#REDIRECT [[Algorithms, theory of]]
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...me alphabet, the concepts of equivalence and complete equivalence of these algorithms with respect to it can be naturally introduced.
...<TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Markov, "Theory of algorithms" , Israel Program Sci. Transl. (1961) (Translated from Russian) (Also: T
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...s [[Optimization of computational algorithms|optimization of computational algorithms]]. Sometimes, however, these concepts are treated differently. For example,
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''for a given class of algorithms''
...machines it is the universal [[Turing machine|Turing machine]], for normal algorithms it is the [[Universal normal algorithm|universal normal algorithm]], etc.
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...olutionary strategies; evolutionary programming; genetic programming. Such algorithms and ideas have found many applications in, e.g.: [[Scheduling theory|schedu
...r><td valign="top">[a5]</td> <td valign="top"> Z. Michalewicz, "Genetic algorithms $+$ data structures $=$ evolution programs" , Springer (1992)</td></tr><tr
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...ho, J.E. Hopcroft, J.D. Ullman, "The design and analysis of computer algorithms" , Addison-Wesley (1976)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Even, "Graph algorithms" , Computer Sci. Press (1979)</TD></TR>
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...basis of which lies a certain synthesis of ideas in the theory of sets and algorithms (see [[#References|[2]]]).
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$#C+1 = 40 : ~/encyclopedia/old_files/data/A011/A.0101890 Algorithms, combinations of
...s for constructing new algorithms (cf. [[Algorithm|Algorithm]]) from given algorithms.
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...connecting algorithms of a fixed type and expressing the fact that any two algorithms thus connected yield the same results if fed with the same given type of in
...he alphabet $A$, then the second algorithm also converts $P$ to $Q$. These algorithms are called completely equivalent with respect to $A$ if for any $P$ in $A$
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...]] used in the numerical research into random processes and phenomena, and algorithms of the [[Monte-Carlo method|Monte-Carlo method]] for solving deterministic
...[#References|[4]]]). Particularly effective are those stochastic numerical algorithms that allow a number of realizations of the algorithm to be made simultaneou
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...t one to use small efficient algorithms as building blocks for larger-size algorithms [[#References|[a3]]].
...ign="top">[a3]</TD> <TD valign="top"> R.C. Agarwal, J.W. Cooley, "New algorithms for digital convolution" ''IEEE Trans. Acoustics, Speech and Signal Proces
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...a sequence representation may define $2^l$ equivalence relations. Genetic algorithms implicitly process these relations in a very parallel manner and detect goo
...neration to generation through selection, crossover, and mutation. Genetic algorithms use such operators for detecting better relations and classes that finally
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...ithm of one kind or another — Turing machines, recursive functions, normal algorithms, and others — one may, by virtue of Church' thesis, restrict oneself to i
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.... Zharov, "The complexity of a universal normal algorithm" , ''Theory of algorithms and mathematical logic'' , Moscow (1974) pp. 34–54 (In Russian)</TD></
...lign="top"> A.A. Markov, N.M. [N.M. Nagornyi] Nagorny, "The theory of algorithms" , Kluwer (1988) pp. Chapt. V (Translated from Russian)</TD></TR></table
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...one of the fundamental results in the general theory of algorithms (cf. [[Algorithms, theory of]]); it may be (is) used in deriving all known negative results f
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...ed "theorems on universal algorithms" , dealing with the realizability of algorithms whose codings model the operation of any algorithm of this type.
...n arises — to wit, how to recognize self-applicable algorithms among other algorithms of this type — is not solvable by an algorithm of this type. This result
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Three algorithms can perform this task.
...ty assumption [[#References|[a4]]]. A general perspective on probabilistic algorithms may be found in [[#References|[a5]]].
