The knapsack problem is the following problem in combinatorial optimization :
152-588: In computer science and operations research , a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to generate high-quality solutions to optimization and search problems via biologically inspired operators such as selection , crossover , and mutation . Some examples of GA applications include optimizing decision trees for better performance, solving sudoku puzzles , hyperparameter optimization , and causal inference . In
304-405: A 1 / 2 {\displaystyle 1/2} -approximation. It can be shown that the average performance converges to the optimal solution in distribution at the error rate n − 1 / 2 {\displaystyle n^{-1/2}} The fully polynomial time approximation scheme (FPTAS) for the knapsack problem takes advantage of the fact that
456-400: A bit string . Typically, numeric parameters can be represented by integers , though it is possible to use floating point representations. The floating point representation is natural to evolution strategies and evolutionary programming . The notion of real-valued genetic algorithms has been offered but is really a misnomer because it does not really represent the building block theory that
608-533: A "child" solution using the above methods of crossover and mutation, a new solution is created which typically shares many of the characteristics of its "parents". New parents are selected for each new child, and the process continues until a new population of solutions of appropriate size is generated. Although reproduction methods that are based on the use of two parents are more "biology inspired", some research suggests that more than two "parents" generate higher quality chromosomes. These processes ultimately result in
760-410: A decision-tree model was given by Meyer auf der Heide who showed that for every n there exists an O ( n ) -deep linear decision tree that solves the subset-sum problem with n items. Note that this does not imply any upper bound for an algorithm that should solve the problem for any given n . Several algorithms are available to solve knapsack problems, based on the dynamic programming approach,
912-570: A discipline, computer science spans a range of topics from theoretical studies of algorithms and the limits of computation to the practical issues of implementing computing systems in hardware and software. CSAB , formerly called Computing Sciences Accreditation Board—which is made up of representatives of the Association for Computing Machinery (ACM), and the IEEE Computer Society (IEEE CS) —identifies four areas that it considers crucial to
1064-724: A distinct academic discipline in the 1950s and early 1960s. The world's first computer science degree program, the Cambridge Diploma in Computer Science , began at the University of Cambridge Computer Laboratory in 1953. The first computer science department in the United States was formed at Purdue University in 1962. Since practical computers became available, many applications of computing have become distinct areas of study in their own rights. Although first proposed in 1956,
1216-510: A floating point representation. An expansion of the Genetic Algorithm accessible problem domain can be obtained through more complex encoding of the solution pools by concatenating several types of heterogenously encoded genes into one chromosome. This particular approach allows for solving optimization problems that require vastly disparate definition domains for the problem parameters. For instance, in problems of cascaded controller tuning,
1368-431: A general rule of thumb genetic algorithms might be useful in problem domains that have a complex fitness landscape as mixing, i.e., mutation in combination with crossover , is designed to move the population away from local optima that a traditional hill climbing algorithm might get stuck in. Observe that commonly used crossover operators cannot change any uniform population. Mutation alone can provide ergodicity of
1520-450: A genetic algorithm, a population of candidate solutions (called individuals, creatures, organisms, or phenotypes ) to an optimization problem is evolved toward better solutions. Each candidate solution has a set of properties (its chromosomes or genotype ) which can be mutated and altered; traditionally, solutions are represented in binary as strings of 0s and 1s, but other encodings are also possible. The evolution usually starts from
1672-534: A knapsack algorithm would determine which subset gives each student the highest possible score. A 1999 study of the Stony Brook University Algorithm Repository showed that, out of 75 algorithmic problems related to the field of combinatorial algorithms and algorithm engineering, the knapsack problem was the 19th most popular and the third most needed after suffix trees and the bin packing problem . The most common problem being solved
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#17328590144481824-471: A less optimal solution. This generational process is repeated until a termination condition has been reached. Common terminating conditions are: Genetic algorithms are simple to implement, but their behavior is difficult to understand. In particular, it is difficult to understand why these algorithms frequently succeed at generating solutions of high fitness when applied to practical problems. The building block hypothesis (BBH) consists of: Goldberg describes
1976-464: A mathematical discipline argue that computer programs are physical realizations of mathematical entities and programs that can be deductively reasoned through mathematical formal methods . Computer scientists Edsger W. Dijkstra and Tony Hoare regard instructions for computer programs as mathematical sentences and interpret formal semantics for programming languages as mathematical axiomatic systems . A number of computer scientists have argued for
2128-443: A mathematics emphasis and with a numerical orientation consider alignment with computational science . Both types of departments tend to make efforts to bridge the field educationally if not across all research. Despite the word science in its name, there is debate over whether or not computer science is a discipline of science, mathematics, or engineering. Allen Newell and Herbert A. Simon argued in 1975, Computer science
2280-434: A maximum possible score of 100 points. However, on tests with a heterogeneous distribution of point values, it is more difficult to provide choices. Feuerman and Weiss proposed a system in which students are given a heterogeneous test with a total of 125 possible points. The students are asked to answer all of the questions to the best of their abilities. Of the possible subsets of problems whose total point values add up to 100,
2432-411: A maximum weight capacity W {\displaystyle W} , Here x i {\displaystyle x_{i}} represents the number of instances of item i {\displaystyle i} to include in the knapsack. Informally, the problem is to maximize the sum of the values of the items in the knapsack so that the sum of the weights is less than or equal to
2584-447: A naive brute force approach (examining all subsets of { 1... n } {\displaystyle \{1...n\}} ), at the cost of using exponential rather than constant space (see also baby-step giant-step ). The current state of the art improvement to the meet-in-the-middle algorithm, using insights from Schroeppel and Shamir's Algorithm for Subset Sum, provides as a corollary a randomized algorithm for Knapsack which preserves
2736-463: A network while using concurrency, this is known as a distributed system. Computers within that distributed system have their own private memory, and information can be exchanged to achieve common goals. This branch of computer science aims to manage networks between computers worldwide. Computer security is a branch of computer technology with the objective of protecting information from unauthorized access, disruption, or modification while maintaining
2888-443: A number of steps from maternal DNA adding a number of steps from paternal DNA and so on. This is like adding vectors that more probably may follow a ridge in the phenotypic landscape. Thus, the efficiency of the process may be increased by many orders of magnitude. Moreover, the inversion operator has the opportunity to place steps in consecutive order or any other suitable order in favour of survival or efficiency. A variation, where
3040-540: A number of terms for the practitioners of the field of computing were suggested in the Communications of the ACM — turingineer , turologist , flow-charts-man , applied meta-mathematician , and applied epistemologist . Three months later in the same journal, comptologist was suggested, followed next year by hypologist . The term computics has also been suggested. In Europe, terms derived from contracted translations of
