Copyright	(c) Andrey Mokhov 2016-2022
License	MIT (see the file LICENSE)
Maintainer	andrey.mokhov@gmail.com
Stability	experimental
Safe Haskell	None
Language	Haskell2010

Algebra.Graph.Label

Contents

Semirings and dioids
Data types for edge labels
Combining edge labels

Description

Alga is a library for algebraic construction and manipulation of graphs in Haskell. See this paper for the motivation behind the library, the underlying theory, and implementation details.

This module provides basic data types and type classes for representing edge labels in edge-labelled graphs, e.g. see Algebra.Graph.Labelled .

Synopsis

class ( Monoid a, Semigroup a) => Semiring a where
- one :: a
- (<.>) :: a -> a -> a
zero :: Monoid a => a
(<+>) :: Semigroup a => a -> a -> a
class Semiring a => StarSemiring a where
- star :: a -> a
class Semiring a => Dioid a
data NonNegative a
finite :: ( Num a, Ord a) => a -> Maybe ( NonNegative a)
finiteWord :: Word -> NonNegative Word
unsafeFinite :: a -> NonNegative a
infinite :: NonNegative a
getFinite :: NonNegative a -> Maybe a
data Distance a
distance :: NonNegative a -> Distance a
getDistance :: Distance a -> NonNegative a
data Capacity a
capacity :: NonNegative a -> Capacity a
getCapacity :: Capacity a -> NonNegative a
data Count a
count :: NonNegative a -> Count a
getCount :: Count a -> NonNegative a
newtype PowerSet a = PowerSet {
- getPowerSet :: Set a
}
data Minimum a
getMinimum :: Minimum a -> Maybe a
noMinimum :: Minimum a
type Path a = [(a, a)]
data Label a
isZero :: Label a -> Bool
type RegularExpression a = Label a
data Optimum o a = Optimum {
- getOptimum :: o
- getArgument :: a
}
type ShortestPath e a = Optimum ( Distance e) ( Minimum ( Path a))
type AllShortestPaths e a = Optimum ( Distance e) ( PowerSet ( Path a))
type CountShortestPaths e = Optimum ( Distance e) ( Count Integer )
type WidestPath e a = Optimum ( Capacity e) ( Minimum ( Path a))

Semirings and dioids

class ( Monoid a, Semigroup a) => Semiring a where Source #

A semiring extends a commutative Monoid with operation <.> that acts similarly to multiplication over the underlying (additive) monoid and has one as the identity. This module also provides two convenient aliases: zero for mempty , and <+> for <> , which makes the interface more uniform.

Instances of this type class must satisfy the following semiring laws:

Associativity of <+> and <.> :

x <+> (y <+> z) == (x <+> y) <+> z
x <.> (y <.> z) == (x <.> y) <.> z

Identities of <+> and <.> :

zero <+> x == x == x <+> zero
 one <.> x == x == x <.> one

Commutativity of <+> :
```
x <+> y == y <+> x
```
Annihilating zero :
```
x <.> zero == zero
zero <.> x == zero
```

Distributivity:

