may have multiple algorithms of differing complexity, while other problems might have no algorithms or no known efficient algorithms. There are also mappings from some problems to other problems. Owing to this, it was found to be more suitable to classify the problems themselves instead of the algorithms into equivalence classes based on the complexity of the best possible algorithms for them.
Continuous algorithms
The adjective "continuous" when applied to the word "algorithm" can mean
An algorithm operating on data that represents continuous quantities, even though this data is represented by discrete approximationssuch algorithms are studied in numerical analysis; or
An algorithm in the form of a differential equation that operates continuously on the data, running on |
an analog computer.
Legal issues
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, and hence algorithms are not patentable as in Gottschalk v. Benson. 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 highly controversial, and there are highly criticized patents involving algorithms, especially data compression algorithms, such as Unisys' LZW patent.
Additionally, some cryptographic algorithms have export restrictions see export of cryptography.
History Development of the notion of "algorithm"
Ancient Near East
The earliest evidence of algorithms is found in the Babylonian mathematics of ancient Mesopotamia modern Iraq. A Sumerian clay tablet found |
in Shuruppak near Baghdad and dated to circa 2500 BC described the earliest division algorithm. During the Hammurabi dynasty circa 18001600 BC, 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 circa 1550 BC. 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 c. 300 BC.
Discrete and distinguishable symbols
Tallymarks To keep track of their flocks, their sacks of grain and their money the ancients used tallying accumulating stones or marks scratched on sticks or making discrete symbols in clay |
. Through the Babylonian and Egyptian use of marks and symbols, eventually Roman numerals and the abacus evolved Dilson, p. 1641. Tally marks appear prominently in unary numeral system arithmetic used in Turing machine and PostTuring machine computations.
Manipulation of symbols as "place holders" for numbers algebra
Muhammad ibn Ms alKhwrizm, a Persian mathematician, wrote the Aljabr in the 9th century. The terms "algorism" and "algorithm" are derived from the name alKhwrizm, while the term "algebra" is derived from the book Aljabr. In Europe, the word "algorithm" was originally used to refer to the sets of rules and techniques used by AlKhwarizmi to solve algebraic equations, before later being generalized to refer to any set of rules or techniques. This eventually culminated in Leibniz's notion of the calculus ratiocinator ca 1680
Cryptographic algorithms
The first cryptographic algorithm for deciphering encrypted code was developed by AlKindi, a 9thcentury Arab mathematician, in A Manuscript On Deciph |
ering Cryptographic Messages. He gave the first description of cryptanalysis by frequency analysis, the earliest codebreaking algorithm.
Mechanical contrivances with discrete states
The clock Bolter credits the invention of the weightdriven clock as "The key invention of Europe in the Middle Ages", in particular, the verge escapement that provides us with the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical automata" beginning in the 13th century and finally to "computational machines"the difference engine and analytical engines of Charles Babbage and Countess Ada Lovelace, mid19th century. Lovelace is credited with the first creation of an algorithm intended for processing on a computerBabbage's analytical engine, the first device considered a real Turingcomplete computer instead of just a calculatorand is sometimes called "history's first programmer" as a result, though a full implementation of Babbage's second device would not be realized until decades |
after her lifetime.
Logical machines 1870 Stanley Jevons' "logical abacus" and "logical machine" The technical problem was to reduce Boolean equations when presented in a form similar to what is now known as Karnaugh maps. Jevons 1880 describes first a simple "abacus" of "slips of wood furnished with pins, contrived so that any part or class of the logical combinations can be picked out mechanically ... More recently, however, I have reduced the system to a completely mechanical form, and have thus embodied the whole of the indirect process of inference in what may be called a Logical Machine" His machine came equipped with "certain moveable wooden rods" and "at the foot are 21 keys like those of a piano etc. ...". With this machine he could analyze a "syllogism or any other simple logical argument".
This machine he displayed in 1870 before the Fellows of the Royal Society. Another logician John Venn, however, in his 1881 Symbolic Logic, turned a jaundiced eye to this effort "I have no high estimate myself |
of the interest or importance of what are sometimes called logical machines ... it does not seem to me that any contrivances at present known or likely to be discovered really deserve the name of logical machines"; see more at Algorithm characterizations. But not to be outdone he too presented "a plan somewhat analogous, I apprehend, to Prof. Jevon's abacus ... And again, corresponding to Prof. Jevons's logical machine, the following contrivance may be described. I prefer to call it merely a logicaldiagram machine ... but I suppose that it could do very completely all that can be rationally expected of any logical machine".
Jacquard loom, Hollerith punch cards, telegraphy and telephony the electromechanical relay Bell and Newell 1971 indicate that the Jacquard loom 1801, precursor to Hollerith cards punch cards, 1887, and "telephone switching technologies" were the roots of a tree leading to the development of the first computers. By the mid19th century the telegraph, the precursor of the telephone, was in |
use throughout the world, its discrete and distinguishable encoding of letters as "dots and dashes" a common sound. By the late 19th century the ticker tape ca 1870s was in use, as was the use of Hollerith cards in the 1890 U.S. census. Then came the teleprinter ca. 1910 with its punchedpaper use of Baudot code on tape.
Telephoneswitching networks of electromechanical relays invented 1835 was behind the work of George Stibitz 1937, the inventor of the digital adding device. As he worked 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".
Davis 2000 observes the particular importance of the electromechanical relay with its two "binary states" open and closed
It was only with the development, beginning in the 1930s, of electromechanical calculators using electrical relays, that machines were built having the scope Babbage had e |
nvisioned."
Mathematics during the 19th century up to the mid20th century
Symbols and rules In rapid succession, the mathematics of George Boole 1847, 1854, Gottlob Frege 1879, and Giuseppe Peano 18881889 reduced arithmetic to a sequence of symbols manipulated by rules. Peano's The principles of arithmetic, presented by a new method 1888 was "the first attempt at an axiomatization of mathematics in a symbolic language".
