Every field of science has its own problems and needs efficient algorithms. Related problems in one field are often studied together. Some example classes are search algorithms, sorting algorithms, merge algorithms, numerical algorithms, graph algorithms, string algorithms, computational geometric algorithms, combinatorial algorithms, machine learning, cryptography, data compression algorithms and parsing techniques.
Fields tend to overlap with each other, and algorithm advances in one field may improve those of other, sometimes completely unrelated, fields. For example, dynamic programming was originally invented for optimisation of resource consumption in industry, but is now used in solving a broad range of problems in many fields.
This is actually problem classification in the strict sense. Some algorithms complete in linear time proportional to input size, and some do in exponential amount of time, and some never do. Some problems may have multiple algorithms, some problems may have no algorithms, and some problems have no known efficient algorithms. There are also mappings from some problems to other problems. So computer scientists found it is suitable to classify the problems rather than algorithms into equivalence classes based on the complexity of the best possible algorithms for them.
Some countries allow algorithms to be patented when embodied in software or in hardware. Patents have long been a controversial issue (see, for example, the software patent debate). Some countries do not allow certain algorithms, such as cryptographic algorithms, to be exported from that country (see export of cryptography).
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