x <.> (y <+> z) == x <.> y <+> x <.> z
(x <+> y) <.> z == x <.> z <+> y <.> z

Methods

one :: a Source #

(<.>) :: a -> a -> a infixr 7 Source #

Instances

Instances details

Semiring Any Source #
Instance details Defined in Algebra.Graph.Label Methods one :: Any Source # (<.>) :: Any -> Any -> Any Source #
Semiring ( Label a) Source #
Instance details Defined in Algebra.Graph.Label Methods one :: Label a Source # (<.>) :: Label a -> Label a -> Label a Source #
( Monoid a, Ord a) => Semiring ( PowerSet a) Source #
Instance details Defined in Algebra.Graph.Label Methods one :: PowerSet a Source # (<.>) :: PowerSet a -> PowerSet a -> PowerSet a Source #
( Monoid a, Ord a) => Semiring ( Minimum a) Source #
Instance details Defined in Algebra.Graph.Label Methods one :: Minimum a Source # (<.>) :: Minimum a -> Minimum a -> Minimum a Source #
( Num a, Ord a) => Semiring ( Distance a) Source #
Instance details Defined in Algebra.Graph.Label Methods one :: Distance a Source # (<.>) :: Distance a -> Distance a -> Distance a Source #
( Num a, Ord a) => Semiring ( Count a) Source #
Instance details Defined in Algebra.Graph.Label Methods one :: Count a Source # (<.>) :: Count a -> Count a -> Count a Source #
( Num a, Ord a) => Semiring ( Capacity a) Source #
Instance details Defined in Algebra.Graph.Label Methods one :: Capacity a Source # (<.>) :: Capacity a -> Capacity a -> Capacity a Source #
( Eq o, Semiring a, Semiring o) => Semiring ( Optimum o a) Source #
Instance details Defined in Algebra.Graph.Label Methods one :: Optimum o a Source # (<.>) :: Optimum o a -> Optimum o a -> Optimum o a Source #

zero :: Monoid a => a Source #

An alias for mempty .

(<+>) :: Semigroup a => a -> a -> a infixr 6 Source #

An alias for <> .

class Semiring a => StarSemiring a where Source #

A star semiring is a Semiring with an additional unary operator star satisfying the following two laws:

star a = one <+> a <.> star a
star a = one <+> star a <.> a

Methods

star :: a -> a Source #

Instances

Instances details

StarSemiring Any Source #
Instance details Defined in Algebra.Graph.Label Methods star :: Any -> Any Source #
StarSemiring ( Label a) Source #
Instance details Defined in Algebra.Graph.Label Methods star :: Label a -> Label a Source #
( Num a, Ord a) => StarSemiring ( Distance a) Source #
Instance details Defined in Algebra.Graph.Label Methods star :: Distance a -> Distance a Source #
( Num a, Ord a) => StarSemiring ( Count a) Source #
Instance details Defined in Algebra.Graph.Label Methods star :: Count a -> Count a Source #
( Num a, Ord a) => StarSemiring ( Capacity a) Source #
Instance details Defined in Algebra.Graph.Label Methods star :: Capacity a -> Capacity a Source #
( Eq o, StarSemiring a, StarSemiring o) => StarSemiring ( Optimum o a) Source #
Instance details Defined in Algebra.Graph.Label Methods star :: Optimum o a -> Optimum o a Source #

class Semiring a => Dioid a Source #

A dioid is an idempotent semiring , i.e. it satisfies the following idempotence law in addition to the Semiring laws:

x <+> x == x

Instances

Instances details

Dioid Any Source #
Instance details Defined in Algebra.Graph.Label
( Monoid a, Ord a) => Dioid ( PowerSet a) Source #
Instance details Defined in Algebra.Graph.Label
( Monoid a, Ord a) => Dioid ( Minimum a) Source #
Instance details Defined in Algebra.Graph.Label
( Num a, Ord a) => Dioid ( Distance a) Source #
Instance details Defined in Algebra.Graph.Label
( Num a, Ord a) => Dioid ( Capacity a) Source #
Instance details Defined in Algebra.Graph.Label
( Eq o, Dioid a, Dioid o) => Dioid ( Optimum o a) Source #
Instance details Defined in Algebra.Graph.Label

Data types for edge labels

data NonNegative a Source #

A non-negative value that can be finite or infinite . Note: the current implementation of the Num instance raises an error on negative literals and on the negate method.