3192-495: A particular kind of mathematically based technique for the specification , development and verification of software and hardware systems. The use of formal methods for software and hardware design is motivated by the expectation that, as in other engineering disciplines, performing appropriate mathematical analysis can contribute to the reliability and robustness of a design. They form an important theoretical underpinning for software engineering, especially where safety or security
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#17328590144483344-441: A population of randomly generated individuals, and is an iterative process , with the population in each iteration called a generation . In each generation, the fitness of every individual in the population is evaluated; the fitness is usually the value of the objective function in the optimization problem being solved. The more fit individuals are stochastically selected from the current population, and each individual's genome
3496-463: A population on each of the computer nodes and migration of individuals among the nodes. Fine-grained parallel genetic algorithms assume an individual on each processor node which acts with neighboring individuals for selection and reproduction. Other variants, like genetic algorithms for online optimization problems, introduce time-dependence or noise in the fitness function. Genetic algorithms with adaptive parameters (adaptive genetic algorithms, AGAs)
3648-688: A second solution S 2 = { k + 1 } {\displaystyle S_{2}=\left\{k+1\right\}} containing the first item that did not fit. Since S 1 ∪ S 2 {\displaystyle S_{1}\cup S_{2}} provides an upper bound for the LP relaxation of the problem, one of the sets must have value at least m / 2 {\displaystyle m/2} ; we thus return whichever of S 1 {\displaystyle S_{1}} and S 2 {\displaystyle S_{2}} has better value to obtain
3800-490: A set of non-divisible projects or tasks under a fixed budget or time constraint, respectively. The knapsack problem has been studied for more than a century, with early works dating as far back as 1897. Knapsack problems appear in real-world decision-making processes in a wide variety of fields, such as finding the least wasteful way to cut raw materials, selection of investments and portfolios , selection of assets for asset-backed securitization , and generating keys for
3952-512: A significant amount of computer science does not involve the study of computers themselves. Because of this, several alternative names have been proposed. Certain departments of major universities prefer the term computing science , to emphasize precisely that difference. Danish scientist Peter Naur suggested the term datalogy , to reflect the fact that the scientific discipline revolves around data and data treatment, while not necessarily involving computers. The first scientific institution to use
4104-407: A slightly worse space complexity of O ∗ ( 2 n / 4 ) {\displaystyle O^{*}(2^{n/4})} . As for most NP-complete problems, it may be enough to find workable solutions even if they are not optimal. Preferably, however, the approximation comes with a guarantee of the difference between the value of the solution found and the value of
4256-601: A solution S 1 {\displaystyle S_{1}} by packing items greedily as long as possible, i.e. S 1 = { 1 , … , k } {\displaystyle S_{1}=\left\{1,\ldots ,k\right\}} where k = max 1 ≤ k ′ ≤ n ∑ i = 1 k ′ w i ≤ W {\displaystyle k=\textstyle \max _{1\leq k'\leq n}\textstyle \sum _{i=1}^{k'}w_{i}\leq W} . Furthermore, construct
4408-410: A specific application. Codes are used for data compression , cryptography , error detection and correction , and more recently also for network coding . Codes are studied for the purpose of designing efficient and reliable data transmission methods. Data structures and algorithms are the studies of commonly used computational methods and their computational efficiency. Programming language theory
4560-574: A sub-field: Evolutionary algorithms is a sub-field of evolutionary computing . Swarm intelligence is a sub-field of evolutionary computing . Evolutionary computation is a sub-field of the metaheuristic methods. Metaheuristic methods broadly fall within stochastic optimisation methods. Computer science Computer science is the study of computation , information , and automation . Computer science spans theoretical disciplines (such as algorithms , theory of computation , and information theory ) to applied disciplines (including
4712-406: A table to store previous computations. The following is pseudocode for the dynamic program: This solution will therefore run in O ( n W ) {\displaystyle O(nW)} time and O ( n W ) {\displaystyle O(nW)} space. (If we only need the value m[n,W], we can modify the code so that the amount of memory required is O(W) which stores
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4864-415: Is a branch of computer science that deals with the design, implementation, analysis, characterization, and classification of programming languages and their individual features . It falls within the discipline of computer science, both depending on and affecting mathematics, software engineering, and linguistics . It is an active research area, with numerous dedicated academic journals. Formal methods are
5016-454: Is a way to improve the running time. Even if P≠NP , the O ( n W ) {\displaystyle O(nW)} complexity does not contradict the fact that the knapsack problem is NP-complete , since W {\displaystyle W} , unlike n {\displaystyle n} , is not polynomial in the length of the input to the problem. The length of the W {\displaystyle W} input to
5168-422: Is an empirical discipline. We would have called it an experimental science, but like astronomy, economics, and geology, some of its unique forms of observation and experience do not fit a narrow stereotype of the experimental method. Nonetheless, they are experiments. Each new machine that is built is an experiment. Actually constructing the machine poses a question to nature; and we listen for the answer by observing
5320-484: Is an open problem in the theory of computation. Information theory, closely related to probability and statistics , is related to the quantification of information. This was developed by Claude Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and communicating data. Coding theory is the study of the properties of codes (systems for converting information from one form to another) and their fitness for
5472-797: Is an optimal solution. Quantum approximate optimization algorithm (QAOA) can be employed to solve Knapsack problem using quantum computation by minimizing the Hamiltonian of the problem. The Knapsack Hamiltonian is constructed via embedding the constraint condition to the cost function of the problem with a penalty term. H = − ∑ i = 1 n v i x i + P ( ∑ i = 1 n w i x i − W ) 2 , {\displaystyle {H}=-\sum _{i=1}^{n}v_{i}x_{i}+P\left(\sum _{i=1}^{n}w_{i}x_{i}-W\right)^{2},} where P {\displaystyle P}
5624-419: Is another significant and promising variant of genetic algorithms. The probabilities of crossover (pc) and mutation (pm) greatly determine the degree of solution accuracy and the convergence speed that genetic algorithms can obtain. Researchers have analyzed GA convergence analytically. Instead of using fixed values of pc and pm , AGAs utilize the population information in each generation and adaptively adjust
5776-405: Is as an array of bits (also called bit set or bit string ). Arrays of other types and structures can be used in essentially the same way. The main property that makes these genetic representations convenient is that their parts are easily aligned due to their fixed size, which facilitates simple crossover operations. Variable length representations may also be used, but crossover implementation
5928-475: Is associated in the popular mind with robotic development , but the main field of practical application has been as an embedded component in areas of software development , which require computational understanding. The starting point in the late 1940s was Alan Turing's question " Can computers think? ", and the question remains effectively unanswered, although the Turing test is still used to assess computer output on
6080-543: Is connected to many other fields in computer science, including computer vision , image processing , and computational geometry , and is heavily applied in the fields of special effects and video games . Information can take the form of images, sound, video or other multimedia. Bits of information can be streamed via signals . Its processing is the central notion of informatics, the European view on computing, which studies information processing algorithms independently of
