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Covering Problem

Covering Problem

In combinatorics and computer science, the covering problem is a type of general question: if a certain structure covers another, or how many structures are required to cover another? For Petri nets, for example, the covering problem is defined as the question if for a given marking, there exists a run of the net, such that some larger (or equal) marking can be reached. Larger means here that all components are at least as large as the ones of the given marking and at least one one is properly larger. See also: set cover problem, vertex cover problem, packing problem. Category:Combinatorics Category:Computer science

CombinaTorics

Combinatorics

Computer science

Computer science, an academic discipline (abbreviated CS or compsci), is a body of knowledge generally about computer hardware, software, computation and its theory. The discipline itself includes, but is not limited to, the fundamentals of computer languages, operating systems and mathematical foundations of computer science. The study of these fundamentals may lead to a wide variety of topics, such as algorithms, formal grammars, programming languages, program design, artificial intelligence and computer engineering. There exist a number of technical definitions of computer science. The status of computer science as a science is often challenged, typically arguing that it is more like mathematics and that it does not follow the scientific method, however these facts are not unanimously accepted. In popular language, the term computer science is often confusingly used to denominate anything related to computers.

History of computer science

Evolutionary

Before the 1920s, computers were human clerks that performed calculations. They were usually under the lead of a physicist. Many thousands of computers were employed in commerce, government, and research establishments. Most of these computers were women, and they were known to have a degree in calculus. After the 1920s, the expression computing machine refered to any machine that performed the work of a human computer, especially those in accordance with effective methods of The Church-Turing Thesis. The thesis states that a mathematical method is effective if it could be set out as a list of instructions able to be followed by a human clerk with paper and pencil, for as long as necessary, and without ingenuity or insight. Machines that computed with discrete values became known as the analog kind. They used machinery that represented discrete numeric quantities, like the angle of a shaft rotation or difference in electrical potential. Digital machinery, in contrast to analog, were able to render a state of a numeric value and store each individual digit. Digital machinery used difference engines or relays before the invention of faster memory devices. The phrase computing machine gradually gave away, after the late 1940s, to just computer as the onset of electronic digital machinery became common. These computers were able to perform the calculations that were performed by the previous human clerks. Since the values stored by digital machines were not bound to physical properties like the analog device, a logical computer, based on digital equipment, was able to do anything that could be described "purely mechanical." Alan Turing, known as the Father of Computer Science, invented such a logical computer, also known as a Turing Machine, that evolved into the modern computer from the tasks performed by the previous human clerks. These new computers were also able to perform non-numeric computations, like music. Computability, by logical computers, began a science by being able to make evident which was not explicitly defined into ordinary sense more immediate.

Academic discipline

Computer science has roots in electrical engineering, logic, mathematics, and linguistics. In the last third of the 20th century computer science emerged as a distinct discipline and developed its own methods and terminology. Originally, CS was taught as part of mathematics or engineering departments, for instance at the University of Cambridge in England and at the Gdansk University of Technology in Poland, respectively. Cambridge claims to have the world's oldest taught qualification in computing. The first computer science department in the United States was founded at Purdue University in 1962, while the first college entirely devoted to computer science was founded at Northeastern University in 1982. Most universities today have specific departments devoted to computer science, while some conjoin it with engineering, with applied mathematics, or other disciplines. Most research in computer science has focused on von Neumann computers or Turing machines (computation models that perform one small, deterministic step at a time). These models resemble, at a basic level, most real computers in use today. Computer scientists also study other models of computation, which includes parallel machines and theoretical models such as probabilistic, oracle, and quantum computers.

Careers

Some of the potential careers for those who study computer science are listed below:

Demographics

- Nearly half of all computer programmers held a bachelor’s degree in 2002; about 1 in 5 held a graduate degree. [http://www.bls.gov/oco/ocos110.htm]
- Computer programmer employment is expected to grow much more slowly than that of other computer specialists.
- Education requirements range from a 2-year degree to a graduate degree. [http://www.bls.gov/oco/ocos042.htm]
- Employment, other than computer programmer, is expected to increase much faster than the average as organizations continue to adopt increasingly sophisticated technologies.

Sub-disciplines of computer science

Computer Science has a number of major sub-fields which can be classified by a number of means (for example the [http://www.acm.org/class/1998/overview.html ACM classification system]).