But Heijenoort gives Frege 1879 this kudos Frege's is "perhaps the most important single work ever written in logic. ... in which we see a " 'formula language', that is a lingua characterica, a language written with special symbols, "for pure thought", that is, free from rhetorical embellishments ... constructed from specific symbols that are manipulated according to definite rules". The work of Frege was further simplified and amplified by Alfred North Whitehead and Bertrand Russell in their Principia Mathematica 19101913.
The paradoxes At the same time a number of disturbing paradoxes ap |
peared in the literature, in particular, the BuraliForti paradox 1897, the Russell paradox 190203, and the Richard Paradox. The resultant considerations led to Kurt Gdel's paper 1931he specifically cites the paradox of the liarthat completely reduces rules of recursion to numbers.
Effective calculability In an effort to solve the Entscheidungsproblem defined precisely by Hilbert in 1928, mathematicians first set about to define what was meant by an "effective method" or "effective calculation" or "effective calculability" i.e., a calculation that would succeed. In rapid succession the following appeared Alonzo Church, Stephen Kleene and J.B. Rosser's calculus a finely honed definition of "general recursion" from the work of Gdel acting on suggestions of Jacques Herbrand cf. Gdel's Princeton lectures of 1934 and subsequent simplifications by Kleene. Church's proof that the Entscheidungsproblem was unsolvable, Emil Post's definition of effective calculability as a worker mindlessly following a list of instruct |
ions to move left or right through a sequence of rooms and while there either mark or erase a paper or observe the paper and make a yesno decision about the next instruction. Alan Turing's proof of that the Entscheidungsproblem was unsolvable by use of his "a automatic machine"in effect almost identical to Post's "formulation", J. Barkley Rosser's definition of "effective method" in terms of "a machine". Kleene's proposal of a precursor to "Church thesis" that he called "Thesis I", and a few years later Kleene's renaming his Thesis "Church's Thesis" and proposing "Turing's Thesis".
Emil Post 1936 and Alan Turing 193637, 1939
Emil Post 1936 described the actions of a "computer" human being as follows
"...two concepts are involved that of a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions.
His symbol space would be
"a twoway infinite sequence of spaces or boxes... The problem solver or worker is to move and work in this symbol spac |
e, being capable of being in, and operating in but one box at a time.... a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke.
"One box is to be singled out and called the starting point. ...a specific problem is to be given in symbolic form by a finite number of boxes i.e., INPUT being marked with a stroke. Likewise, the answer i.e., OUTPUT is to be given in symbolic form by such a configuration of marked boxes...
"A set of directions applicable to a general problem sets up a deterministic process when applied to each specific problem. This process terminates only when it comes to the direction of type C i.e., STOP". See more at PostTuring machine
Alan Turing's work preceded that of Stibitz 1937; it is unknown whether Stibitz knew of the work of Turing. Turing's biographer believed that Turing's use of a typewriterlike model derived from a youthful interest "Alan had dreamt of inventing typewriters as a boy; Mrs. Turing had |
a typewriter, and he could well have begun by asking himself what was meant by calling a typewriter 'mechanical'". Given the prevalence of Morse code and telegraphy, ticker tape machines, and teletypewriters we might conjecture that all were influences.
Turinghis model of computation is now called a Turing machinebegins, as did Post, with an analysis of a human computer that he whittles down to a simple set of basic motions and "states of mind". But he continues a step further and creates a machine as a model of computation of numbers.
"Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book...I assume then that the computation is carried out on onedimensional paper, i.e., on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite...
"The behavior of the computer at any moment is determined by the symbols which he is observing, and his "state of mind" at that moment. |
We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite...
"Let us imagine that the operations performed by the computer to be split up into 'simple operations' which are so elementary that it is not easy to imagine them further divided."
Turing's reduction yields the following
"The simple operations must therefore include
"a Changes of the symbol on one of the observed squares
"b Changes of one of the squares observed to another square within L squares of one of the previously observed squares.
"It may be that some of these change necessarily invoke a change of state of mind. The most general single operation must, therefore, be taken to be one of the following
"A A possible change a of symbol together with a possible change of state of mind.
"B A possible change b of observed sq |
uares, together with a possible change of state of mind"
"We may now construct a machine to do the work of this computer."
A few years later, Turing expanded his analysis thesis, definition with this forceful expression of it
"A function is said to be "effectively calculable" if its values can be found by some purely mechanical process. Though it is fairly easy to get an intuitive grasp of this idea, it is nevertheless desirable to have some more definite, mathematical expressible definition ... he discusses the history of the definition pretty much as presented above with respect to Gdel, Herbrand, Kleene, Church, Turing, and Post ... We may take this statement literally, understanding by a purely mechanical process one which could be carried out by a machine. It is possible to give a mathematical description, in a certain normal form, of the structures of these machines. The development of these ideas leads to the author's definition of a computable function, and to an identification of computability wit |
h effective calculability ... .
" We shall use the expression "computable function" to mean a function calculable by a machine, and we let "effectively calculable" refer to the intuitive idea without particular identification with any one of these definitions".
J.B. Rosser 1939 and S.C. Kleene 1943
J. Barkley Rosser defined an 'effective mathematical method' in the following manner italicization added
"'Effective method' is used here in the rather special sense of a method each step of which is precisely determined and which is certain to produce the answer in a finite number of steps. With this special meaning, three different precise definitions have been given to date. his footnote 5; see discussion immediately below. The simplest of these to state due to Post and Turing says essentially that an effective method of solving certain sets of problems exists if one can build a machine which will then solve any problem of the set with no human intervention beyond inserting the question and later reading the a |
nswer. All three definitions are equivalent, so it doesn't matter which one is used. Moreover, the fact that all three are equivalent is a very strong argument for the correctness of any one." Rosser 1939225226
Rosser's footnote No. 5 references the work of 1 Church and Kleene and their definition of definability, in particular, Church's use of it in his An Unsolvable Problem of Elementary Number Theory 1936; 2 Herbrand and Gdel and their use of recursion, in particular, Gdel's use in his famous paper On Formally Undecidable Propositions of Principia Mathematica and Related Systems I 1931; and 3 Post 1936 and Turing 193637 in their mechanismmodels of computation.