6232-409: Is considered by some to have a much closer relationship with mathematics than many scientific disciplines, with some observers saying that computing is a mathematical science. Early computer science was strongly influenced by the work of mathematicians such as Kurt Gödel , Alan Turing , John von Neumann , Rózsa Péter and Alonzo Church and there continues to be a useful interchange of ideas between
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6384-571: Is correct within a factor of (1-ε) of the optimal solution. Theorem: The set S ′ {\displaystyle S'} computed by the algorithm above satisfies p r o f i t ( S ′ ) ≥ ( 1 − ε ) ⋅ p r o f i t ( S ∗ ) {\displaystyle \mathrm {profit} (S')\geq (1-\varepsilon )\cdot \mathrm {profit} (S^{*})} , where S ∗ {\displaystyle S^{*}}
6536-508: Is determining what can and cannot be automated. The Turing Award is generally recognized as the highest distinction in computer science. The earliest foundations of what would become computer science predate the invention of the modern digital computer . Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, aiding in computations such as multiplication and division. Algorithms for performing computations have existed since antiquity, even before
6688-495: Is employed. An adequate population size ensures sufficient genetic diversity for the problem at hand, but can lead to a waste of computational resources if set to a value larger than required. In addition to the main operators above, other heuristics may be employed to make the calculation faster or more robust. The speciation heuristic penalizes crossover between candidate solutions that are too similar; this encourages population diversity and helps prevent premature convergence to
6840-453: Is exponential in the number of different items but may be preferable to the DP algorithm when W {\displaystyle W} is large compared to n . In particular, if the w i {\displaystyle w_{i}} are nonnegative but not integers, we could still use the dynamic programming algorithm by scaling and rounding (i.e. using fixed-point arithmetic ), but if
6992-459: Is for their use in public-key cryptography systems, such as the Merkle–Hellman knapsack cryptosystem . More generally, better understanding of the structure of the space of instances of an optimization problem helps to advance the study of the particular problem and can improve algorithm selection. Furthermore, notable is the fact that the hardness of the knapsack problem depends on the form of
7144-630: Is generally considered the province of disciplines other than computer science. For example, the study of computer hardware is usually considered part of computer engineering , while the study of commercial computer systems and their deployment is often called information technology or information systems . However, there has been exchange of ideas between the various computer-related disciplines. Computer science research also often intersects other disciplines, such as cognitive science , linguistics , mathematics , physics , biology , Earth science , statistics , philosophy , and logic . Computer science
7296-426: Is greater than the value of i {\displaystyle i} . Then i {\displaystyle i} cannot appear in the optimal solution, because we could always improve any potential solution containing i {\displaystyle i} by replacing i {\displaystyle i} with the set J {\displaystyle J} . Therefore, we can disregard
7448-584: Is intended to organize, store, and retrieve large amounts of data easily. Digital databases are managed using database management systems to store, create, maintain, and search data, through database models and query languages . Data mining is a process of discovering patterns in large data sets. The philosopher of computing Bill Rapaport noted three Great Insights of Computer Science : Programming languages can be used to accomplish different tasks in different ways. Common programming paradigms include: Many languages offer support for multiple paradigms, making
7600-423: Is interesting from the perspective of computer science for many reasons: There is a link between the "decision" and "optimization" problems in that if there exists a polynomial algorithm that solves the "decision" problem, then one can find the maximum value for the optimization problem in polynomial time by applying this algorithm iteratively while increasing the value of k. On the other hand, if an algorithm finds
7752-426: Is involved. Formal methods are a useful adjunct to software testing since they help avoid errors and can also give a framework for testing. For industrial use, tool support is required. However, the high cost of using formal methods means that they are usually only used in the development of high-integrity and life-critical systems , where safety or security is of utmost importance. Formal methods are best described as
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#17328590144487904-545: Is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. It aims to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. According to Peter Denning, the fundamental question underlying computer science is, "What can be automated?" Theory of computation is focused on answering fundamental questions about what can be computed and what amount of resources are required to perform those computations. In an effort to answer
8056-441: Is modified ( recombined and possibly randomly mutated) to form a new generation. The new generation of candidate solutions is then used in the next iteration of the algorithm . Commonly, the algorithm terminates when either a maximum number of generations has been produced, or a satisfactory fitness level has been reached for the population. A typical genetic algorithm requires: A standard representation of each candidate solution
8208-440: Is more complex in this case. Tree-like representations are explored in genetic programming and graph-form representations are explored in evolutionary programming ; a mix of both linear chromosomes and trees is explored in gene expression programming . Once the genetic representation and the fitness function are defined, a GA proceeds to initialize a population of solutions and then to improve it through repetitive application of
8360-519: Is of high quality, affordable, maintainable, and fast to build. It is a systematic approach to software design, involving the application of engineering practices to software. Software engineering deals with the organizing and analyzing of software—it does not just deal with the creation or manufacture of new software, but its internal arrangement and maintenance. For example software testing , systems engineering , technical debt and software development processes . Artificial intelligence (AI) aims to or
8512-460: Is often employed. In this way, small changes in the integer can be readily affected through mutations or crossovers. This has been found to help prevent premature convergence at so-called Hamming walls , in which too many simultaneous mutations (or crossover events) must occur in order to change the chromosome to a better solution. Other approaches involve using arrays of real-valued numbers instead of bit strings to represent chromosomes. Results from
8664-474: Is quite unnatural to model applications in terms of genetic operators like mutation and crossover on bit strings. The pseudobiology adds another level of complexity between you and your problem. Second, genetic algorithms take a very long time on nontrivial problems. [...] [T]he analogy with evolution—where significant progress require [sic] millions of years—can be quite appropriate. [...] I have never encountered any problem where genetic algorithms seemed to me
8816-575: Is required to synthesize goal-orientated processes such as problem-solving, decision-making, environmental adaptation, learning, and communication found in humans and animals. From its origins in cybernetics and in the Dartmouth Conference (1956), artificial intelligence research has been necessarily cross-disciplinary, drawing on areas of expertise such as applied mathematics , symbolic logic, semiotics , electrical engineering , philosophy of mind , neurophysiology , and social intelligence . AI
8968-482: Is the 0-1 knapsack problem , which restricts the number x i {\displaystyle x_{i}} of copies of each kind of item to zero or one. Given a set of n {\displaystyle n} items numbered from 1 up to n {\displaystyle n} , each with a weight w i {\displaystyle w_{i}} and a value v i {\displaystyle v_{i}} , along with