Instances

Instances details

Monad NonNegative Source #
Instance details Defined in Algebra.Graph.Label Methods (>>=) :: NonNegative a -> (a -> NonNegative b) -> NonNegative b Source # (>>) :: NonNegative a -> NonNegative b -> NonNegative b Source # return :: a -> NonNegative a Source #
Functor NonNegative Source #
Instance details Defined in Algebra.Graph.Label Methods fmap :: (a -> b) -> NonNegative a -> NonNegative b Source # (<$) :: a -> NonNegative b -> NonNegative a Source #
Applicative NonNegative Source #
Instance details Defined in Algebra.Graph.Label Methods pure :: a -> NonNegative a Source # (<>) :: NonNegative (a -> b) -> NonNegative a -> NonNegative b Source # liftA2 :: (a -> b -> c) -> NonNegative a -> NonNegative b -> NonNegative c Source # (>) :: NonNegative a -> NonNegative b -> NonNegative b Source # (<*) :: NonNegative a -> NonNegative b -> NonNegative a Source #
Num a => Bounded ( NonNegative a) Source #
Instance details Defined in Algebra.Graph.Label Methods minBound :: NonNegative a Source # maxBound :: NonNegative a Source #
Eq a => Eq ( NonNegative a) Source #
Instance details Defined in Algebra.Graph.Label Methods (==) :: NonNegative a -> NonNegative a -> Bool Source # (/=) :: NonNegative a -> NonNegative a -> Bool Source #
( Num a, Ord a) => Num ( NonNegative a) Source #
Instance details Defined in Algebra.Graph.Label Methods (+) :: NonNegative a -> NonNegative a -> NonNegative a Source # (-) :: NonNegative a -> NonNegative a -> NonNegative a Source # (*) :: NonNegative a -> NonNegative a -> NonNegative a Source # negate :: NonNegative a -> NonNegative a Source # abs :: NonNegative a -> NonNegative a Source # signum :: NonNegative a -> NonNegative a Source # fromInteger :: Integer -> NonNegative a Source #
Ord a => Ord ( NonNegative a) Source #
Instance details Defined in Algebra.Graph.Label Methods compare :: NonNegative a -> NonNegative a -> Ordering Source # (<) :: NonNegative a -> NonNegative a -> Bool Source # (<=) :: NonNegative a -> NonNegative a -> Bool Source # (>) :: NonNegative a -> NonNegative a -> Bool Source # (>=) :: NonNegative a -> NonNegative a -> Bool Source # max :: NonNegative a -> NonNegative a -> NonNegative a Source # min :: NonNegative a -> NonNegative a -> NonNegative a Source #
( Num a, Show a) => Show ( NonNegative a) Source #
Instance details Defined in Algebra.Graph.Label Methods showsPrec :: Int -> NonNegative a -> ShowS Source # show :: NonNegative a -> String Source # showList :: [ NonNegative a] -> ShowS Source #

finite :: ( Num a, Ord a) => a -> Maybe ( NonNegative a) Source #

A finite non-negative value or Nothing if the argument is negative.

finiteWord :: Word -> NonNegative Word Source #

A finite Word .

unsafeFinite :: a -> NonNegative a Source #

A non-negative finite value, created unsafely : the argument is not checked for being non-negative, so unsafeFinite (-1) compiles just fine.

infinite :: NonNegative a Source #

The (non-negative) infinite value.

getFinite :: NonNegative a -> Maybe a Source #

Get a finite value or Nothing if the value is infinite.

data Distance a Source #

A distance is a non-negative value that can be finite or infinite . Distances form a Dioid as follows:

zero  = distance infinite
one   = 0
(<+>) = min
(<.>) = (+)