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...lign="top"> A.A. Markov, N.M. [N.M. Nagornyi] Nagorny, "The theory of algorithms" , Kluwer (1988) (Translated from Russian)</TD></TR></table>
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* Ivan Rival (ed.), "Algorithms and Order", Kluwer (1989) {{ISBN|940107691X}} {{ZBL|0709.68004}}
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...rier transform algorithms, such as the Cooley–Tukey fast Fourier transform algorithms {{Cite|CoTu}}, depend on the existence of non-trivial divisors of the trans
...m computations of prime power size $p^N$ and introduced cyclic convolution algorithms based on the polynomial version of the [[Chinese remainder theorem|Chinese
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...(ed.) ''The Schur complement and its applications'', Numerical Methods and Algorithms '''4''' Springer (2005) {{ISBN|0-387-24271-6}} {{ZBL|1075.15002}}
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A branch of mathematics and computer science concerned with finding efficient algorithms, or computational procedures, for solving geometric problems. Such problems
...metric configurations. It is impossible to list all important problems and algorithms of computational geometry, and some basic concepts and results are consider
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...to find primality proofs for numbers of $ 1000 $ decimal digits. For these algorithms, see the overview in [[#References|[a4]]] and the original references [[#Re
..."> E. Bach, J. Shallit, "Algorithmic number theory" , '''1: Efficient Algorithms''' , MIT (1996)</TD></TR>
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...D valign="top">[2]</TD> <TD valign="top"> N.M. Nagornyi, "The theory of algorithms" , Kluwer (1988) (Translated from Russian)</TD></TR></table>
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...ssical real numbers that are computable in some sense or other by means of algorithms. (See also [[Constructive analysis|Constructive analysis]].)
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The comparison of algorithms and the analysis of numerical problems in a Bayesian setting, cf. also [[Ba
...for inputs sharing some common properties. In a theoretical comparison of algorithms one either selects a class $P$ of inputs and defines the cost and the error
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...ociative calculus]]). The detailed definition and general theory of normal algorithms are set forth in [[#References|[1]]] (Chapts. I–V); see also [[#Reference
...riginal algorithm $\mathfrak A$. This convention is known in the theory of algorithms as the normalization principle. The formalization of the concept of an algo
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...ty, for designing and analyzing parallel combinatorial and graph-theoretic algorithms. The model focuses on the inherent level of computational parallelism witho
...n $O(\log n)$ time using $O(n)$ processors, parallel matrix multiplication algorithms running in $O(\log n)$ time using $O(n^3/\log n)$ processors, and string ma
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...(i.e. through schemes of general recursion). Let the complexity of normal algorithms and of Turing machines mean, respectively, the length of the encoding and t
The study of the complexity of algorithms solving the finite restrictions of algorithmically unsolvable problems (the
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...tions has been an object of study in the [[Algorithms, theory of|theory of algorithms]] from the very beginning of its creation. In particular, this analogy lies
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Genetic algorithms (GAs) (also known as evolutionary algorithms [EAs]) assume a discrete search space $H$ and a function $f:H\to \R$. The
[1] Back, T., Evolutionary Algorithms in Theory and Practice, Oxford University Press, 1996
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* Ivan Rival (ed.),"Algorithms and Order", Kluwer (1989) {{ISBN|940107691X}} {{ZBL|0709.68004}}
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...ic countries. It is supposed to be a standard language for the exchange of algorithms between socialistic countries. It is based on the language Algol-60 (cf. [[
..."top">[1]</TD> <TD valign="top"> "Description of the language ALGAMS" , ''Algorithms and algorithmic languages'' : 3 , Moscow (1968) pp. 3–56 (In Russian)
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Some authors make a specific distinction between algorithms and procedures. Algorithms are prescriptions of sequences of actions which should always terminate aft
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...~/encyclopedia/old_files/data/O068/O.0608550 Optimization of computational algorithms
...possessing these properties. For this reason, in the theoretical study of algorithms one introduces a class of problems $ P $
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...e treated as an aspect of the [[idempotent correspondence principle]]. For algorithms, this heuristic principle means that "if one has an important and interest
...algorithms can often be approximated by linear ones. It is well-known that algorithms from linear algebra are convenient for parallel computations (see, e.g. [[#
7 KB (1,079 words) - 20:39, 16 November 2023
...osely connected with the theory of algorithms (cf. [[Algorithms, theory of|Algorithms, theory of]]), particularly so with the theory of abstract machines, since
...an automaton and that of obtaining corresponding estimates. Fairly simple algorithms have been developed for finite automata making it possible, with the aid of
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...rithm. Exact definitions of computable functions, effective procedures and algorithms were given in various forms by D. Hilbert, K. Gödel, A. Church, S.C. Kleen
...an algorithm are Turing machines, partial recursive functions, the normal algorithms of Markov, and others.