Stephen C. Kleene defined as his nowfamous "Thesis I" known as the ChurchTuring thesis. But he did this in the following context boldface in original
"12. Algorithmic theories... In setting up a complete algorithmic theory, what we do is to describe a procedure, performable for each set of values of the independent variables, which procedure necess |
arily terminates and in such manner that from the outcome we can read a definite answer, "yes" or "no," to the question, "is the predicate value true?"" Kleene 1943273
History after 1950
A number of efforts have been directed toward further refinement of the definition of "algorithm", and activity is ongoing because of issues surrounding, in particular, foundations of mathematics especially the ChurchTuring thesis and philosophy of mind especially arguments about artificial intelligence. For more, see Algorithm characterizations.
See also
Abstract machine
Algorithm engineering
Algorithm characterizations
Algorithmic bias
Algorithmic composition
Algorithmic entities
Algorithmic synthesis
Algorithmic technique
Algorithmic topology
Garbage in, garbage out
Introduction to Algorithms textbook
List of algorithms
List of algorithm general topics
List of important publications in theoretical computer science Algorithms
Regulation of algorithms
Theory of computation
Computability theory
Computa |
tional complexity theory
Notes
Bibliography
Bell, C. Gordon and Newell, Allen 1971, Computer Structures Readings and Examples, McGrawHill Book Company, New York. .
Includes an excellent bibliography of 56 references.
,
cf. Chapter 3 Turing machines where they discuss "certain enumerable sets not effectively mechanically enumerable".
Campagnolo, M.L., Moore, C., and Costa, J.F. 2000 An analog characterization of the subrecursive functions. In Proc. of the 4th Conference on Real Numbers and Computers, Odense University, pp. 91109
Reprinted in The Undecidable, p. 89ff. The first expression of "Church's Thesis". See in particular page 100 The Undecidable where he defines the notion of "effective calculability" in terms of "an algorithm", and he uses the word "terminates", etc.
Reprinted in The Undecidable, p. 110ff. Church shows that the Entscheidungsproblem is unsolvable in about 3 pages of text and 3 pages of footnotes.
Davis gives commentary before each article. Papers of Gdel, Alonzo C |
hurch, Turing, Rosser, Kleene, and Emil Post are included; those cited in the article are listed here by author's name.
Davis offers concise biographies of Leibniz, Boole, Frege, Cantor, Hilbert, Gdel and Turing with von Neumann as the showstealing villain. Very brief bios of JosephMarie Jacquard, Babbage, Ada Lovelace, Claude Shannon, Howard Aiken, etc.
,
Yuri Gurevich, Sequential Abstract State Machines Capture Sequential Algorithms, ACM Transactions on Computational Logic, Vol 1, no 1 July 2000, pp. 77111. Includes bibliography of 33 sources.
, 3rd edition 1976?, pbk.
, . Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.
Presented to the American Mathematical Society, September 1935. Reprinted in The Undecidable, p. 237ff. Kleene's definition of "general recursion" known now as murecursion 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 res |
ult.
Reprinted in The Undecidable, p. 255ff. Kleene refined his definition of "general recursion" and proceeded in his chapter "12. Algorithmic theories" to posit "Thesis I" p. 274; he would later repeat this thesis in Kleene 1952300 and name it "Church's Thesis"Kleene 1952317 i.e., the Church thesis.
Kosovsky, N.K. Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms, LSU Publ., Leningrad, 1981
A.A. Markov 1954 Theory of algorithms. Translated by Jacques J. SchorrKon and PST staff Imprint Moscow, Academy of Sciences of the USSR, 1954 i.e., Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title Teoriya algerifmov. QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS .
|
Minsky expands his "...idea of an algorithm an effective procedure..." in chapter 5.1 Computability, Effective Procedures and Algorithms. Infinite machines.
Reprinted in The Undecidable, pp. 289ff. Post defines a simple algorithmiclike process of a man writing marks or erasing marks and going from box to box and eventually halting, as he follows a list of simple instructions. This is cited by Kleene as one source of his "Thesis I", the socalled ChurchTuring thesis.
Reprinted in The Undecidable, p. 223ff. Herein is Rosser's famous definition of "effective method" "...a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps... a machine which will then solve any problem of the set with no human intervention beyond inserting the question and later reading the answer" p. 225226, The Undecidable
Cf. in particular the first chapter titled Algorithms, Turing Machines, and Programs. His succinct informal definition "...any sequenc |
e of instructions that can be obeyed by a robot, is called an algorithm" p. 4.
. Corrections, ibid, vol. 431937 pp. 544546. Reprinted in The Undecidable, p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.
Reprinted in The Undecidable, pp. 155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton.
United States Patent and Trademark Office 2006, 2106.02 Mathematical Algorithms 2100 Patentability, Manual of Patent Examining Procedure MPEP. Latest revision August 2006
Further reading
Knuth, Donald E. 2000. Selected Papers on Analysis of Algorithms. Stanford, California Center for the Study of Language and Information.
Knuth, Donald E. 2010. Selected Papers on Design of Algorithms. Stanford, California Center for the Study of Language and Information.
External links
Dictionary of Algorithms and Data Structures National Institute of Standards and Technology
Algorithm repositories
The Stony Brook Alg |
orithm Repository State University of New York at Stony Brook
Collected Algorithms of the ACM Association for Computing Machinery
The Stanford GraphBase Stanford University
Articles with example pseudocode
Mathematical logic
Theoretical computer science |
An annual plant is a plant that completes its life cycle, from germination to the production of seeds, within one growing season, and then dies. The length of growing seasons and period in which they take place vary according to geographical location, and may not correspond to the four traditional seasonal divisions of the year. With respect to the traditional seasons, annual plants are generally categorized into summer annuals and winter annuals. Summer annuals germinate during spring or early summer and mature by autumn of the same year. Winter annuals germinate during the autumn and mature during the spring or summer of the following calendar year.
One seedtoseed life cycle for an annual can occur in as little as a month in some species, though most last several months. Oilseed rapa can go from seedtoseed in about five weeks under a bank of fluorescent lamps. This style of growing is often used in classrooms for education. Many desert annuals are therophytes, because their seedtoseed life cycle is only we |
eks and they spend most of the year as seeds to survive dry conditions.