Algorithms

The study of algorithms is aimed at creating techniques that will enable a computer to perform a certain task in an efficient manner. An algorithm is a set of well-defined instructions for accomplishing some task, often explained by analogy with a culinary recipe. Algorithms are often implemented in software, and advancing the state of the art in algorithms is responsible for many of the most spectacular successes in computing. An algorithms specialist may come up with methods to accomplish new tasks, but just as often, they will work on improving the efficiency of an existing algorithm. These improvements can come in "time" (the length of time it takes for the algorithm to work) and "space" (the amount of computer memory the algorithm consumes. The field of algorithms is highly formal and many things can be proved about a given algorithm (using complexity theory), including roughly how long it will take to complete, as compared to the size of its input (the number of options it must consider). One interesting open question in algorithms concerns the complexity classes P and NP, and whether or not P = NP. If it does, then a whole range of seemingly difficult algorithms can in theory be performed quickly.

Data structures

A data structure is a way to store data so that it can be remembered and retrieved efficiently. A well-designed data structure means that an algorithm can be performed using as little memory space and time as possible. Some data structures are very simple: the simplest is an array which is simply a numbered list of items. Since the memory of a computer is usually modelled as an array (of bytes), this is also one of the most important data structures in computer science. For example, strings of text are usually modelled as arrays. But there are many other data structures such as linked lists, trees, hash tables and many others that are quite different but critical to the science. Type theory classifies data at a most basic (mathematical) level into different types, such as integers, complex numbers, strings, etc. and deals with how those types can interact. Abstract data types, at a more concrete level, deal with how types and data structures are used in software programming.

Listing of sub-disciplines

Computer science is closely related to a number of fields. These fields overlap considerably, though important differences exist:

Major fields of importance for computer science

External links

- [http://www.dmoz.org/Computers/Computer_Science/ Open Directory Project: Computer Science]
- [http://www.techbooksforfree.com/science.shtml Downloadable Science and Computer Science books]
- [http://liinwww.ira.uka.de/bibliography/ Collection of Computer Science Bibliographies]
- [http://www.geocities.com/tablizer/science.htm Belief that title "science" in "computer science" is inappropriate] Category:Computer science ko:컴퓨터 과학 ja:情報工学 simple:Computer science th:วิทยาการคอมพิวเตอร์

Set cover problem

The set cover problem (also set covering) is a classical question in computer science and complexity theory. As input you are given several sets. They may have some elements in common. You must select a minimum number of these sets so that the sets you have picked contain all the elements that are contained in any of the sets in the input. It was one of Karp's 21 NP-complete problems shown to be NP-complete in 1972. More formally, the input is a set of sets denoted

S_i

, so the first set is

S_0

, the second

S_1

, and so on. Use n to indicate the number of input sets, so the last input set is

S_

. A solution to the set covering problem is a selection of a minimum number of the input sets, which we can indicate by giving the index numbers of each set that we chose. That is, a solution is a subset

C

of . The solution

C

has two properties. First, it includes all elements of the input sets, so :

\bigcup_ S_j = \bigcup_ S_i

. Second, there is no way to pick fewer of the input sets and still cover all the elements: :

\not\exists D : |D|<|C| \mbox \bigcup_ S_j = \bigcup_ S_i

The most direct way (known as "brute force") of solving set cover is to try all combinations of sets. This can be improved upon, for example by trying the larger sets first, an approach known as "greedy set cover". It has been shown that the set cover problem is NP-hard. Related problems are vertex cover, set packing, and edge cover. The set cover problem can be seen as a finite version of the notion of compactness in topology, where the elements of certain infinite families of sets can be covered by choosing only finitely many of them.

References

- Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0262032937. Section 35.3, The set-covering problem, pp.1033–1038. Category:Set families ja:集合被覆問題

Vertex cover problem

In computer science, the vertex cover problem or node cover problem is an NP-complete problem in complexity theory, and was one of Karp's 21 NP-complete problems. Karp's 21 NP-complete problems A vertex cover of an undirected graph

G = (V, E)

is a subset

V'

of the vertices of the graph which contains at least one of the two endpoints of each edge: :

V' \subseteq V: \forall \ \in E : a \in V' \or b \in V'

. In the graph at the right, is an example of a vertex cover. The vertex cover problem is the optimization problem of finding a vertex cover of minimum size in a graph. The problem can also be stated as a decision problem: :INSTANCE: A graph