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..., "Separability with respect to invariants" , ''Studies on the theory of algorithms and mathematical logic'' , '''1''' , Moscow (1973) pp. 205–210 (In Rus
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...The following universal functions play an important role in the theory of algorithms: 1) universal partial recursive functions for the class of all $ n $-
...TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.I. Mal'tsev, "Algorithms and recursive functions" , Wolters-Noordhoff (1970) (Translated from Russ
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...TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Mal'tsev, "Algorithms and recursive functions" , Wolters-Noordhoff (1970) (Translated from Russ
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...d decoding algorithms that are close in their possibilities to the optimal algorithms whose existence has been proved in the theory of information transmission.
...bed on an intuitive level and have no clear mathematical description. Such algorithms always include learning processes with respect to a list of objects that ar
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In the theory of algorithms more special kinds of $tt$-reducibility are also considered, for instance b
...gn="top"> A.N. Degtev, "Reducibilities of tabular type in the theory of algorithms" ''Russian Math. Surveys'' , '''34''' : 3 (1979) pp. 155–192 ''Uspek
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...(ed.) ''The Schur complement and its applications'', Numerical Methods and Algorithms '''4''' Springer (2005) {{ISBN|0-387-24271-6}}{{ZBL|1075.15002}}
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..., W.H. Payne, "Generalized feedback shift register pseudorandom number algorithms" ''J. ACM'' , '''20''' (1973) pp. 456–468</td></tr><tr><td valign="top
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The use of adaptive algorithms is now (1998) very widespread across such varied applications as system ide
The study of adaptive algorithms is generally made through the theory of [[Stochastic approximation|stochast
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...n the computational complexity of a problem; the analysis of all permitted algorithms together with the structure of a problem can provide lower bounds (cf. also
Consider a computational model for uniform algorithms over a set $\Sigma$. Two prominent examples are
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.../TD> <TD valign="top"> M. Minoux, "Mathematical programming: theory and algorithms" , Wiley (1986)</TD></TR>
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...hms with regular closure does not coincide with the set of actually stable algorithms, construction of a closed algorithm frequently helps in investigating the s
...>[6]</TD> <TD valign="top"> A.F. Shapkin, "Closure of two computational algorithms, based on the idea of orthogonalization" ''Zh. Vychisl. Mat. i Mat. Fiz.''
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...<TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Markov, "Theory of algorithms" , Israel Program Sci. Transl. (1961) (Translated from Russian) (Also: T
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...ly-increasing function from the natural numbers onto $A$. In the theory of algorithms the important characteristics of a direct counting of a set are recursivene
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...the transport problem and the assignment problem makes it possible to use algorithms that are more efficient than the [[Simplex method|simplex method]]. Some of
...<TD valign="top"> M. Grötschel, L. Lovász, A. Schrijver, "Geometric algorithms and combinatorial optimization" , Springer (1987)</TD></TR><TR><TD valign=
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...ersion of the first one. The methods of solution here consist of arbitrary algorithms and non-deterministic formally-describable procedures.
==Equivalent transformations of algorithms.==
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...discrete optimization problems are naturally formulated in terms of local algorithms.