Cultivation
In cultivation, many food plants are, or are grown as, annuals, including virtually all domesticated grains. Some perennials and biennials are grown in gardens as annuals for convenience, particularly if they are not considered cold hardy for the local climate. Carrot, celery and parsley are true biennials that are usually grown as annual crops for their edible roots, petioles and leaves, respectively. Tomato, sweet potato and bell pepper are tender perennials usually grown as annuals. Ornamental perennials commonly grown as annuals are impatiens, mirabilis, wax begonia, snapdragon, pelargonium, coleus and petunia. Examples of true annuals include corn, wheat, rice, lettuce, peas, watermelon, beans, zinnia and marigold.
Summer
Summer annuals sprout, flower, produce seed, and die, during the warmer months of the year.
The lawn weed crabgrass is a summer annual.
Winter
Winter annuals germinate in autumn or winter, live thr |
ough the winter, then bloom in winter or spring.
The plants grow and bloom during the cool season when most other plants are dormant or other annuals are in seed form waiting for warmer weather to germinate. Winter annuals die after flowering and setting seed. The seeds germinate in the autumn or winter when the soil temperature is cool.
Winter annuals typically grow low to the ground, where they are usually sheltered from the coldest nights by snow cover, and make use of warm periods in winter for growth when the snow melts. Some common winter annuals include henbit, deadnettle, chickweed, and winter cress.
Winter annuals are important ecologically, as they provide vegetative cover that prevents soil erosion during winter and early spring when no other cover exists and they provide fresh vegetation for animals and birds that feed on them. Although they are often considered to be weeds in gardens, this viewpoint is not always necessary, as most of them die when the soil temperature warms up again in early |
to late spring when other plants are still dormant and have not yet leafed out.
Even though they do not compete directly with cultivated plants, sometimes winter annuals are considered a pest in commercial agriculture, because they can be hosts for insect pests or fungal diseases such as ovary smut, Microbotryum sp. which attack crops being cultivated. The property that they prevent the soil from drying out can also be problematic for commercial agriculture.
Molecular genetics
In 2008, it was discovered that the inactivation of only two genes in one species of annual plant leads to its conversion into a perennial plant. Researchers deactivated the SOC1 and FUL genes which control flowering time of Arabidopsis thaliana. This switch established phenotypes common in perennial plants, such as wood formation.
See also
References
External links
Garden plants |
The anthophytes are a grouping of plant taxa bearing flowerlike reproductive structures. They were formerly thought to be a clade comprising plants bearing flowerlike structures. The group contained the angiosperms the extant flowering plants, such as roses and grasses as well as the Gnetales and the extinct Bennettitales.
Detailed morphological and molecular studies have shown that the group is not actually monophyletic, with proposed floral homologies of the gnetophytes and the angiosperms having evolved in parallel. This makes it easier to reconcile molecular clock data that suggests that the angiosperms diverged from the gymnosperms around 320300 mya.
Some more recent studies have used the word anthophyte to describe a group which includes the angiosperms and a variety of fossils glossopterids, Pentoxylon, Bennettitales, and Caytonia, but not the Gnetales.
References
Historically recognized plant taxa |
An atlas is a collection of maps, originally named after the Ancient Greek deity.
Atlas may also refer to
Mythology
Atlas mythology, an Ancient Greek Titanic deity who held up the celestial sphere
Atlas, the first legendary king of Atlantis and further variant of the mythical Titan
Atlas of Mauretania, a legendary king of Mauretania and variant of the mythical Titan
Places
United States
Atlas, California
Atlas, Illinois
Atlas, Texas
Atlas, West Virginia
Atlas, Wisconsin
Atlas District, an area in Washington, D.C.
Atlas Peak AVA, a California wine region
Atlas Township, Michigan
Other
Atlas Cinema, a historic movie theatre in Istanbul, Turkey
Atlas Mountains, a set of mountain ranges in northwestern Africa
Atlas, Nilfer, a village in Nilfer district of Bursa Province, Turkey
People with the name
Atlas graffiti artist, American graffiti artist
Atlas DaBone, American wrestler and football player
Charles Atlas 18921972, ItalianAmerican bodybuilder
Charles Atlas artist
David Atlas born 19 |
24, American meteorologist who pioneered weather radar
James Atlas 19492019, American writer, editor and publisher
Meir Atlas 18481926, Lithuanian rabbi
Natacha Atlas born 1964, Belgian singer
Nava Atlas, American book artist and author
Omar Atlas born 1938, former Venezuelan professional wrestler
Scott Atlas born 1955, American conservative health care policy advisor
Teddy Atlas born 1956, American boxing trainer and commentator
Tony Atlas born 1954, American wrestler and bodybuilder
Arts, entertainment, and media
Comics
Atlas Drawn and Quarterly, a comic book series by Dylan Horrocks
Agents of Atlas, a Marvel Comics miniseries
Atlas Comics 1950s, a publisher
AtlasSeaboard Comics, a 1970s line of comics
Fictional characters
Atlas DC Comics, the name of several of DC Comics' fictional characters, comic book superheroes, and deities
Atlas Teen Titans, Teen Titans character
Atlas, an Astro Boy character
Atlas, a BioShock character
Atlas, a BattleMech in the BattleTech universe
Atlas, an ant |
agonist in Mega Man ZX Advent
Atlas, a Portal 2 character
Atlas, a PS238 character
Erik Josten, a.k.a. Atlas, a Marvel Comics supervillain
The Atlas, a strong driving force from No Man's Sky
Literature
Atlas, a photography book by Gerhard Richter
ATLAS of Finite Groups, a group theory book
Atlas Shrugged, a novel by Ayn Rand
The Atlas novel, by William T. Vollmann
Music
Groups
Atlas band, a New Zealand rock band
Atlas Sound, the solo musical project of Bradford Cox, lead singer and guitarist of the indie rock band Deerhunter
Musicians
Black Atlass, a Canadian musician
Albums
Atlas Kinky album
Atlas Parkway Drive album, Parkway Drive's fourth album
Atlas Real Estate album
Atlas RFS album
Operas
Atlas opera, 1991 opera by Meredith Monk
Atlas An Opera in Three Parts, 1993 recording of Monk's opera
Songs
"Atlas" Battles song, 2007 song by Battles on the album Mirrored
"Atlas" Coldplay song, 2013 song by Coldplay from The Hunger Games Catching Fire soundtrack
"Atlas", a song by Caligula's |