9120-432: Is the field of study and research concerned with the design and use of computer systems , mainly based on the analysis of the interaction between humans and computer interfaces . HCI has several subfields that focus on the relationship between emotions , social behavior and brain activity with computers . Software engineering is the study of designing, implementing, and modifying the software in order to ensure it
9272-783: Is the field of study concerned with constructing mathematical models and quantitative analysis techniques and using computers to analyze and solve scientific problems. A major usage of scientific computing is simulation of various processes, including computational fluid dynamics , physical, electrical, and electronic systems and circuits, as well as societies and social situations (notably war games) along with their habitats, among many others. Modern computers enable optimization of such designs as complete aircraft. Notable in electrical and electronic circuit design are SPICE, as well as software for physical realization of new (or modified) designs. The latter includes essential design software for integrated circuits . Human–computer interaction (HCI)
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#17328590144489424-457: Is the penalty constant which is determined by case-specific fine-tuning. Solving the unbounded knapsack problem can be made easier by throwing away items which will never be needed. For a given item i {\displaystyle i} , suppose we could find a set of items J {\displaystyle J} such that their total weight is less than the weight of i {\displaystyle i} , and their total value
9576-431: Is the quick development of this relatively new field requires rapid review and distribution of results, a task better handled by conferences than by journals. Knapsack problem It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items. The problem often arises in resource allocation where the decision-makers have to choose from
9728-399: Is the sum of values of all objects in the knapsack if the representation is valid, or 0 otherwise. In some problems, it is hard or even impossible to define the fitness expression; in these cases, a simulation may be used to determine the fitness function value of a phenotype (e.g. computational fluid dynamics is used to determine the air resistance of a vehicle whose shape is encoded as
9880-686: Is the value of the i {\displaystyle i} -th kind of item. The second property needs to be explained in detail. During the process of the running of this method, how do we get the weight w {\displaystyle w} ? There are only i {\displaystyle i} ways and the previous weights are w − w 1 , w − w 2 , . . . , w − w i {\displaystyle w-w_{1},w-w_{2},...,w-w_{i}} where there are total i {\displaystyle i} kinds of different item (by saying different, we mean that
10032-457: Is worth tuning parameters such as the mutation probability, crossover probability and population size to find reasonable settings for the problem class being worked on. A very small mutation rate may lead to genetic drift (which is non- ergodic in nature). A recombination rate that is too high may lead to premature convergence of the genetic algorithm. A mutation rate that is too high may lead to loss of good solutions, unless elitist selection
10184-544: The O ∗ ( 2 n / 2 ) {\displaystyle O^{*}(2^{n/2})} (up to polynomial factors) running time and reduces the space requirements to O ∗ ( 2 0.249999 n ) {\displaystyle O^{*}(2^{0.249999n})} (see Corollary 1.4). In contrast, the best known deterministic algorithm runs in O ∗ ( 2 n / 2 ) {\displaystyle O^{*}(2^{n/2})} time with
10336-421: The i {\displaystyle i} -th item altogether. In such cases, J {\displaystyle J} is said to dominate i {\displaystyle i} . (Note that this does not apply to bounded knapsack problems, since we may have already used up the items in J {\displaystyle J} .) Finding dominance relations allows us to significantly reduce
10488-444: The Merkle–Hellman and other knapsack cryptosystems . One early application of knapsack algorithms was in the construction and scoring of tests in which the test-takers have a choice as to which questions they answer. For small examples, it is a fairly simple process to provide the test-takers with such a choice. For example, if an exam contains 12 questions each worth 10 points, the test-taker need only answer 10 questions to achieve
10640-418: The Turing machine ). In contrast, decision trees count each decision as a single step. Dobkin and Lipton show an 1 2 n 2 {\displaystyle {1 \over 2}n^{2}} lower bound on linear decision trees for the knapsack problem, that is, trees where decision nodes test the sign of affine functions . This was generalized to algebraic decision trees by Steele and Yao. If
10792-525: The branch and bound approach or hybridizations of both approaches. The unbounded knapsack problem ( UKP ) places no restriction on the number of copies of each kind of item. Besides, here we assume that x i > 0 {\displaystyle x_{i}>0} Observe that m [ w ] {\displaystyle m[w]} has the following properties: 1. m [ 0 ] = 0 {\displaystyle m[0]=0\,\!} (the sum of zero items, i.e.,
10944-406: The pc and pm in order to maintain the population diversity as well as to sustain the convergence capacity. In AGA (adaptive genetic algorithm), the adjustment of pc and pm depends on the fitness values of the solutions. There are more examples of AGA variants: Successive zooming method is an early example of improving convergence. In CAGA (clustering-based adaptive genetic algorithm), through
11096-475: The "technocratic paradigm" (which might be found in engineering approaches, most prominently in software engineering), and the "scientific paradigm" (which approaches computer-related artifacts from the empirical perspective of natural sciences , identifiable in some branches of artificial intelligence ). Computer science focuses on methods involved in design, specification, programming, verification, implementation and testing of human-made computing systems. As
11248-901: The 0-1 knapsack problem also runs in pseudo-polynomial time. Assume w 1 , w 2 , … , w n , W {\displaystyle w_{1},\,w_{2},\,\ldots ,\,w_{n},\,W} are strictly positive integers. Define m [ i , w ] {\displaystyle m[i,w]} to be the maximum value that can be attained with weight less than or equal to w {\displaystyle w} using items up to i {\displaystyle i} (first i {\displaystyle i} items). We can define m [ i , w ] {\displaystyle m[i,w]} recursively as follows: (Definition A) The solution can then be found by calculating m [ n , W ] {\displaystyle m[n,W]} . To do this efficiently, we can use
11400-554: The 100th anniversary of the invention of the arithmometer, Torres presented in Paris the Electromechanical Arithmometer , a prototype that demonstrated the feasibility of an electromechanical analytical engine, on which commands could be typed and the results printed automatically. In 1937, one hundred years after Babbage's impossible dream, Howard Aiken convinced IBM, which was making all kinds of punched card equipment and
11552-413: The 1960s and early 1970s – Rechenberg's group was able to solve complex engineering problems through evolution strategies . Another approach was the evolutionary programming technique of Lawrence J. Fogel , which was proposed for generating artificial intelligence. Evolutionary programming originally used finite state machines for predicting environments, and used variation and selection to optimize
11704-451: The 2nd of the only two designs for mechanical analytical engines in history. In 1914, the Spanish engineer Leonardo Torres Quevedo published his Essays on Automatics , and designed, inspired by Babbage, a theoretical electromechanical calculating machine which was to be controlled by a read-only program. The paper also introduced the idea of floating-point arithmetic . In 1920, to celebrate
11856-618: The Analytical Engine, Ada Lovelace wrote, in one of the many notes she included, an algorithm to compute the Bernoulli numbers , which is considered to be the first published algorithm ever specifically tailored for implementation on a computer. Around 1885, Herman Hollerith invented the tabulator , which used punched cards to process statistical information; eventually his company became part of IBM . Following Babbage, although unaware of his earlier work, Percy Ludgate in 1909 published
12008-551: The Building Block Hypothesis in adaptively reducing disruptive recombination. Prominent examples of this approach include the mGA, GEMGA and LLGA. Problems which appear to be particularly appropriate for solution by genetic algorithms include timetabling and scheduling problems , and many scheduling software packages are based on GAs. GAs have also been applied to engineering . Genetic algorithms are often applied as an approach to solve global optimization problems. As
