Truncated minplus matrices¶
Defined in matrix.hpp
.
This page describes the functionality for \(n \times n\) matrices over the finite quotient of the minplus semiring by the congruence \(t = t + 1\) for arbitrary \(n\) and \(t\). The value \(t\) is referred to as the threshold.
There are three types of such matrices where:
the dimension is known at compiletime;
the dimension is to be defined a run time but the arithmetic operations are known at compiletime (i.e. the value of \(t\) is known at compile time)
both the dimension and the arithmetic operations (i.e. \(t\)) are to be defined a run time.
All three of these types can be accessed via the alias template
MinPlusTruncMat<T, R, C, Scalar>
: if T
has value 0
, then the
threshold can be set at run time, and if R
or C
is 0
, then the
dimension can be set at run time. The default value of T
is 0
, R
is 0
, and of C
is R
.
The alias MinPlusTruncMat<T, R, C, Scalar>
is either
StaticMatrix
or DynamicMatrix
, please refer to the
documentation of these class templates for more details. The only substantial
difference in the interface of StaticMatrix
and
DynamicMatrix
is that the former can be default constructed and the
latter should be constructed using the dimensions.
Example
MinPlusTruncMat<11, 3> m; // default construct an uninitialized 3 x 3 static matrix with threshold 11
MinPlusTruncMat<11> m(4, 4); // construct an uninitialized 4 x 4 dynamic matrix with threshold 11
MinPlusTruncSemiring sr(11); // construct a truncated minplus semiring with threshold 11
MinPlusTruncMat<> m(sr, 5, 5); // construct an uninitialized 5 x 5 dynamic matrix with threshold 11 (defined at run time)

template<size_t T, typename Scalar>
struct MinPlusTruncProd¶ This is a stateless struct with a single call operator of signature:
Scalar operator()(Scalar const x, Scalar const y) const noexcept
that returns \(x \otimes y\) which is defined by\[\begin{split}x\otimes y = \begin{cases} \min\{x + y, T\} & \text{if } x \neq \infty\text{ and }y \neq \infty \\ \infty & \text{if } x = \infty \text{ or }y = \infty; \\ \end{cases}\end{split}\]representing multiplication in the quotient of the minplus semiring by the congruence \(T = T + 1\).

template<typename Scalar = int>
class MinPlusTruncSemiring final¶ This class represents the minplus truncated semiring consists of the integers \(\{0, \ldots , t\}\) for some value \(t\) (called the threshold of the semiring) and \(\infty\). Instances of this class can be used to define the value of the threshold \(t\) at run time.
 Template Parameters
Scalar – the type of the elements of the semiring. This must be an integral type.

MinPlusTruncSemiring() = delete¶
Deleted default constructor.

MinPlusTruncSemiring(MinPlusTruncSemiring const&) = default¶
Default copy constructor.

MinPlusTruncSemiring(MinPlusTruncSemiring&&) = default¶
Default move constructor.

MinPlusTruncSemiring &operator=(MinPlusTruncSemiring const&) = default¶
Default copy assignment operator.

MinPlusTruncSemiring &operator=(MinPlusTruncSemiring&&) = default¶
Default move assignment operator.

explicit MinPlusTruncSemiring(Scalar const threshold)¶
Construct from threshold.
 Parameters
threshold – the threshold.
 Throws
LibsemigroupsException
ifScalar
is a signed type andthreshold
is less than zero. Complexity
Constant.

Scalar zero() const noexcept¶
Returns \(\infty\); representing the additive identity of the quotient of the minplus semiring.
 Parameters
(None)
 Returns
A value of type
Scalar
. Exceptions
This function is
noexcept
and is guaranteed never to throw. Complexity
Constant.

Scalar one() const noexcept¶
Returns \(0\); representing the multiplicative identity of the quotient of the minplus semiring.
 Parameters
(None)
 Returns
A value of type
Scalar
. Exceptions
This function is
noexcept
and is guaranteed never to throw. Complexity
Constant.

Scalar plus(Scalar const x, Scalar const y) const noexcept¶
Returns \(x \oplus y\) which is defined by
\[\begin{split}x\oplus y = \begin{cases} \min\{x, y\} & \text{if } x \neq \infty\text{ and }y \neq \infty \\ \infty & \text{if } x = \infty \text{ or }y = \infty; \\ \end{cases}\end{split}\]representing addition in the minplus semiring (and its quotient).
 Parameters
x – scalar
y – scalar
 Returns
A value of type
Scalar
. Exceptions
This function is
noexcept
and is guaranteed never to throw. Complexity
Constant.

Scalar prod(Scalar const x, Scalar const y) const noexcept¶
Returns \(x \otimes y\) which is defined by
\[\begin{split}x\otimes y = \begin{cases} \min\{x + y, t\} & \text{if } x \neq \infty\text{ and }y \neq \infty \\ \infty & \text{if } x = \infty \text{ or }y = \infty; \\ \end{cases}\end{split}\]where \(t\) is the threshold; representing multiplication in the quotient of the minplus semiring.
 Parameters
x – scalar
y – scalar
 Returns
A value of type
Scalar
. Exceptions
This function is
noexcept
and is guaranteed never to throw. Complexity
Constant.

template<typename Scalar>
using DynamicMinPlusTruncMatSR = DynamicMatrix<MinPlusTruncSemiring<Scalar>, Scalar>¶ Alias for truncated minplus matrices with dimensions and threshold defined at runtime.
 Template Parameters
Scalar – The type of the entries in the matrix.

template<typename T, typename Scalar>
using DynamicMinPlusTruncMat = DynamicMatrix<MinPlusPlus<Scalar>, MinPlusTruncProd<T, Scalar>, MinPlusZero<Scalar>, IntegerZero<Scalar>, Scalar>¶ Alias for the type of dynamic minplus matrices where the dimension is defined at run time, but the threshold is defined at compiletime.
 Template Parameters
T – the threshold.
Scalar – the type of the entries in the matrix.

template<size_t T, size_t R, size_t C, typename Scalar>
using StaticMinPlusTruncMat = StaticMatrix<MinPlusPlus<Scalar>, MinPlusTruncProd<T, Scalar>, MinPlusZero<Scalar>, IntegerZero<Scalar>, R, C, Scalar>¶ Alias for static minplus truncated matrices where the threshold and dimensions are defined at compiletime.
 Template Parameters
T – the threshold.
R – the number of rows.
C – the number of columns.
Scalar – The type of the entries in the matrix.

template<size_t T = 0, size_t R = 0, size_t C = R, typename Scalar = int>
using MinPlusTruncMat = std::conditional_t<R == 0  C == 0, std::conditional_t<T == 0, DynamicMinPlusTruncMatSR<Scalar>, DynamicMinPlusTruncMat<T, Scalar>>, StaticMinPlusTruncMat<T, R, C, Scalar>>¶  Template Parameters
T – the threshold. A value of
0
indicates that the value will be set at run time (default:0
).R – the number of rows. A value of
0
indicates that the value will be set at run time (default:0
).C – the number of columns. A value of
0
indicates that the value will be set at run time (default:R
).Scalar – The type of the entries in the matrix (default:
int
).

template<typename T>
static constexpr bool IsMinPlusTruncMat¶ This variable has value
true
if the template parameterT
is the same asMinPlusTruncMat<T, R, C, Scalar>
for some values ofT
,R
,C
, andScalar
; andfalse
if it is not.