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Random Number Generator Library Concepts

Introduction

Random numbers are required in a number of different problem domains, such as

The Boost Random Number Generator Library provides a framework for random number generators with well-defined properties so that the generators can be used in the demanding numerics and security domains. For a general introduction to random numbers in numerics, see
"Numerical Recipes in C: The art of scientific computing", William H. Press, Saul A. Teukolsky, William A. Vetterling, Brian P. Flannery, 2nd ed., 1992, pp. 274-328
Depending on the requirements of the problem domain, different variations of random number generators are appropriate: All variations have some properties in common, these concepts (in the STL sense) are called NumberGenerator and UniformRandomNumberGenerator. Each concept will be defined in a subsequent section.

The goals for this library are the following:

Number Generator

A number generator is a function object (std:20.3 [lib.function.objects]) that takes zero arguments. Each call to operator() returns a number. In the following table, X denotes a number generator class returning objects of type T, and u is a value of X.

NumberGenerator requirements
expression return type pre/post-condition
X::result_type T std::numeric_limits<T>::is_specialized is true, T is LessThanComparable
u.operator()() T -

Note: The NumberGenerator requirements do not impose any restrictions on the characteristics of the returned numbers.

Uniform Random Number Generator

A uniform random number generator is a NumberGenerator that provides a sequence of random numbers uniformly distributed on a given range. The range can be compile-time fixed or available (only) after run-time construction of the object.

The tight lower bound of some (finite) set S is the (unique) member l in S, so that for all v in S, l <= v holds. Likewise, the tight upper bound of some (finite) set S is the (unique) member u in S, so that for all v in S, v <= u holds.

In the following table, X denotes a number generator class returning objects of type T, and v is a const value of X.

UniformRandomNumberGenerator requirements
expression return type pre/post-condition
X::has_fixed_range bool compile-time constant; if true, the range on which the random numbers are uniformly distributed is known at compile-time and members min_value and max_value exist. Note: This flag may also be false due to compiler limitations.
X::min_value T compile-time constant; min_value is equal to v.min()
X::max_value T compile-time constant; max_value is equal to v.max()
v.min() T tight lower bound on the set of all values returned by operator(). The return value of this function shall not change during the lifetime of the object.
v.max() T if std::numeric_limits<T>::is_integer, tight upper bound on the set of all values returned by operator(), otherwise, the smallest representable number larger than the tight upper bound on the set of all values returned by operator(). In any case, the return value of this function shall not change during the lifetime of the object.

The member functions min, max, and operator() shall have amortized constant time complexity.

Note: For integer generators (i.e. integer T), the generated values x fulfill min() <= x <= max(), for non-integer generators (i.e. non-integer T), the generated values x fulfill min() <= x < max().
Rationale: The range description with min and max serves two purposes. First, it allows scaling of the values to some canonical range, such as [0..1). Second, it describes the significant bits of the values, which may be relevant for further processing.
The range is a closed interval [min,max] for integers, because the underlying type may not be able to represent the half-open interval [min,max+1). It is a half-open interval [min, max) for non-integers, because this is much more practical for borderline cases of continuous distributions.

Note: The UniformRandomNumberGenerator concept does not require operator()(long) and thus it does not fulfill the RandomNumberGenerator (std:25.2.11 [lib.alg.random.shuffle]) requirements. Use the random_number_generator adapter for that.
Rationale: operator()(long) is not provided, because mapping the output of some generator with integer range to a different integer range is not trivial.

Non-deterministic Uniform Random Number Generator

A non-deterministic uniform random number generator is a UniformRandomNumberGenerator that is based on some stochastic process. Thus, it provides a sequence of truly-random numbers. Examples for such processes are nuclear decay, noise of a Zehner diode, tunneling of quantum particles, rolling a die, drawing from an urn, and tossing a coin. Depending on the environment, inter-arrival times of network packets or keyboard events may be close approximations of stochastic processes.

The class random_device is a model for a non-deterministic random number generator.

Note: This type of random-number generator is useful for security applications, where it is important to prevent that an outside attacker guesses the numbers and thus obtains your encryption or authentication key. Thus, models of this concept should be cautious not to leak any information, to the extent possible by the environment. For example, it might be advisable to explicitly clear any temporary storage as soon as it is no longer needed.

Pseudo-Random Number Generator

A pseudo-random number generator is a UniformRandomNumberGenerator which provides a deterministic sequence of pseudo-random numbers, based on some algorithm and internal state. Linear congruential and inversive congruential generators are examples of such pseudo-random number generators. Often, these generators are very sensitive to their parameters. In order to prevent wrong implementations from being used, an external testsuite should check that the generated sequence and the validation value provided do indeed match.