Horse from the album The Tide, the Thief River's End
"Atlas", the titular song from Parkway Drive's fourth album
"Atlas", a song by Man Overboard from Man Overboard
"Atlas", a song by Jake Chudnow used as main theme in the YouTube series Mind Field
Periodicals
Atlas magazine
The Atlas, a newspaper published in England from 1826 to 1869
Other uses in arts, entertainment, and media
Atlas film
Atlas statue, iconic statue by Lee Lawrie in Rockefeller Center
Atlas, a book about flora andor fauna of a region, such as atlases of the flora and fauna of Britain and Ireland
Atlas Entertainment, a film production company
Atlas folio, a book size
Atlas Media Corp., a nonfiction entertainment company
Atlas Press, a UK publisher
RTV Atlas, a broadcaster in Montenegro
Atlas Sound, a solo musical project by Bradford Cox
The Atlas video game, a 1991 multiplatform strategy video game
Atlas video game, an upcoming massivelymultiplayer online video game
Atlas Corporation, a fictional arms manufacturer in the |
video game series Borderlands series
Brands and enterprises
Atlas appliance company, a Belarusian company
Atlas Consortium, a group of technology companies
Atlas Copco, Swedish company founded in 1873
Atlas Corporation, an investment company
Atlas Elektronik, a German navalmarine electronics and systems business
Atlas Group, a Pakistani business group
Atlas Mara Limited, formerly Atlas Mara CoNvest Limited, a financial holding company that owns banks in Africa
Atlas Model Railroad, American maker of model trains and accessories
Atlas Network, formerly Atlas Economic Research Foundation
Atlas Press tool company
Atlas Solutions, a subsidiary of Facebook for digital online advertising, formerly owned by Microsoft
Atlas Telecom, a worldwide communications company
Atlas Van Lines, a moving company
AtlasImperial, an American diesel engine manufacturer
Dresser Atlas, a provider of oilfield and factory automation services
Tele Atlas, a Dutch mapping company
Western Atlas, an oilfield services compan |
y
Computing and technology
Atlas computer, an early supercomputer, built in the 1960s
Atlas robot, a humanoid robot developed by Boston Dynamics and DARPA
ATLAS software, a software flagging naturalized American for denaturalization
Atlas, a computer used at the Lawrence Livermore National Laboratory in 2006
Abbreviated Test Language for All Systems, or ATLAS, a MILSPEC language for avionics equipment testing
Advanced Technology Leisure Application Simulator, or ATLAS, a hydraulic motion simulator used in theme parks
ASP.NET AJAX formerly "Atlas", a set of ASP.NET extensions
ATLAS Transformation Language, programming language
Atlas.ti, a qualitative analysis program
Automatically Tuned Linear Algebra Software, or ATLAS,
Texture atlas, or image sprite sheet
UNIVAC 1101, an early American computer, built in the 1950s
Science
Astronomy
Atlas comet C2019 Y4
Atlas crater on the near side of the Moon
Atlas moon, a satellite of Saturn
Atlas star, also designated 27 Tauri, a triple star system in |
the constellation of Taurus and a member of the Pleiades
Advanced Technology LargeAperture Space Telescope ATLAST
Advanced Topographic Laser Altimeter System ATLAS, a spacebased lidar instrument on ICESat2
Asteroid Terrestrialimpact Last Alert System ATLAS
Mathematics
Atlas manifolds, a set of smooth charts
Atlas topology, a set of charts
Smooth atlas
Physics
Argonne Tandem Linear Accelerator System, or ATLAS, a linear accelerator at the Argonne National Laboratory
ATLAS experiment, a particle detector for the Large Hadron Collider at CERN
Atomicterrace lowangle shadowing, or ATLAS, a nanofabrication technique
Biology and healthcare
Atlas anatomy, part of the spine
Atlas personality, a term used in psychology to describe the personality of someone whose childhood was characterized by excessive responsibilities
Brain atlas, a neuroanatomical map of the brain of a human or other animal
Animals and plants
Atlas bear
Atlas beetle
Atlas cedar
Atlas moth
Atlas pied flycatcher, a bird
Atlas tur |
tle
Sport
Atlas Delmenhorst, a German association football club
Atlas F.C., a Mexican professional football club
Club Atltico Atlas, an Argentine amateur football club
KK Atlas, a former men's professional basketball club based in Belgrade today's Serbia
Transport
Aerospace
Atlas rocket family
SM65 Atlas intercontinental ballistic missile ICBM
AeroVelo Atlas, a humanpowered helicopter
Airbus A400M Atlas, a military aircraft produced 2007present
Armstrong Whitworth Atlas, a British military aeroplane produced 19271933
Atlas Air, an American cargo airline
Atlas Aircraft, a 1940s aircraft manufacturer
Atlas Aircraft Corporation, a South African military aircraft manufacturer
Atlas Aviation, an aircraft maintenance firm
Atlas Blue, a Moroccan lowcost airline
Atlasjet, a Turkish airline
Birdman Atlas, an ultralight aircraft
HMLAT303, U.S. Marine Corps helicopter training squadron
La Mouette Atlas, a French hang glider design
Automotive
Atlas 1951 automobile, a French minicar
Atlas light tru |
cks, a Greek motor vehicle manufacturer
Atlas Pittsburgh automobile, produced 19061907
Atlas Springfield automobile, produced 19071913
Atlas, a British van by the Standard Motor Company produced 19581962
Atlas Drop Forge Company, a parts subsidiary of REO Motor Car Company
Atlas Motor Buggy, an American highwheeler produced in 1909
General Motors Atlas engine
Honda Atlas Cars Pakistan, a Pakistani car manufacturer
Nissan Atlas, a Japanese light truck
Volkswagen Atlas, a sport utility vehicle
Geely Atlas, a sport utility vehicle
Ships and boats
Atlas Werke, a former German shipbuilding company
, the name of several Royal Navy ships
ST Atlas, a Swedish tugboat
, the name of several U.S. Navy ships
Trains
Atlas, an 18631885 South Devon Railway Dido class locomotive
Atlas, a 19271962 LMS Royal Scot Class locomotive
Atlas Car and Manufacturing Company, a locomotive manufacturer
Atlas Model Railroad
Other uses
Atlas architecture
ATLAS simulation Army Tactical Level Advanced Simulation, a Thai |
military system
Atlas storm, which hit the Midwestern United States in October 2013, named by The Weather Channel
Agrupacin de Trabajadores Latinoamericanos Sindicalistas, or ATLAS, a former Latin American trade union confederation in the early 1950s
Atlas languages, Berber languages spoken in the Atlas Mountains of Morocco
ATLAS Network, a network of European special police units
Atlas Uranium Mill
See also
Altas disambiguation |
Mouthwash, mouth rinse, oral rinse, or mouth bath is a liquid which is held in the mouth passively or swilled around the mouth by contraction of the perioral muscles andor movement of the head, and may be gargled, where the head is tilted back and the liquid bubbled at the back of the mouth.