12160-547: The Machine Organization department in IBM's main research center in 1959. Concurrency is a property of systems in which several computations are executing simultaneously, and potentially interacting with each other. A number of mathematical models have been developed for general concurrent computation including Petri nets , process calculi and the parallel random access machine model. When multiple computers are connected in
12312-546: The UK (as in the School of Informatics, University of Edinburgh ). "In the U.S., however, informatics is linked with applied computing, or computing in the context of another domain." A folkloric quotation, often attributed to—but almost certainly not first formulated by— Edsger Dijkstra , states that "computer science is no more about computers than astronomy is about telescopes." The design and deployment of computers and computer systems
12464-516: The accessibility and usability of the system for its intended users. Historical cryptography is the art of writing and deciphering secret messages. Modern cryptography is the scientific study of problems relating to distributed computations that can be attacked. Technologies studied in modern cryptography include symmetric and asymmetric encryption , digital signatures , cryptographic hash functions , key-agreement protocols , blockchain , zero-knowledge proofs , and garbled circuits . A database
12616-433: The application of a fairly broad variety of theoretical computer science fundamentals, in particular logic calculi, formal languages , automata theory , and program semantics , but also type systems and algebraic data types to problems in software and hardware specification and verification. Computer graphics is the study of digital visual contents and involves the synthesis and manipulation of image data. The study
12768-410: The binary number system. In 1820, Thomas de Colmar launched the mechanical calculator industry when he invented his simplified arithmometer , the first calculating machine strong enough and reliable enough to be used daily in an office environment. Charles Babbage started the design of the first automatic mechanical calculator , his Difference Engine , in 1822, which eventually gave him the idea of
12920-401: The bounded problem, where the supply of each kind of item is limited, the above algorithm may be far from optimal. Nevertheless, a simple modification allows us to solve this case: Assume for simplicity that all items individually fit in the sack ( w i ≤ W {\displaystyle w_{i}\leq W} for all i {\displaystyle i} ). Construct
13072-471: The design and implementation of hardware and software ). Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and preventing security vulnerabilities . Computer graphics and computational geometry address
13224-467: The development of sophisticated computing equipment. Wilhelm Schickard designed and constructed the first working mechanical calculator in 1623. In 1673, Gottfried Leibniz demonstrated a digital mechanical calculator, called the Stepped Reckoner . Leibniz may be considered the first computer scientist and information theorist, because of various reasons, including the fact that he documented
13376-583: The discipline of computer science: theory of computation , algorithms and data structures , programming methodology and languages , and computer elements and architecture . In addition to these four areas, CSAB also identifies fields such as software engineering, artificial intelligence, computer networking and communication, database systems, parallel computation, distributed computation, human–computer interaction, computer graphics, operating systems, and numerical and symbolic computation as being important areas of computer science. Theoretical computer science
13528-424: The distinction more a matter of style than of technical capabilities. Conferences are important events for computer science research. During these conferences, researchers from the public and private sectors present their recent work and meet. Unlike in most other academic fields, in computer science, the prestige of conference papers is greater than that of journal publications. One proposed explanation for this
13680-459: The distinction of three separate paradigms in computer science. Peter Wegner argued that those paradigms are science, technology, and mathematics. Peter Denning 's working group argued that they are theory, abstraction (modeling), and design. Amnon H. Eden described them as the "rationalist paradigm" (which treats computer science as a branch of mathematics, which is prevalent in theoretical computer science, and mainly employs deductive reasoning),
13832-423: The distribution of the sampling probability tuned to focus in those areas of greater interest. During each successive generation, a portion of the existing population is selected to reproduce for a new generation. Individual solutions are selected through a fitness-based process, where fitter solutions (as measured by a fitness function ) are typically more likely to be selected. Certain selection methods rate
13984-431: The early 1960s, and the methods were described in books by Fraser and Burnell (1970) and Crosby (1973). Fraser's simulations included all of the essential elements of modern genetic algorithms. In addition, Hans-Joachim Bremermann published a series of papers in the 1960s that also adopted a population of solution to optimization problems, undergoing recombination, mutation, and selection. Bremermann's research also included
14136-537: The elements in the problem are real numbers or rationals , the decision-tree lower bound extends to the real random-access machine model with an instruction set that includes addition, subtraction and multiplication of real numbers, as well as comparison and either division or remaindering ("floor"). This model covers more algorithms than the algebraic decision-tree model, as it encompasses algorithms that use indexing into tables. However, in this model all program steps are counted, not just decisions. An upper bound for
14288-451: The elements of modern genetic algorithms. Other noteworthy early pioneers include Richard Friedberg, George Friedman, and Michael Conrad. Many early papers are reprinted by Fogel (1998). Although Barricelli, in work he reported in 1963, had simulated the evolution of ability to play a simple game, artificial evolution only became a widely recognized optimization method as a result of the work of Ingo Rechenberg and Hans-Paul Schwefel in
14440-511: The expression "automatic information" (e.g. "informazione automatica" in Italian) or "information and mathematics" are often used, e.g. informatique (French), Informatik (German), informatica (Italian, Dutch), informática (Spanish, Portuguese), informatika ( Slavic languages and Hungarian ) or pliroforiki ( πληροφορική , which means informatics) in Greek . Similar words have also been adopted in
14592-457: The first programmable mechanical calculator , his Analytical Engine . He started developing this machine in 1834, and "in less than two years, he had sketched out many of the salient features of the modern computer". "A crucial step was the adoption of a punched card system derived from the Jacquard loom " making it infinitely programmable. In 1843, during the translation of a French article on
14744-477: The first question, computability theory examines which computational problems are solvable on various theoretical models of computation . The second question is addressed by computational complexity theory , which studies the time and space costs associated with different approaches to solving a multitude of computational problems. The famous P = NP? problem, one of the Millennium Prize Problems ,
14896-417: The fitness of each solution and preferentially select the best solutions. Other methods rate only a random sample of the population, as the former process may be very time-consuming. The fitness function is defined over the genetic representation and measures the quality of the represented solution. The fitness function is always problem-dependent. For instance, in the knapsack problem one wants to maximize
15048-475: The general process of constructing a new population is to allow the best organism(s) from the current generation to carry over to the next, unaltered. This strategy is known as elitist selection and guarantees that the solution quality obtained by the GA will not decrease from one generation to the next. Parallel implementations of genetic algorithms come in two flavors. Coarse-grained parallel genetic algorithms assume