The starting theorems for local algorithms are the uniqueness theorem and the theorem on the existence of a best algor
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...many geometric and analytical properties. There are elegant and efficient algorithms for evaluation, differentiation, subdivision of the curves, and conversion
...s that allows exact representation of conic sections. These properties and algorithms make the curves popular in areas where shape is important. In addition to C
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.... Combinatorial Gray codes are useful because they often lead to efficient algorithms for generating (i.e., listing) the objects. Some of the more important comb
...TD valign="top"> C.D. Savage, "Gray code sequences of partitions" ''J. Algorithms'' , '''10''' (1989) pp. 577–595</TD></TR><TR><TD valign="top">[a6]</TD>
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...hastic Lyapunov functions have been used to prove convergence of recursive algorithms driven by stochastic processes. Convergence problems of this type arise in
...Schuppen, "Convergence results for continuous-time stochastic filtering algorithms" ''J. Math. Anal. Appl.'' , '''96''' (1983) pp. 209–225</TD></TR></tab
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A basic concept in the branch of the theory of algorithms called enumeration theory, which investigates general properties of classes
...ation of this idea it was noticed that many known results in the theory of algorithms turn out to be consequences of general regularities in enumeration theory.
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...of abstract computers by other classes (e.g. in order to simulate parallel algorithms by sequential ones).
...ty of their structure and also by the type of information processed by the algorithms. The main classes of abstract computers are listed below.
9 KB (1,295 words) - 17:46, 4 June 2020
$#C+1 = 33 : ~/encyclopedia/old_files/data/A011/A.0101910 Algorithms, theory of
...[Turing machine|Turing machine]]). Subsequent development of the theory of algorithms is due to the studies of Kleene, Post [[#References|[6]]], [[#References|[7
19 KB (2,846 words) - 16:10, 1 April 2020
...programming language has a universal character, allowing one to formulate algorithms for solving various problems that are to be solved on different computers.
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* Rajeev Motwani, Prabhakar Raghavan, "Randomized Algorithms", Cambridge University Press (1995) {{ISBN|978-0-521-47465-8}} {{ZBL|0849.6
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...ctionals. In discrete programming the algorithms employed are analogues of algorithms of local optimization, descent, random search, etc. For a review of approxi
...to find the extremum or a good approximation to it in the class of simple algorithms; this collection is often effectively described. This was first noted in so
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...such as the memory management aspects and computational complexity of such algorithms (cf. also [[Complexity theory]] and [[Algorithm, computational complexity o
<TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Edelsbrunner, "Algorithms in combinatorial geometry" , Springer (1987)</TD></TR>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Markov, "Theory of algorithms" , Israel Program Sci. Transl. (1961) (Translated from Russian) (Also: T
...lign="top"> A.A. Markov, N.M. [N.M. Nagornyi] Nagorny, "The theory of algorithms" , Kluwer (1988) (Translated from Russian) {{ZBL|0663.03023}}</TD></TR>
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...used method for abbreviating proofs, and is often useful in improving the algorithms for the establishment of derivability. One of the most important results in
1 KB (207 words) - 19:21, 17 October 2014
...s present no difficulties). Also, rational numbers, algebraic polynomials, algorithms, and calculi of various well-defined types (cf. [[Algorithm|Algorithm]]; [[
...lign="top"> A.A. Markov, N.M. [N.M. Nagornyi] Nagorny, "The theory of algorithms" , Reidel (1988) (Translated from Russian) {{ZBL|0663.03023}}</TD></TR>
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There are efficient algorithms to solve this problem. Let $ X $
...valign="top">[a9]</TD> <TD valign="top"> J. Kennington, R. Helgason, "Algorithms for network programming" , Wiley (1980)</TD></TR></table>
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...mathematical logic — the theory of algorithms (cf. [[Algorithms, theory of|Algorithms, theory of]]). The growing stream of information and the related problems o
...re of discrete analysis, related to problems in finite structures, is that algorithms by which these problems may be solved exist in most cases, while in classic
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* Eric Bach, Jeffrey O. Shallit, ''Algorithmic Number Theory: Efficient algorithms, Volume 1'', MIT Press (1996) {{ISBN|0262024055}}
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...been constructed, based on certain intuitive principles, i.e. families of algorithms for solving classification problems. The most widely used are: models based
...ication is a priori known. In developing the general theory of recognition algorithms the most complete results have been obtained within an algebraic framework.