Usually mouthwashes are antiseptic solutions intended to reduce the microbial load in the mouth, although other mouthwashes might be given for other reasons such as for their analgesic, antiinflammatory or antifungal action. Additionally, some rinses act as saliva substitutes to neutralize acid and keep the mouth moist in xerostomia dry mouth. Cosmetic mouthrinses temporarily control or reduce bad breath and leave the mouth with a pleasant taste.
Rinsing with water or mouthwash after brushing with a fluoride toothpaste can reduce the availability of salivary fluoride. This can lower the anticavity remineralization and antibacterial effects of fluoride. Fluoridated mouthwash may mitigate this effect or in high concentrat |
ions increase available fluoride, but is not as cost effective as leaving the fluoride toothpaste on the teeth after brushing. A group of experts discussing post brushing rinsing in 2012 found that although there was clear guidance given in many public health advice publications to "spit, avoid rinsing with waterexcessive rinsing with water" they believed there was a limited evidence base for best practice.
Use
Common use involves rinsing the mouth with about 2050 ml 23 fl oz of mouthwash. The wash is typically swished or gargled for about half a minute and then spat out. Most companies suggest not drinking water immediately after using mouthwash. In some brands, the expectorate is stained, so that one can see the bacteria and debris.
Mouthwash should not be used immediately after brushing the teeth so as not to wash away the beneficial fluoride residue left from the toothpaste. Similarly, the mouth should not be rinsed out with water after brushing. Patients were told to "spit don't rinse" after toothbrushi |
ng as part of a National Health Service campaign in the UK. A fluoride mouthrinse can be used at a different time of the day to brushing.
Gargling is where the head is tilted back, allowing the mouthwash to sit in the back of the mouth while exhaling, causing the liquid to bubble. Gargling is practiced in Japan for perceived prevention of viral infection. One commonly used way is with infusions or tea. In some cultures, gargling is usually done in private, typically in a bathroom at a sink so the liquid can be rinsed away.
Effects
The mostcommonlyused mouthwashes are commercial antiseptics, which are used at home as part of an oral hygiene routine. Mouthwashes combine ingredients to treat a variety of oral conditions. Variations are common, and mouthwash has no standard formulation, so its use and recommendation involves concerns about patient safety. Some manufacturers of mouthwash state that their antiseptic and antiplaque mouthwashes kill the bacterial plaque that causes cavities, gingivitis, and bad bre |
ath. It is, however, generally agreed that the use of mouthwash does not eliminate the need for both brushing and flossing. The American Dental Association asserts that regular brushing and proper flossing are enough in most cases, in addition to regular dental checkups, although they approve many mouthwashes.
For many patients, however, the mechanical methods could be tedious and timeconsuming, and, additionally, some local conditions may render them especially difficult. Chemotherapeutic agents, including mouthwashes, could have a key role as adjuncts to daily home care, preventing and controlling supragingival plaque, gingivitis and oral malodor.
Minor and transient side effects of mouthwashes are very common, such as taste disturbance, tooth staining, sensation of a dry mouth, etc. Alcoholcontaining mouthwashes may make dry mouth and halitosis worse, as they dry out the mouth. Soreness, ulceration and redness may sometimes occur e.g., aphthous stomatitis or allergic contact stomatitis if the person is al |
lergic or sensitive to mouthwash ingredients, such as preservatives, coloring, flavors and fragrances. Such effects might be reduced or eliminated by diluting the mouthwash with water, using a different mouthwash e.g. saltwater, or foregoing mouthwash entirely.
Prescription mouthwashes are used prior to and after oral surgery procedures, such as tooth extraction, or to treat the pain associated with mucositis caused by radiation therapy or chemotherapy. They are also prescribed for aphthous ulcers, other oral ulcers, and other mouth pain. "Magic mouthwashes" are prescription mouthwashes compounded in a pharmacy from a list of ingredients specified by a doctor. Despite a lack of evidence that prescription mouthwashes are more effective in decreasing the pain of oral lesions, many patients and prescribers continue to use them. There has been only one controlled study to evaluate the efficacy of magic mouthwash; it shows no difference in efficacy between the most common magicmouthwash formulation, on the one ha |
nd, and commercial mouthwashes such as chlorhexidine or a salinebaking soda solution, on the other. Current guidelines suggest that saline solution is just as effective as magic mouthwash in pain relief and in shortening the healing time of oral mucositis from cancer therapies.
History
The first known references to mouth rinsing is in Ayurveda for treatment of gingivitis. Later, in the Greek and Roman periods, mouth rinsing following mechanical cleansing became common among the upper classes, and Hippocrates recommended a mixture of salt, alum, and vinegar. The Jewish Talmud, dating back about 1,800 years, suggests a cure for gum ailments containing "dough water" and olive oil.