15200-450: The generality and/or practicality of the building-block hypothesis as an explanation for GAs' efficiency still remains. Indeed, there is a reasonable amount of work that attempts to understand its limitations from the perspective of estimation of distribution algorithms. The practical use of a genetic algorithm has limitations, especially as compared to alternative optimization algorithms: The simplest algorithm represents each chromosome as
15352-461: The generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of repositories of data. Human–computer interaction investigates the interfaces through which humans and computers interact, and software engineering focuses on the design and principles behind developing software. Areas such as operating systems , networks and embedded systems investigate
15504-478: The genetic diversity of the subsequent generation of children. Opinion is divided over the importance of crossover versus mutation. There are many references in Fogel (2006) that support the importance of mutation-based search. Although crossover and mutation are known as the main genetic operators, it is possible to use other operators such as regrouping, colonization-extinction, or migration in genetic algorithms. It
15656-440: The heuristic as follows: Despite the lack of consensus regarding the validity of the building-block hypothesis, it has been consistently evaluated and used as reference throughout the years. Many estimation of distribution algorithms , for example, have been proposed in an attempt to provide an environment in which the hypothesis would hold. Although good results have been reported for some classes of problems, skepticism concerning
15808-491: The individual filling the knapsack. Instead of one objective, such as maximizing the monetary profit, the objective could have several dimensions. For example, there could be environmental or social concerns as well as economic goals. Problems frequently addressed include portfolio and transportation logistics optimizations. As an example, suppose you ran a cruise ship. You have to decide how many famous comedians to hire. This boat can handle no more than one ton of passengers and
15960-471: The input. If the weights and profits are given as integers, it is weakly NP-complete , while it is strongly NP-complete if the weights and profits are given as rational numbers. However, in the case of rational weights and profits it still admits a fully polynomial-time approximation scheme . The NP-hardness of the Knapsack problem relates to computational models in which the size of integers matters (such as
16112-498: The internal loop controller structure can belong to a conventional regulator of three parameters, whereas the external loop could implement a linguistic controller (such as a fuzzy system) which has an inherently different description. This particular form of encoding requires a specialized crossover mechanism that recombines the chromosome by section, and it is a useful tool for the modelling and simulation of complex adaptive systems, especially evolution processes. A practical variant of
16264-5156: The items themselves that we chose are fixed. That is to say, the program above computes more than necessary because the weight changes from 0 to W often. From this perspective, we can program this method so that it runs recursively. For example, there are 10 different items and the weight limit is 67. So, w [ 1 ] = 23 , w [ 2 ] = 26 , w [ 3 ] = 20 , w [ 4 ] = 18 , w [ 5 ] = 32 , w [ 6 ] = 27 , w [ 7 ] = 29 , w [ 8 ] = 26 , w [ 9 ] = 30 , w [ 10 ] = 27 v [ 1 ] = 505 , v [ 2 ] = 352 , v [ 3 ] = 458 , v [ 4 ] = 220 , v [ 5 ] = 354 , v [ 6 ] = 414 , v [ 7 ] = 498 , v [ 8 ] = 545 , v [ 9 ] = 473 , v [ 10 ] = 543 {\displaystyle {\begin{aligned}&w[1]=23,w[2]=26,w[3]=20,w[4]=18,w[5]=32,w[6]=27,w[7]=29,w[8]=26,w[9]=30,w[10]=27\\&v[1]=505,v[2]=352,v[3]=458,v[4]=220,v[5]=354,v[6]=414,v[7]=498,v[8]=545,v[9]=473,v[10]=543\\\end{aligned}}} If you use above method to compute for m ( 10 , 67 ) {\displaystyle m(10,67)} , you will get this, excluding calls that produce m ( i , j ) = 0 {\displaystyle m(i,j)=0} : m ( 10 , 67 ) = 1270 m ( 9 , 67 ) = 1270 , m ( 9 , 40 ) = 678 m ( 8 , 67 ) = 1270 , m ( 8 , 40 ) = 678 , m ( 8 , 37 ) = 545 m ( 7 , 67 ) = 1183 , m ( 7 , 41 ) = 725 , m ( 7 , 40 ) = 678 , m ( 7 , 37 ) = 505 m ( 6 , 67 ) = 1183 , m ( 6 , 41 ) = 725 , m ( 6 , 40 ) = 678 , m ( 6 , 38 ) = 678 , m ( 6 , 37 ) = 505 m ( 5 , 67 ) = 1183 , m ( 5 , 41 ) = 725 , m ( 5 , 40 ) = 678 , m ( 5 , 38 ) = 678 , m ( 5 , 37 ) = 505 m ( 4 , 67 ) = 1183 , m ( 4 , 41 ) = 725 , m ( 4 , 40 ) = 678 , m ( 4 , 38 ) = 678 , m ( 4 , 37 ) = 505 , m ( 4 , 35 ) = 505 m ( 3 , 67 ) = 963 , m ( 3 , 49 ) = 963 , m ( 3 , 41 ) = 505 , m ( 3 , 40 ) = 505 , m ( 3 , 38 ) = 505 , m ( 3 , 37 ) = 505 , m ( 3 , 35 ) = 505 , m ( 3 , 23 ) = 505 , m ( 3 , 22 ) = 458 , m ( 3 , 20 ) = 458 m ( 2 , 67 ) = 857 , m ( 2 , 49 ) = 857 , m ( 2 , 47 ) = 505 , m ( 2 , 41 ) = 505 , m ( 2 , 40 ) = 505 , m ( 2 , 38 ) = 505 , m ( 2 , 37 ) = 505 , m ( 2 , 35 ) = 505 , m ( 2 , 29 ) = 505 , m ( 2 , 23 ) = 505 m ( 1 , 67 ) = 505 , m ( 1 , 49 ) = 505 , m ( 1 , 47 ) = 505 , m ( 1 , 41 ) = 505 , m ( 1 , 40 ) = 505 , m ( 1 , 38 ) = 505 , m ( 1 , 37 ) = 505 , m ( 1 , 35 ) = 505 , m ( 1 , 29 ) = 505 , m ( 1 , 23 ) = 505 {\displaystyle {\begin{aligned}&m(10,67)=1270\\&m(9,67)=1270,m(9,40)=678\\&m(8,67)=1270,m(8,40)=678,m(8,37)=545\\&m(7,67)=1183,m(7,41)=725,m(7,40)=678,m(7,37)=505\\&m(6,67)=1183,m(6,41)=725,m(6,40)=678,m(6,38)=678,m(6,37)=505\\&m(5,67)=1183,m(5,41)=725,m(5,40)=678,m(5,38)=678,m(5,37)=505\\&m(4,67)=1183,m(4,41)=725,m(4,40)=678,m(4,38)=678,m(4,37)=505,m(4,35)=505\\&m(3,67)=963,m(3,49)=963,m(3,41)=505,m(3,40)=505,m(3,38)=505,m(3,37)=505,m(3,35)=505,m(3,23)=505,m(3,22)=458,m(3,20)=458\\&m(2,67)=857,m(2,49)=857,m(2,47)=505,m(2,41)=505,m(2,40)=505,m(2,38)=505,m(2,37)=505,m(2,35)=505,m(2,29)=505,m(2,23)=505\\&m(1,67)=505,m(1,49)=505,m(1,47)=505,m(1,41)=505,m(1,40)=505,m(1,38)=505,m(1,37)=505,m(1,35)=505,m(1,29)=505,m(1,23)=505\\\end{aligned}}} Besides, we can break
16416-511: The knapsack problem has a fully polynomial time approximation scheme (FPTAS). George Dantzig proposed a greedy approximation algorithm to solve the unbounded knapsack problem. His version sorts the items in decreasing order of value per unit of weight, v 1 / w 1 ≥ ⋯ ≥ v n / w n {\displaystyle v_{1}/w_{1}\geq \cdots \geq v_{n}/w_{n}} . It then proceeds to insert them into
16568-400: The knapsack's capacity. The bounded knapsack problem ( BKP ) removes the restriction that there is only one of each item, but restricts the number x i {\displaystyle x_{i}} of copies of each kind of item to a maximum non-negative integer value c {\displaystyle c} : The unbounded knapsack problem ( UKP ) places no upper bound on
16720-498: The last few mutations to find the absolute optimum. Other techniques (such as simple hill climbing ) are quite efficient at finding absolute optimum in a limited region. Alternating GA and hill climbing can improve the efficiency of GA while overcoming the lack of robustness of hill climbing. This means that the rules of genetic variation may have a different meaning in the natural case. For instance – provided that steps are stored in consecutive order – crossing over may sum
16872-502: The machine in operation and analyzing it by all analytical and measurement means available. It has since been argued that computer science can be classified as an empirical science since it makes use of empirical testing to evaluate the correctness of programs , but a problem remains in defining the laws and theorems of computer science (if any exist) and defining the nature of experiments in computer science. Proponents of classifying computer science as an engineering discipline argue that
17024-413: The mutation, crossover, inversion and selection operators. The population size depends on the nature of the problem, but typically contains hundreds or thousands of possible solutions. Often, the initial population is generated randomly, allowing the entire range of possible solutions (the search space ). Occasionally, the solutions may be "seeded" in areas where optimal solutions are likely to be found or
17176-427: The next generation population of chromosomes that is different from the initial generation. Generally, the average fitness will have increased by this procedure for the population, since only the best organisms from the first generation are selected for breeding, along with a small proportion of less fit solutions. These less fit solutions ensure genetic diversity within the genetic pool of the parents and therefore ensure