8 KB (1,189 words) - 13:43, 17 April 2014
...<TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Markov, "Theory of algorithms" , Israel Program Sci. Transl. (1961) (Translated from Russian) (Also: T
2 KB (216 words) - 17:08, 7 February 2011
One of the basic concepts in the [[Algorithms, theory of|theory of algorithms]] and its applications. Its background was the fact that solvability (and u
...so-called priority method, which is extensively employed in the theory of algorithms.
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...ecial method for constructing algorithms based on the requirement that the algorithms should be exact or have an error of prescribed order of accuracy on some se
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...[[Automatic programming|automatic programming]] and for the publication of algorithms (it is an abbreviation of "ALGOrithmic Language" ).
...lly refers to "Algol-60" . It is especially suited for the description of algorithms of numerical analysis. The language is machine-independent and does not pro
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...]. The Scarf algorithm and other related ones rapidly evolved into various algorithms for solving equations, for finding extrema and for tracing bifurcating solu
...as a very extensive bibliography. A substantial number of previously known algorithms for solving equations can be obtained as particular cases of the homotopy c
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...nald E. Knuth, ''The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1'', Addison-Wesley (2014) {{ISBN|0133488853}}
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there are algorithms which are particularly convenient in practice. There is a considerable numb
...>[a4]</TD> <TD valign="top"> D.F. Elliot, K.R. Rao, "Fast transforms: algorithms, analysis, applications" , Acad. Press (1982)</TD></TR>
4 KB (602 words) - 14:32, 17 March 2023
...computable functions and theory of algorithms (cf. [[Algorithms, theory of|Algorithms, theory of]]; [[Computable function|Computable function]]) is any mapping o
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* Rajeev Motwani, Prabhakar Raghavan, "Randomized Algorithms", Cambridge University Press (1995) {{ISBN|978-0-521-47465-8}} {{ZBL|0849.6
1 KB (226 words) - 07:28, 14 November 2023
...o [[Optimization of computational algorithms|Optimization of computational algorithms]]; [[Index transform|Information-based complexity]]; [[Curse of dimension|C
...gn="top"> G.W. Wasilkowski, H. Woźniakowski, "Weighted tensor-product algorithms for linear multivariate problems" ''Preprint'' (1998)</td></tr><tr><td va
8 KB (1,086 words) - 17:43, 1 July 2020
...first such algorithm was proposed by H. Scarf, [[#References|[a3]]]. Such algorithms later developed in the so-called homotopy or continuation methods for calcu
...n="top">[a4]</TD> <TD valign="top"> S. Karamadian (ed.) , ''Fixed points. Algorithms and applications'' , Acad. Press (1977)</TD></TR>
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The accuracy of the algorithms based on the interpolation formulas of Lagrange, Newton and others strongly
...tly smooth and if the number of nodal points is large [[#References|[1]]]. Algorithms of numerical differentiation involving spline-interpolation [[#References|[
5 KB (779 words) - 19:35, 5 June 2020
...</TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Edelsbrunner, "Algorithms in combinatorial geometry" , Springer (1987)</TD></TR><TR><TD valign="top"
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...tape scanned by the head of the machine. Time and storage costs for normal algorithms (cf. [[Normal algorithm|Normal algorithm]]), iterative arrays, multi-head a
better than other algorithms" . However, as exemplified by the speed-up theorem (see below), such a form
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1) Provide algorithms turning $ R [ {\mathcal X} ] $
By keeping track of intermediate results in the algorithms, it is possible to express the Gröbner basis $ C $
10 KB (1,471 words) - 14:52, 30 May 2020
...theory of calculi is the theory of algorithms (cf. [[Algorithms, theory of|Algorithms, theory of]]). This is clear from the fact that the notion of a calculus is
...<TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Markov, "Theory of algorithms" , Israel Program Sci. Transl. (1961) (Translated from Russian) (Also: T
8 KB (1,204 words) - 18:51, 13 November 2014
...to use approximate solution methods of (2) in order to construct iteration algorithms for solving equation (1) (cf. [[Iteration algorithm|Iteration algorithm]]).