Before Europeans came to the Americas, Native North American and Mesoamerican cultures used mouthwashes, often made from plants such as Coptis trifolia. Indeed, Aztec dentistry was more advanced than European dentistry of the age. Peoples of the Americas used salt water mouthwashes for sore throats, and other mouthwashes for proble |
ms such as teething and mouth ulcers.
Anton van Leeuwenhoek, the famous 17th century microscopist, discovered living organisms living, because they were mobile in deposits on the teeth what we now call dental plaque. He also found organisms in water from the canal next to his home in Delft. He experimented with samples by adding vinegar or brandy and found that this resulted in the immediate immobilization or killing of the organisms suspended in water. Next he tried rinsing the mouth of himself and somebody else with a mouthwash containing vinegar or brandy and found that living organisms remained in the dental plaque. He concludedcorrectlythat the mouthwash either did not reach, or was not present long enough, to kill the plaque organisms.
In 1892, German Richard Seifert invented mouthwash product Odol, which was produced by company founder Karl August Lingner 18611916 in Dresden.
That remained the state of affairs until the late 1960s when Harald Loe at the time a professor at the Royal Dental College in |
Aarhus, Denmark demonstrated that a chlorhexidine compound could prevent the buildup of dental plaque. The reason for chlorhexidine's effectiveness is that it strongly adheres to surfaces in the mouth and thus remains present in effective concentrations for many hours.
Since then commercial interest in mouthwashes has been intense and several newer products claim effectiveness in reducing the buildup in dental plaque and the associated severity of gingivitis, in addition to fighting bad breath. Many of these solutions aim to control the Volatile Sulfur Compound VSCcreating anaerobic bacteria that live in the mouth and excrete substances that lead to bad breath and unpleasant mouth taste. For example, the number of mouthwash variants in the United States of America has grown from 15 1970 to 66 1998 to 113 2012.
Research
Research in the field of microbiotas shows that only a limited set of microbes cause tooth decay, with most of the bacteria in the human mouth being harmless. Focused attention on cavitycaus |
ing bacteria such as Streptococcus mutans has led research into new mouthwash treatments that prevent these bacteria from initially growing. While current mouthwash treatments must be used with a degree of frequency to prevent this bacteria from regrowing, future treatments could provide a viable longterm solution.
Ingredients
Alcohol
Alcohol is added to mouthwash not to destroy bacteria but to act as a carrier agent for essential active ingredients such as menthol, eucalyptol and thymol, which help to penetrate plaque. Sometimes a significant amount of alcohol up to 27 vol is added, as a carrier for the flavor, to provide "bite". Because of the alcohol content, it is possible to fail a breathalyzer test after rinsing, although breath alcohol levels return to normal after 10 minutes. In addition, alcohol is a drying agent, which encourages bacterial activity in the mouth, releasing more malodorous volatile sulfur compounds. Therefore, alcoholcontaining mouthwash may temporarily worsen halitosis in those wh |
o already have it, or, indeed, be the sole cause of halitosis in other individuals.
It is hypothesized that alcohol in mouthwashes acts as a carcinogen cancerinducing agent. Generally, there is no scientific consensus about this. One review stated
The same researchers also state that the risk of acquiring oral cancer rises almost five times for users of alcoholcontaining mouthwash who neither smoke nor drink with a higher rate of increase for those who do. In addition, the authors highlight side effects from several mainstream mouthwashes that included dental erosion and accidental poisoning of children. The review garnered media attention and conflicting opinions from other researchers. Yinka Ebo of Cancer Research UK disputed the findings, concluding that "there is still not enough evidence to suggest that using mouthwash that contains alcohol will increase the risk of mouth cancer". Studies conducted in 1985, 1995, 2003, and 2012 did not support an association between alcoholcontaining mouth rinses and o |
ral cancer. Andrew Penman, chief executive of The Cancer Council New South Wales, called for further research on the matter. In a March 2009 brief, the American Dental Association said "the available evidence does not support a connection between oral cancer and alcoholcontaining mouthrinse". Many newer brands of mouthwash are alcohol free, not just in response to consumer concerns about oral cancer, but also to cater for religious groups who abstain from alcohol consumption.
Benzydamine analgesic
In painful oral conditions such as aphthous stomatitis, analgesic mouthrinses e.g. benzydamine mouthwash, or "Difflam" are sometimes used to ease pain, commonly used before meals to reduce discomfort while eating.
Benzoic acid
Benzoic acid acts as a buffer.
Betamethasone
Betamethasone is sometimes used as an antiinflammatory, corticosteroid mouthwash. It may be used for severe inflammatory conditions of the oral mucosa such as the severe forms of aphthous stomatitis.
Cetylpyridinium chloride antiseptic, antimalo |
dor
Cetylpyridinium chloride containing mouthwash e.g. 0.05 is used in some specialized mouthwashes for halitosis. Cetylpyridinium chloride mouthwash has less antiplaque effect than chlorhexidine and may cause staining of teeth, or sometimes an oral burning sensation or ulceration.
Chlorhexidine digluconate and hexetidine antiseptic
Chlorhexidine digluconate is a chemical antiseptic and is used in a 0.120.2 solution as a mouthwash. However, there is no evidence to support that higher concentrations are more effective in controlling dental plaque and gingivitis. It has antiplaque action, but also some antifungal action. It is especially effective against Gramnegative rods. The proportion of Gramnegative rods increase as gingivitis develops, so it is also used to reduce gingivitis. It is sometimes used as an adjunct to prevent dental caries and to treat gingivitis periodontal disease, although it does not penetrate into periodontal pockets well. Chlorhexidine mouthwash alone is unable to prevent plaque, so it |
is not a substitute for regular toothbrushing and flossing. Instead, chlorhexidine mouthwash is more effective when used as an adjunctive treatment with toothbrushing and flossing. In the short term, if toothbrushing is impossible due to pain, as may occur in primary herpetic gingivostomatitis, chlorhexidine mouthwash is used as a temporary substitute for other oral hygiene measures. It is not suited for use in acute necrotizing ulcerative gingivitis, however. Rinsing with chlorhexidine mouthwash before a tooth extraction reduces the risk of a dry socket, a painful condition where the blood clot is lost from an extraction socket and bone is exposed to the oral cavity. Other uses of chlorhexidine mouthwash include prevention of oral candidiasis in immunocompromised persons, treatment of denturerelated stomatitis, mucosal ulcerationerosions and oral mucosal lesions, general burning sensation and many other uses.