G

and a positive integer

k

. :QUESTION: Is there a vertex cover of size

k

or less for

G

? Vertex cover is NP-complete, which means it is unlikely that there is an efficient algorithm to solve it. NP-completeness can be proven by reduction from 3-satisfiability or, as Karp did, by reduction from the clique problem. As shown by Garey and Johnson in 1974, vertex cover remains NP-complete even in cubic graphs and even in planar graphs of degree at most 6. Vertex cover is closely related to Independent Set problem by this theorem: a graph with

n

vertices has a vertex cover of size

k

if and only if the graph has an independent set of size

n-k

. One can find a factor-2 approximation by repeatedly taking both endpoints of an edge into the vertex cover, removing them from the graph. No better constant-factor approximation is known; the problem is APX-complete, i.e., it cannot be approximated arbitrarily well. A brute force algorithm to find a vertex cover in a graph is to choose some vertex and recursively branch into two cases: either take this vertex into the vertex cover, or all its neighbors. This algorithm is exponential in

k

, but not in the size of the graph, i.e., vertex cover is fixed-parameter tractable with respect to

k

References

- Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York, 1979.
- Preeti Patil & Sangita Patil. (Lecturer in Vartak Polytechnic) A Guide to the Theory of NP-Completeness. BPB & Co., Mumbai, 2004.
- M. R. Garey, D. S. Johnson, and L. Stockmeyer. [http://portal.acm.org/citation.cfm?id=803884 Some simplified NP-complete problems]. Proceedings of the sixth annual ACM symposium on Theory of computing, p.47-63. 1974.
- Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0262032937. Section 35.1, pp.1024–1027.
- A1.1: GT1, pg.190. Category:Graph algorithms Category:NP-complete problems ja:頂点被覆問題

Packing problem

Packing problems are one area where mathematics meets puzzles (recreational mathematics). Many of these problems stem from real-life packing problems. In a packing problem, you are given:
- one or more (usually two-or three-dimensional) containers
- several 'goods', some or all of which must be packed into this container Usually the packing must be without gaps or overlaps, but in some packing problems the overlapping (of goods with each other and/or with the boundary of the container) is allowed but should be minimised. In others, gaps are allowed, but overlaps are not (usually the total area of gaps has to be minimised).

Problems

One classic problem is to fit as many circles of 1 cm diameter as possible into a strip of dimensions 2 cm × n, where n is any integer greater than or equal to zero. At least 2n circles can fit (side by side, in perfect rows) but the solution is that if n > 63, then at least one more circle can fit than the formula 2n suggests . In fact, for every added length of 64, an additional circle can fit . Another classic problem is the sphere packing problem, where one must determine how many spherical objects of given diameter d can you pack into a box of size a × b × c. This is one of the hardest problems in this category .

Puzzles

A classic puzzle of this kind is pentomino, where the task is to arrange all twelve pentominoes into rectangles sized 3×20, 4×15, 5×12 or 6×10, respectively.

External links

Many puzzle books as well as mathematical journals contain articles on packing problems.
- [http://www.baywood.com/journals/previewjournals.asp?id=0022-412x Journal of Recreational Mathematics] - easy read, many articles.
- [http://www.elversonpuzzle.com Log Stacker]
- [http://mathworld.wolfram.com/CirclePacking.html The math of circle packing] Category:Discrete geometry Category:Recreational mathematics Category:Puzzles

Category:Combinatorics

Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with "counting" the objects in those collections (enumerative combinatorics) and with deciding whether certain "optimal" objects exist (extremal combinatorics). One of the most prominent combinatorialists of recent times was Gian-Carlo Rota, who helped formalize the subject beginning in the 1960s. The prolific problem-solver Paul Erdős worked mainly on extremal questions. The study of how to count objects is sometimes thought of separately as the field of enumeration. Category:Mathematics th:Category:คณิตศาสตร์เชิงการจัด

Category:Computer science

In its most general sense, computer science (CS or compsci) is the study of computation and information processing, both in hardware and in software. In practice, computer science includes a variety of topics relating to computers, which range from the abstract analysis of algorithms, formal grammars, etc. to more concrete subjects like programming languages, software, and computer hardware. As a scientific discipline, it is distinct from mathematics, programming, software engineering, informatics, and computer engineering, although there are significant overlaps and no clear demarcation. Category:Computing Category:Academic disciplines Category:Information technology Category:Science ja:Category:情報工学 th:Category:วิทยาการคอมพิวเตอร์

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