...m state is the desired optimum. Again this leads to all kinds of iteration algorithms, as in [[#References|[a3]]].
4 KB (649 words) - 05:25, 19 March 2022
<TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Mal'tsev, "Algorithms and recursive functions" , Wolters-Noordhoff (1970) (Translated from Russ
2 KB (250 words) - 20:46, 23 November 2023
...m the previous one, usually by a small amount. The convergence analysis of algorithms like (a7) is treated in e.g. [[#References|[a6]]], [[#References|[a1]]], [[
...</td> <td valign="top"> V. Solo, X. Kong, "Adaptive signal processing algorithms" , Prentice-Hall (1995)</td></tr></table>
7 KB (1,131 words) - 14:45, 16 December 2023
...t in real problems the computations may be too laborious. If so, heuristic algorithms are used to generate better patterns: as a rule, it is enough to consider c
...isted. For linear and rectangular parts applicable heuristic approximation algorithms have been programmed (see [[#References|[2]]]).
8 KB (1,179 words) - 17:25, 27 October 2014
...algorithms to test primality of numbers and, if composite, to factor them. Algorithms of this kind which can handle numbers of 100 or 200 digits have recently at
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...rocedures that solve every instance of the problem? Which problems possess algorithms for their solution?
...ion is of definite practical significance: One should not try to construct algorithms for an unsolvable problem. (There are some notorious examples of such attem
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This approach makes it possible to construct numerical algorithms for a wide class of differential equations, including partial differential
...polation of operators and functionals in the construction of computational algorithms for solving concrete problems is based on the use of interpolation formulas
26 KB (3,882 words) - 22:13, 5 June 2020
...heorem, one asks to find these factors algorithmically, and to devise such algorithms with low cost.
...lled polynomial representation of $\sigma$ as its basic tool. The last two algorithms are based on Gauss' theorem that $x ^ { q ^ { d } } - x$ is the product of
12 KB (1,739 words) - 13:11, 26 March 2023
...le computation one has to use specialized [[Software|software]]. To encode algorithms on a computer, special language means are required. These are also consider
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.... As shown in [[#References|[a3]]], this suffices to guarantee linear-time algorithms for many otherwise hard graph problems when instances are confined to Halin
...Borie, R.G. Parker, C.A. Tovey, "Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed
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* Donald E. Knuth, "Introduction to combinatorial algorithms and Boolean functions", ser. The Art of Computer Programming 4.0 (2008) pp.
2 KB (261 words) - 18:11, 14 October 2023
...for the modern precise statement of this problem as given by the theory of algorithms (cf. [[Church thesis|Church thesis]]). Other examples of such group calculi
...lign="top"> A.A. Markov, N.M. [N.M. Nagornyi] Nagorny, "The theory of algorithms" , Kluwer (1988) (Translated from Russian)</TD></TR></table>
9 KB (1,334 words) - 19:42, 5 June 2020
...nts take the values 0 or 1 only. Conditions a) and b) enable one to devise algorithms for the solution of the transportation problem that are essentially simpler
The best known of these algorithms are the method of potentials and the so-called Hungarian method. The method
5 KB (746 words) - 14:56, 7 June 2020