17328-527: The next generation, known as Holland's Schema Theorem . Research in GAs remained largely theoretical until the mid-1980s, when The First International Conference on Genetic Algorithms was held in Pittsburgh, Pennsylvania . In the late 1980s, General Electric started selling the world's first genetic algorithm product, a mainframe-based toolkit designed for industrial processes. In 1989, Axcelis, Inc. released Evolver ,
17480-431: The number of copies of each kind of item and can be formulated as above except that the only restriction on x i {\displaystyle x_{i}} is that it is a non-negative integer. One example of the unbounded knapsack problem is given using the figure shown at the beginning of this article and the text "if any number of each book is available" in the caption of that figure. The knapsack problem
17632-409: The number of copies of each member of J {\displaystyle J} . There are many variations of the knapsack problem that have arisen from the vast number of applications of the basic problem. The main variations occur by changing the number of some problem parameter such as the number of items, number of objectives, or even the number of knapsacks. This variation changes the goal of
17784-401: The optimal solution. As with many useful but computationally complex algorithms, there has been substantial research on creating and analyzing algorithms that approximate a solution. The knapsack problem, though NP-Hard, is one of a collection of algorithms that can still be approximated to any specified degree. This means that the problem has a polynomial time approximation scheme. To be exact,
17936-588: The optimal value of the optimization problem in polynomial time, then the decision problem can be solved in polynomial time by comparing the value of the solution output by this algorithm with the value of k. Thus, both versions of the problem are of similar difficulty. One theme in research literature is to identify what the "hard" instances of the knapsack problem look like, or viewed another way, to identify what properties of instances in practice might make them more amenable than their worst-case NP-complete behaviour suggests. The goal in finding these "hard" instances
18088-435: The overall genetic algorithm process (seen as a Markov chain ). Examples of problems solved by genetic algorithms include: mirrors designed to funnel sunlight to a solar collector, antennae designed to pick up radio signals in space, walking methods for computer figures, optimal design of aerodynamic bodies in complex flowfields In his Algorithm Design Manual , Skiena advises against genetic algorithms for any task: [I]t
18240-407: The phenotype), or even interactive genetic algorithms are used. The next step is to generate a second generation population of solutions from those selected, through a combination of genetic operators : crossover (also called recombination), and mutation . For each new solution to be produced, a pair of "parent" solutions is selected for breeding from the pool selected previously. By producing
18392-457: The population as a whole is evolved rather than its individual members, is known as gene pool recombination. A number of variations have been developed to attempt to improve performance of GAs on problems with a high degree of fitness epistasis, i.e. where the fitness of a solution consists of interacting subsets of its variables. Such algorithms aim to learn (before exploiting) these beneficial phenotypic interactions. As such, they are aligned with
18544-458: The predictive logics. Genetic algorithms in particular became popular through the work of John Holland in the early 1970s, and particularly his book Adaptation in Natural and Artificial Systems (1975). His work originated with studies of cellular automata , conducted by Holland and his students at the University of Michigan . Holland introduced a formalized framework for predicting the quality of
18696-478: The principal focus of computer science is studying the properties of computation in general, while the principal focus of software engineering is the design of specific computations to achieve practical goals, making the two separate but complementary disciplines. The academic, political, and funding aspects of computer science tend to depend on whether a department is formed with a mathematical emphasis or with an engineering emphasis. Computer science departments with
18848-615: The principles and design behind complex systems . Computer architecture describes the construction of computer components and computer-operated equipment. Artificial intelligence and machine learning aim to synthesize goal-orientated processes such as problem-solving, decision-making, environmental adaptation, planning and learning found in humans and animals. Within artificial intelligence, computer vision aims to understand and process image and video data, while natural language processing aims to understand and process textual and linguistic data. The fundamental concern of computer science
19000-418: The problem is proportional to the number of bits in W {\displaystyle W} , log W {\displaystyle \log W} , not to W {\displaystyle W} itself. However, since this runtime is pseudopolynomial , this makes the (decision version of the) knapsack problem a weakly NP-complete problem . A similar dynamic programming solution for
19152-691: The problem requires d {\displaystyle d} fractional digits of precision to arrive at the correct answer, W {\displaystyle W} will need to be scaled by 10 d {\displaystyle 10^{d}} , and the DP algorithm will require O ( W 10 d ) {\displaystyle O(W10^{d})} space and O ( n W 10 d ) {\displaystyle O(nW10^{d})} time. The algorithm takes O ( 2 n / 2 ) {\displaystyle O(2^{n/2})} space, and efficient implementations of step 3 (for instance, sorting
19304-401: The reason the problem has no known polynomial time solutions is because the profits associated with the items are not restricted. If one rounds off some of the least significant digits of the profit values then they will be bounded by a polynomial and 1/ε where ε is a bound on the correctness of the solution. This restriction then means that an algorithm can find a solution in polynomial time that
19456-410: The recent two lines of the array "m".) However, if we take it a step or two further, we should know that the method will run in the time between O ( n W ) {\displaystyle O(nW)} and O ( 2 n ) {\displaystyle O(2^{n})} . From Definition A , we know that there is no need to compute all the weights when the number of items and
19608-429: The recursion and convert it into a tree. Then we can cut some leaves and use parallel computing to expedite the running of this method. To find the actual subset of items, rather than just their total value, we can run this after running the function above: Another algorithm for 0-1 knapsack, discovered in 1974 and sometimes called "meet-in-the-middle" due to parallels to a similarly named algorithm in cryptography ,
19760-484: The reliability of computational systems is investigated in the same way as bridges in civil engineering and airplanes in aerospace engineering . They also argue that while empirical sciences observe what presently exists, computer science observes what is possible to exist and while scientists discover laws from observation, no proper laws have been found in computer science and it is instead concerned with creating phenomena. Proponents of classifying computer science as
19912-429: The right way to attack it. Further, I have never seen any computational results reported using genetic algorithms that have favorably impressed me. Stick to simulated annealing for your heuristic search voodoo needs. In 1950, Alan Turing proposed a "learning machine" which would parallel the principles of evolution. Computer simulation of evolution started as early as in 1954 with the work of Nils Aall Barricelli , who
20064-437: The sack, starting with as many copies as possible of the first kind of item until there is no longer space in the sack for more. Provided that there is an unlimited supply of each kind of item, if m {\displaystyle m} is the maximum value of items that fit into the sack, then the greedy algorithm is guaranteed to achieve at least a value of m / 2 {\displaystyle m/2} . For
20216-409: The scale of human intelligence. But the automation of evaluative and predictive tasks has been increasingly successful as a substitute for human monitoring and intervention in domains of computer application involving complex real-world data. Computer architecture, or digital computer organization, is the conceptual design and fundamental operational structure of a computer system. It focuses largely on