Donald E. Knuth gives an extensive overview on pseudo-random number generation in his book "The Art of Computer Programming, Vol. 2, 3rd edition, Addison-Wesley, 1997". The descriptions for the specific generators contain additional references.

Note: Because the state of a pseudo-random number generator is necessarily finite, the sequence of numbers returned by the generator will loop eventually.

In addition to the UniformRandomNumberGenerator requirements, a pseudo-random number generator has some additional requirements. In the following table, X denotes a pseudo-random number generator class returning objects of type T, x is a value of T, u is a value of X, and v is a const value of X.

PseudoRandomNumberGenerator requirements
expression return type pre/post-condition
X() - creates a generator in some implementation-defined state. Note: Several generators thusly created may possibly produce dependent or identical sequences of random numbers.
explicit X(...) - creates a generator with user-provided state; the implementation shall specify the constructor argument(s)
u.seed(...) void sets the current state according to the argument(s); at least functions with the same signature as the non-default constructor(s) shall be provided.
X::validation(x) bool compares the pre-computed and hardcoded 10001th element in the generator's random number sequence with x. The generator must have been constructed by its default constructor and seed must not have been called for the validation to be meaningful.

Note: The seed member function is similar to the assign member function in STL containers. However, the naming did not seem appropriate.

Classes which model a pseudo-random number generator shall also model EqualityComparable, i.e. implement operator==. Two pseudo-random number generators are defined to be equivalent if they both return an identical sequence of numbers starting from a given state.

Classes which model a pseudo-random number generator should also model the Streamable concept, i.e. implement operator<< and operator>>. If so, operator<< writes all current state of the pseudo-random number generator to the given ostream so that operator>> can restore the state at a later time. The state shall be written in a platform-independent manner, but it is assumed that the locales used for writing and reading be the same. The pseudo-random number generator with the restored state and the original at the just-written state shall be equivalent.

Classes which model a pseudo-random number generator may also model the CopyConstructible and Assignable concepts. However, note that the sequences of the original and the copy are strongly correlated (in fact, they are identical), which may make them unsuitable for some problem domains. Thus, copying pseudo-random number generators is discouraged; they should always be passed by (non-const) reference.

The classes rand48, minstd_rand, and mt19937 are models for a pseudo-random number generator.

Note: This type of random-number generator is useful for numerics, games and testing. The non-zero arguments constructor(s) and the seed() member function(s) allow for a user-provided state to be installed in the generator. This is useful for debugging Monte-Carlo algorithms and analyzing particular test scenarios. The Streamable concept allows to save/restore the state of the generator, for example to re-run a test suite at a later time.

Random Distribution

A radom distribution produces random numbers distributed according to some distribution, given uniformly distributed random values as input. In the following table, X denotes a random distribution class returning objects of type T, u is a value of X, x is a (possibly const) value of X, and e is an lvalue of an arbitrary type that meets the requirements of a uniform random number generator, returning values of type U.

Random distribution requirements (in addition to number generator, CopyConstructible, and Assignable)
expression return type pre/post-condition complexity
X::input_type U - compile-time
u.reset() void subsequent uses of u do not depend on values produced by e prior to invoking reset. constant
u(e) T the sequence of numbers returned by successive invocations with the same object e is randomly distributed with some probability density function p(x) amortized constant number of invocations of e
os << x std::ostream& writes a textual representation for the parameters and additional internal data of the distribution x to os.
post: The os.fmtflags and fill character are unchanged.
O(size of state)
is >> u std::istream& restores the parameters and additional internal data of the distribution u.
pre: is provides a textual representation that was previously written by operator<<
post: The is.fmtflags are unchanged.
O(size of state)

Additional requirements: The sequence of numbers produced by repeated invocations of x(e) does not change whether or not os << x is invoked between any of the invocations x(e). If a textual representation is written using os << x and that representation is restored into the same or a different object y of the same type using is >> y, repeated invocations of y(e) produce the same sequence of random numbers as would repeated invocations of x(e).

Quasi-Random Number Generators

A quasi-random number generator is a Number Generator which provides a deterministic sequence of numbers, based on some algorithm and internal state. The numbers do not have any statistical properties (such as uniform distribution or independence of successive values).

Note: Quasi-random number generators are useful for Monte-Carlo integrations where specially crafted sequences of random numbers will make the approximation converge faster.

[Does anyone have a model?]


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