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| Category: algorithms | | Component type: function |
Prototype
Power is an overloaded name; there are actually two power functions.
template <class T, class Integer>
inline T power(T x, Integer n);
template <class T, class Integer, class MonoidOperation>
T power(T x, Integer n, MonoidOperation op);
Description
Power is generalized exponentiation: it raises the value x to the power n, where n is a non-negative integer.
The first version of power returns x raised to the nth power; that is, it returns x * x ... * x, where x is repeated n times. [1] If n == 0, then it returns MonoidOperation(multiplies<T>()).
The second version of power is just like the first except that it uses an arbitrary MonoidOperation instead of multiplies<T>, returning MonoidOperation(op) when n == 0.
Important note: power does not assume that multiplication is commutative, but it does rely crucially on the fact that multiplication is associative. If you have defined * or MonoidOperation to be a non-associative operation, then power will give you the wrong answer. [2]
Definition
Defined in the standard header numeric, and in the nonstandard backward-compatibility header algo.h. This function is an SGI extension; it is not part of the C++ standard.
Requirements on types
For the first version:
For the second version:
-
MonoidOperation is a model of MonoidOperation.
-
T is MonoidOperation's argument type.
-
n is an integral type.
Preconditions
Complexity
The number of multiplications (or, in the case of the second version, the number of applications of MonoidOperation) is lg(n) + nu(n) where lg is the base 2 logarithm and nu(n) is the number of 1s in the binary representation of n. [3]
Example
int main() {
cout << "2 ** 30 = " << power(2, 30) << endl;
}
Notes
[1] This is a conceptual description of what power's return value is; it is not how power is actually implemented. If power were implemented that way then it would require n-1 multiplications, which would be grossly inefficient. Power is implemented using the "Russian peasant algorithm", which requires only O(log n) multiplications. See section 4.6.3 of Knuth (D. E. Knuth, The Art of Computer Programming. Volume 2: Seminumerical Algorithms, Addison-Wesley, 1981) for a discussion.
[2] See the MonoidOperation requirements for a discussion of associativity.
[3] This is in fact not the minimum possible number of multiplications: it is possible to compute the fifteenth power of x using only five multiplications, but power(x, 15) uses six.
See also
MonoidOperation, multiplies, plus