20368-1249: The size of the search space. There are several different types of dominance relations , which all satisfy an inequality of the form: ∑ j ∈ J w j x j ≤ α w i {\displaystyle \qquad \sum _{j\in J}w_{j}\,x_{j}\ \leq \alpha \,w_{i}} , and ∑ j ∈ J v j x j ≥ α v i {\displaystyle \sum _{j\in J}v_{j}\,x_{j}\ \geq \alpha \,v_{i}\,} for some x ∈ Z + n {\displaystyle x\in Z_{+}^{n}} where α ∈ Z + , J ⊊ N {\displaystyle \alpha \in Z_{+}\,,J\subsetneq N} and i ∉ J {\displaystyle i\not \in J} . The vector x {\displaystyle x} denotes
20520-690: The solution. Since the calculation of each m [ w ] {\displaystyle m[w]} involves examining at most n {\displaystyle n} items, and there are at most W {\displaystyle W} values of m [ w ] {\displaystyle m[w]} to calculate, the running time of the dynamic programming solution is O ( n W ) {\displaystyle O(nW)} . Dividing w 1 , w 2 , … , w n , W {\displaystyle w_{1},\,w_{2},\,\ldots ,\,w_{n},\,W} by their greatest common divisor
20672-470: The subsets of B by weight, discarding subsets of B which weigh more than other subsets of B of greater or equal value, and using binary search to find the best match) result in a runtime of O ( n 2 n / 2 ) {\displaystyle O(n2^{n/2})} . As with the meet in the middle attack in cryptography, this improves on the O ( n 2 n ) {\displaystyle O(n2^{n})} runtime of
20824-587: The summation of the empty set). 2. m [ w ] = max ( v 1 + m [ w − w 1 ] , v 2 + m [ w − w 2 ] , . . . , v n + m [ w − w n ] ) {\displaystyle m[w]=\max(v_{1}+m[w-w_{1}],v_{2}+m[w-w_{2}],...,v_{n}+m[w-w_{n}])} , w i ≤ w {\displaystyle w_{i}\leq w} , where v i {\displaystyle v_{i}}
20976-559: The term computer came to refer to the machines rather than their human predecessors. As it became clear that computers could be used for more than just mathematical calculations, the field of computer science broadened to study computation in general. In 1945, IBM founded the Watson Scientific Computing Laboratory at Columbia University in New York City . The renovated fraternity house on Manhattan's West Side
21128-743: The term "computer science" appears in a 1959 article in Communications of the ACM , in which Louis Fein argues for the creation of a Graduate School in Computer Sciences analogous to the creation of Harvard Business School in 1921. Louis justifies the name by arguing that, like management science , the subject is applied and interdisciplinary in nature, while having the characteristics typical of an academic discipline. His efforts, and those of others such as numerical analyst George Forsythe , were rewarded: universities went on to create such departments, starting with Purdue in 1962. Despite its name,
21280-561: The term was the Department of Datalogy at the University of Copenhagen, founded in 1969, with Peter Naur being the first professor in datalogy. The term is used mainly in the Scandinavian countries. An alternative term, also proposed by Naur, is data science ; this is now used for a multi-disciplinary field of data analysis, including statistics and databases. In the early days of computing,
21432-436: The theory of schemata suggest that in general the smaller the alphabet, the better the performance, but it was initially surprising to researchers that good results were obtained from using real-valued chromosomes. This was explained as the set of real values in a finite population of chromosomes as forming a virtual alphabet (when selection and recombination are dominant) with a much lower cardinality than would be expected from
21584-409: The total value of objects that can be put in a knapsack of some fixed capacity. A representation of a solution might be an array of bits, where each bit represents a different object, and the value of the bit (0 or 1) represents whether or not the object is in the knapsack. Not every such representation is valid, as the size of objects may exceed the capacity of the knapsack. The fitness of the solution
21736-443: The two fields in areas such as mathematical logic , category theory , domain theory , and algebra . The relationship between computer science and software engineering is a contentious issue, which is further muddied by disputes over what the term "software engineering" means, and how computer science is defined. David Parnas , taking a cue from the relationship between other engineering and science disciplines, has claimed that
21888-481: The type of information carrier – whether it is electrical, mechanical or biological. This field plays important role in information theory , telecommunications , information engineering and has applications in medical image computing and speech synthesis , among others. What is the lower bound on the complexity of fast Fourier transform algorithms? is one of the unsolved problems in theoretical computer science . Scientific computing (or computational science)
22040-605: The use of clustering analysis to judge the optimization states of the population, the adjustment of pc and pm depends on these optimization states. Recent approaches use more abstract variables for deciding pc and pm . Examples are dominance & co-dominance principles and LIGA (levelized interpolative genetic algorithm), which combines a flexible GA with modified A* search to tackle search space anisotropicity. It can be quite effective to combine GA with other optimization methods. A GA tends to be quite good at finding generally good global solutions, but quite inefficient at finding
22192-438: The way by which the central processing unit performs internally and accesses addresses in memory. Computer engineers study computational logic and design of computer hardware, from individual processor components, microcontrollers , personal computers to supercomputers and embedded systems . The term "architecture" in computer literature can be traced to the work of Lyle R. Johnson and Frederick P. Brooks Jr. , members of
22344-498: The weight and the value are not completely the same). If we know each value of these i {\displaystyle i} items and the related maximum value previously, we just compare them to each other and get the maximum value ultimately and we are done. Here the maximum of the empty set is taken to be zero. Tabulating the results from m [ 0 ] {\displaystyle m[0]} up through m [ W ] {\displaystyle m[W]} gives
22496-592: The world's first commercial GA product for desktop computers. The New York Times technology writer John Markoff wrote about Evolver in 1990, and it remained the only interactive commercial genetic algorithm until 1995. Evolver was sold to Palisade in 1997, translated into several languages, and is currently in its 6th version. Since the 1990s, MATLAB has built in three derivative-free optimization heuristic algorithms (simulated annealing, particle swarm optimization, genetic algorithm) and two direct search algorithms (simplex search, pattern search). Genetic algorithms are
22648-440: Was IBM's first laboratory devoted to pure science. The lab is the forerunner of IBM's Research Division, which today operates research facilities around the world. Ultimately, the close relationship between IBM and Columbia University was instrumental in the emergence of a new scientific discipline, with Columbia offering one of the first academic-credit courses in computer science in 1946. Computer science began to be established as
22800-536: Was also in the calculator business to develop his giant programmable calculator, the ASCC/Harvard Mark I , based on Babbage's Analytical Engine, which itself used cards and a central computing unit. When the machine was finished, some hailed it as "Babbage's dream come true". During the 1940s, with the development of new and more powerful computing machines such as the Atanasoff–Berry computer and ENIAC ,
22952-730: Was proposed by John Henry Holland in the 1970s. This theory is not without support though, based on theoretical and experimental results (see below). The basic algorithm performs crossover and mutation at the bit level. Other variants treat the chromosome as a list of numbers which are indexes into an instruction table, nodes in a linked list , hashes , objects , or any other imaginable data structure . Crossover and mutation are performed so as to respect data element boundaries. For most data types, specific variation operators can be designed. Different chromosomal data types seem to work better or worse for different specific problem domains. When bit-string representations of integers are used, Gray coding
23104-584: Was using the computer at the Institute for Advanced Study in Princeton, New Jersey . His 1954 publication was not widely noticed. Starting in 1957, the Australian quantitative geneticist Alex Fraser published a series of papers on simulation of artificial selection of organisms with multiple loci controlling a measurable trait. From these beginnings, computer simulation of evolution by biologists became more common in
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