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Computational complexity theory

Computational complexity theory focuses on classifying computational problems according to their inherent difficulty, relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used; the theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e. the amount of resources needed to solve them, such as time and storage. Other measures of complexity are used, such as the amount of communication, the number of gates in a circuit and the number of processors. One of the roles of computational complexity theory is to determine the practical limits on what computers can and cannot do; the P versus NP problem, one of the seven Millennium Prize Problems, is dedicated to the field of computational complexity.

Related fields in theoretical computer science are analysis of algorithms and computability theory. A key distinction between analysis of algorithms and computational complexity theory is that the former is devoted to analyzing the amount of resources needed by a particular algorithm to solve a problem, whereas the latter asks a more general question about all possible algorithms that could be used to solve the same problem. More computational complexity theory tries to classify problems that can or cannot be solved with appropriately restricted resources. In turn, imposing restrictions on the available resources is what distinguishes computational complexity from computability theory: the latter theory asks what kind of problems can, in principle, be solved algorithmically. A computational problem can be viewed as an infinite collection of instances together with a solution for every instance; the input string for a computational problem is referred to as a problem instance, should not be confused with the problem itself.

In computational complexity theory, a problem refers to the abstract question to be solved. In contrast, an instance of this problem is a rather concrete utterance, which can serve as the input for a decision problem. For example, consider the problem of primality testing; the instance is a number and the solution is "yes" if the number is prime and "no" otherwise. Stated another way, the instance is a particular input to the problem, the solution is the output corresponding to the given input. To further highlight the difference between a problem and an instance, consider the following instance of the decision version of the traveling salesman problem: Is there a route of at most 2000 kilometres passing through all of Germany's 15 largest cities? The quantitative answer to this particular problem instance is of little use for solving other instances of the problem, such as asking for a round trip through all sites in Milan whose total length is at most 10 km. For this reason, complexity theory addresses computational problems and not particular problem instances.

When considering computational problems, a problem instance is a string over an alphabet. The alphabet is taken to be the binary alphabet, thus the strings are bitstrings; as in a real-world computer, mathematical objects other than bitstrings must be suitably encoded. For example, integers can be represented in binary notation, graphs can be encoded directly via their adjacency matrices, or by encoding their adjacency lists in binary. Though some proofs of complexity-theoretic theorems assume some concrete choice of input encoding, one tries to keep the discussion abstract enough to be independent of the choice of encoding; this can be achieved by ensuring that different representations can be transformed into each other efficiently. Decision problems are one of the central objects of study in computational complexity theory. A decision problem is a special type of computational problem whose answer is either yes or no, or alternately either 1 or 0. A decision problem can be viewed as a formal language, where the members of the language are instances whose output is yes, the non-members are those instances whose output is no.

The objective is to decide, with the aid of an algorithm, whether a given input string is a member of the formal language under consideration. If the algorithm deciding this problem returns the answer yes, the algorithm is said to accept the input string, otherwise it is said to reject the input. An example of a decision problem is the following; the input is an arbitrary graph. The problem consists in deciding; the formal language associated with this decision problem is the set of all connected graphs — to obtain a precise definition of this language, one has to decide how graphs are encoded as binary strings. A function problem is a computational problem where a single output is expected for every input, but the output is more complex than that of a decision problem—that is, the output isn't just yes or no. Notable examples include the integer factorization problem, it is tempting to think that the notion of function problems is much richer than the notion of decision problems. However, this is not the case, since function problems can be recast as decision problems.

For example, the multiplication of two integers can be expressed as the set of triples such that the relation a × b = c holds. Deciding whether a given triple is a member of this set corresponds to solving

Bennie Borgmann

Bernhard Borgmann was a professional basketball player. Born in Haledon, New Jersey, he played for 17 years between 1919 and 1936, is known for his time with the Kingston Colonials and Original Celtics. Borgmann is regarded as the best offensive player of his era, he served as the first coach of the Syracuse Nationals of the National Basketball League—now the NBA's Philadelphia 76ers—from 1946 to 1948. He was inducted into the Naismith Memorial Basketball Hall of Fame in 1961. Borgmann played as a middle infielder in the baseball minor leagues from 1928 to 1942, managed in the minors from until 1950, scouted until 1974, he died in New Jersey. Borgmann was featured in the book Basketball History in Syracuse, Hoops Roots by Mark Allen Baker, published by The History Press in 2010. Basketball Hall of Fame profile Career statistics and player information from Baseball-Reference

University of Public Health, Yangon

The University of Public Health, Yangon is the premier university of public health in Myanmar. Founded in 2007, the university offers only graduate and post-graduate degree programs: Master of Public Health, PhD in Public Health, Diploma in Medical Science and Diploma in Medical Education; the university is a member of the South-East Asia Public Health Education Institution Network. It is located at the heart of Yangon; the full address is 246, Myo Ma Kyaung street, corner of Bo Gyoke Aung San Road and Lamadaw street, Yangon. It is just across the road in front of University of Medicine 1, the oldest medical school in Myanmar; the university was founded in 2007 to strengthen the public health system and infrastructure in Myanmar. It is to help strengthen referral systems by providing technical back-up. Referral facilities for primary and tertiary care are important to effective community health work. Current dean of the school is Professor Nay Soe Maung who has served as an army doctor up to the rank of a colonel.

He is ex-husband of one of the daughters of former Myanmar Junta head. Physicians, dentists and health assistants are allowed to enter this school. Medical doctors and community/public health professions who are working in government service can enroll the UOPH via entrance exam. Applicant must be under 50 years old, they finished the training of Central Institute of Civil Service in Myanmar. It is under authority of Ministry of Health in Myanmar. University of Community Health, Magway Official website 4. Myanma Alinn Daily Newspaper in 2013-06-15