Chlorhexidine has good substantivity the ability of a mouthwash to bind to hard and soft tissues in |
the mouth. However, chlorhexidine binds to tannins, meaning that prolonged use in persons who consume coffee, tea or red wine is associated with extrinsic staining i.e. removable staining of teeth. Chlorhexidine mouthwash can also cause taste disturbance or alteration. Chlorhexidine is rarely associated with other issues like overgrowth of enterobacteria in persons with leukemia, desquamation and irritation of oral mucosa, salivary gland pain and swelling, and hypersensitivity reactions including anaphylaxis. A randomized clinical trial conducted in Rabat University in Morocco found better results in plaque inhibition when chlorohexidine with alcohol base 0.12 was used, when compared to an alcoholfree 0.1 chlorhexidine mouthrinse. Chlorhexidine mouthwashes increase staining of teeth over a period of time. However, many publications, and, in recent times, also a systematic review van Swaaij 2020 revealed that an AntiDiscoloration System ADS based on Lascorbic acid and sodium metabisulfite is able to reduce to |
oth staining without affecting the antibacterial effect of chlorhexidine.
Hexetidine also has antiplaque, analgesic, astringent and antimalodor properties, but is considered an inferior alternative to chlorhexidine.
Edible oils
In traditional Ayurvedic medicine, the use of oil mouthwashes is called "Kavala" "oil swishing" or "Gandusha", and this practice has more recently been remarketed by the complementary and alternative medicine industry as "oil pulling". Its promoters claim it works by "pulling out" "toxins", which are known as ama in Ayurvedic medicine, and thereby reducing inflammation. Ayurvedic literature claims that oil pulling is capable of improving oral and systemic health, including a benefit in conditions such as headaches, migraines, diabetes mellitus, asthma, and acne, as well as whitening teeth.
Oil pulling has received little study and there is little evidence to support claims made by the technique's advocates. When compared with chlorhexidine in one small study, it was found to be les |
s effective at reducing oral bacterial load, and the other health claims of oil pulling have failed scientific verification or have not been investigated. There is a report of lipid pneumonia caused by accidental inhalation of the oil during oil pulling.
The mouth is rinsed with approximately one tablespoon of oil for 1020 minutes then spat out. Sesame oil, coconut oil and ghee are traditionally used, but newer oils such as sunflower oil are also used.
Essential oils
Phenolic compounds and monoterpenes include essential oil constituents that have some antibacterial properties, such as eucalyptol, eugenol, hinokitiol, menthol, phenol, or thymol.
Essential oils are oils which have been extracted from plants. Mouthwashes based on essential oils could be more effective than traditional mouthcare as antigingival treatments. They have been found effective in reducing halitosis, and are being used in several commercial mouthwashes.
Fluoride anticavity
Anticavity mouthwashes use sodium fluoride to protect against |
tooth decay. Fluoridecontaining mouthwashes are used as prevention for dental caries for individuals who are considered at higher risk for tooth decay, whether due to xerostomia related to salivary dysfunction or side effects of medication, to not drinking fluoridated water, or to being physically unable to care for their oral needs brushing and flossing, and as treatment for those with dentinal hypersensitivity, gingival recession root exposure.
Flavoring agents and Xylitol
Flavoring agents include sweeteners such as sorbitol, sucralose, sodium saccharin, and xylitol, which stimulate salivary function due to their sweetness and taste and helps restore the mouth to a neutral level of acidity.
Xylitol rinses double as a bacterial inhibitor, and have been used as substitute for alcohol to avoid dryness of mouth associated with alcohol.
Hydrogen peroxide
Hydrogen peroxide can be used as an oxidizing mouthwash e.g. Peroxyl, 1.5. It kills anaerobic bacteria, and also has a mechanical cleansing action when it fr |
oths as it comes into contact with debris in mouth. It is often used in the short term to treat acute necrotising ulcerative gingivitis. Side effects can occur with prolonged use, including hypertrophy of the lingual papillae.
Lactoperoxidase saliva substitute
Enzymes and nonenzymatic proteins, such as lactoperoxidase, lysozyme, and lactoferrin, have been used in mouthwashes e.g., Biotene to reduce levels of oral bacteria, and, hence, of the acids produced by these bacteria.
Lidocainexylocaine
Oral lidocaine is useful for the treatment of mucositis symptoms inflammation of mucous membranes induced by radiation or chemotherapy. There is evidence that lidocaine anesthetic mouthwash has the potential to be systemically absorbed, when it was tested in patients with oral mucositis who underwent a bone marrow transplant.
Methyl salicylate
Methyl salicylate functions as an antiseptic, antiinflammatory, and analgesic agent, a flavoring, and a fragrance. Methyl salicylate has some antiplaque action, but less than c |
hlorhexidine. Methyl salicylate does not stain teeth.
Nystatin
Nystatin suspension is an antifungal ingredient used for the treatment of oral candidiasis.
Potassium oxalate
A randomized clinical trial found promising results in controlling and reducing dentine hypersensitivity when potassium oxalate mouthwash was used in conjugation with toothbrushing.
Povidoneiodine PVPI
A 2005 study found that gargling three times a day with simple water or with a povidoneiodine solution was effective in preventing upper respiratory infection and decreasing the severity of symptoms if contracted. Other sources attribute the benefit to a simple placebo effect.
PVPI in general covers "a wider virucidal spectrum, covering both enveloped and nonenveloped viruses, than the other commercially available antiseptics", which also includes the novel SARSCoV2 Virus.
Sanguinarine
Sanguinarinecontaining mouthwashes are marketed as antiplaque and antimalodor treatments. Sanguinarine is a toxic alkaloid herbal extract, obtained from |
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