Applications of Complexity
Computational complexity theory is the study of the complexity of problems—that is, the difficulty of solving them. Problems can be classified by complexity class according to the time it takes for an algorithm—usually a computer program—to solve them as a function of the problem size. Some problems are difficult to solve, while others are easy. For example, some difficult problems need algorithms that take an exponential amount of time in terms of the size of the problem to solve. Take the travelling salesman problem, for example. It can be solved in time (where n is the size of the network to visit—let's say the number of cities the travelling salesman must visit exactly once). As the size of the network of cities grows, the time needed to find the route grows (more than) exponentially.
Even though a problem may be computationally solvable in principle, in actual practice it may not be that simple. These problems might require large amounts of time or an inordinate amount of space. Computational complexity may be approached from many different aspects. Computational complexity can be investigated on the basis of time, memory or other resources used to solve the problem. Time and space are two of the most important and popular considerations when problems of complexity are analyzed.
There exist a certain class of problems that although they are solvable in principle they require so much time or space that it is not practical to attempt to solve them. These problems are called intractable.
There is another form of complexity called hierarchical complexity. It is orthogonal to the forms of complexity discussed so far, which are called horizontal complexity
Bejan and Lorente showed that complexity is modest (not maximum, not increasing), and is a feature of the natural phenomenon of design generation in nature, which is predicted by the Constructal law.
Bejan and Lorente also showed that all the optimality (max,min) statements have limited ad-hoc applicability, and are unified under the Constructal law of design and evolution in nature.
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Famous quotes containing the word complexity:
“It is not only their own need to mother that takes some women by surprise; there is also the shock of discovering the complexity of alternative child-care arrangements that have been made to sound so simple. Those for whom the intended solution is equal parenting have found that some parents are more equal than others.”
—Elaine Heffner (20th century)