Editing Computational complexity (section)

==Resources==
===Time===
The resource that is most commonly considered is time. When "complexity" is used without qualification, this generally means time complexity.

The usual units of time (seconds, minutes etc.) are not used in [[computational complexity theory|complexity theory]] because they are too dependent on the choice of a specific computer and on the evolution of technology. For instance, a computer today can execute an algorithm significantly faster than a computer from the 1960s; however, this is not an intrinsic feature of the algorithm but rather a consequence of technological advances in [[computer hardware]]. Complexity theory seeks to quantify the intrinsic time requirements of algorithms, that is, the basic time constraints an algorithm would place on ''any'' computer. This is achieved by counting the number of ''elementary operations'' that are executed during the computation. These operations are assumed to take constant time (that is, not affected by the size of the input) on a given machine, and are often called ''steps''.

===Bit complexity===
{{anchor|bit complexity}}
Formally, the ''bit complexity'' refers to the number of operations on [[bit]]s that are needed for running an algorithm. With most [[models of computation]], it equals the time complexity up to a constant factor. On [[computer]]s, the number of operations on [[machine word]]s that are needed is also proportional to the bit complexity. So, the ''time complexity'' and the ''bit complexity'' are equivalent  for realistic models of computation.

===Space===
Another important resource is the size of [[computer memory]] that is needed for running algorithms.

===Communication===
{{Main article|communication complexity}}
For the class of [[distributed computation|distributed algorithms]] that are commonly executed by multiple, interacting parties, the resource that is of most interest is the communication complexity. It is the necessary amount of communication between the executing parties.

===Others===
The number of [[arithmetic operations]] is another resource that is commonly used. In this case, one talks of '''arithmetic complexity'''. If one knows an [[upper bound]] on the size of the [[binary representation]] of the numbers that occur during a computation, the time complexity is generally the product of the arithmetic complexity by a constant factor.

{{anchor|bit complexity}}
For many algorithms the size of the integers that are used during a computation is not bounded, and it is not realistic to consider that arithmetic operations take a constant time. Therefore, the time complexity, generally called '''bit complexity''' in this context, may be much larger than the arithmetic complexity. For example, the arithmetic complexity of the computation of the [[determinant]] of a {{math|''n''×''n''}} [[integer matrix]] is <math>O(n^3)</math> for the usual algorithms ([[Gaussian elimination]]). The bit complexity of the same algorithms is [[exponential function|exponential]] in {{mvar|n}}, because the size of the coefficients may grow exponentially during the computation. On the other hand, if these algorithms are coupled with [[modular arithmetic|multi-modular arithmetic]], the bit complexity may be reduced to {{math|[[soft O notation|''O''<sup>~</sup>(''n''<sup>4</sup>)]]}}.

In [[sorting]] and [[search algorithm|searching]], the resource that is generally considered is the number of entry comparisons. This is generally a good measure of the time complexity if data are suitably organized.