Instances

Instances details

Num a => Bounded ( Distance a) Source #
Instance details Defined in Algebra.Graph.Label Methods minBound :: Distance a Source # maxBound :: Distance a Source #
Eq a => Eq ( Distance a) Source #
Instance details Defined in Algebra.Graph.Label Methods (==) :: Distance a -> Distance a -> Bool Source # (/=) :: Distance a -> Distance a -> Bool Source #
( Num a, Ord a) => Num ( Distance a) Source #
Instance details Defined in Algebra.Graph.Label Methods (+) :: Distance a -> Distance a -> Distance a Source # (-) :: Distance a -> Distance a -> Distance a Source # (*) :: Distance a -> Distance a -> Distance a Source # negate :: Distance a -> Distance a Source # abs :: Distance a -> Distance a Source # signum :: Distance a -> Distance a Source # fromInteger :: Integer -> Distance a Source #
Ord a => Ord ( Distance a) Source #
Instance details Defined in Algebra.Graph.Label Methods compare :: Distance a -> Distance a -> Ordering Source # (<) :: Distance a -> Distance a -> Bool Source # (<=) :: Distance a -> Distance a -> Bool Source # (>) :: Distance a -> Distance a -> Bool Source # (>=) :: Distance a -> Distance a -> Bool Source # max :: Distance a -> Distance a -> Distance a Source # min :: Distance a -> Distance a -> Distance a Source #
Show a => Show ( Distance a) Source #
Instance details Defined in Algebra.Graph.Label Methods showsPrec :: Int -> Distance a -> ShowS Source # show :: Distance a -> String Source # showList :: [ Distance a] -> ShowS Source #
Ord a => Semigroup ( Distance a) Source #
Instance details Defined in Algebra.Graph.Label Methods (<>) :: Distance a -> Distance a -> Distance a Source # sconcat :: NonEmpty ( Distance a) -> Distance a Source # stimes :: Integral b => b -> Distance a -> Distance a Source #
( Ord a, Num a) => Monoid ( Distance a) Source #
Instance details Defined in Algebra.Graph.Label Methods mempty :: Distance a Source # mappend :: Distance a -> Distance a -> Distance a Source # mconcat :: [ Distance a] -> Distance a Source #
( Num a, Ord a) => Dioid ( Distance a) Source #
Instance details Defined in Algebra.Graph.Label
( Num a, Ord a) => StarSemiring ( Distance a) Source #
Instance details Defined in Algebra.Graph.Label Methods star :: Distance a -> Distance a Source #
( Num a, Ord a) => Semiring ( Distance a) Source #
Instance details Defined in Algebra.Graph.Label Methods one :: Distance a Source # (<.>) :: Distance a -> Distance a -> Distance a Source #

distance :: NonNegative a -> Distance a Source #

A non-negative distance.

getDistance :: Distance a -> NonNegative a Source #

Get the value of a distance.

data Capacity a Source #

A capacity is a non-negative value that can be finite or infinite . Capacities form a Dioid as follows:

zero  = 0
one   = capacity infinite
(<+>) = max
(<.>) = min

Instances

Instances details

Num a => Bounded ( Capacity a) Source #
Instance details Defined in Algebra.Graph.Label Methods minBound :: Capacity a Source # maxBound :: Capacity a Source #
Eq a => Eq ( Capacity a) Source #
Instance details Defined in Algebra.Graph.Label Methods (==) :: Capacity a -> Capacity a -> Bool Source # (/=) :: Capacity a -> Capacity a -> Bool Source #
( Num a, Ord a) => Num ( Capacity a) Source #
Instance details Defined in Algebra.Graph.Label Methods (+) :: Capacity a -> Capacity a -> Capacity a Source # (-) :: Capacity a -> Capacity a -> Capacity a Source # (*) :: Capacity a -> Capacity a -> Capacity a Source # negate :: Capacity a -> Capacity a Source # abs :: Capacity a -> Capacity a Source # signum :: Capacity a -> Capacity a Source # fromInteger :: Integer -> Capacity a Source #
Ord a => Ord ( Capacity a) Source #
Instance details Defined in Algebra.Graph.Label Methods compare :: Capacity a -> Capacity a -> Ordering Source # (<) :: Capacity a -> Capacity a -> Bool Source # (<=) :: Capacity a -> Capacity a -> Bool Source # (>) :: Capacity a -> Capacity a -> Bool Source # (>=) :: Capacity a -> Capacity a -> Bool Source # max :: Capacity a -> Capacity a -> Capacity a Source # min :: Capacity a -> Capacity a -> Capacity a Source #
Show a => Show ( Capacity a) Source #
Instance details Defined in Algebra.Graph.Label Methods showsPrec :: Int -> Capacity a -> ShowS Source # show :: Capacity a -> String Source # showList :: [ Capacity a] -> ShowS Source #
Ord a => Semigroup ( Capacity a) Source #
Instance details Defined in Algebra.Graph.Label Methods (<>) :: Capacity a -> Capacity a -> Capacity a Source # sconcat :: NonEmpty ( Capacity a) -> Capacity a Source # stimes :: Integral b => b -> Capacity a -> Capacity a Source #
( Ord a, Num a) => Monoid ( Capacity a) Source #
Instance details Defined in Algebra.Graph.Label Methods mempty :: Capacity a Source # mappend :: Capacity a -> Capacity a -> Capacity a Source # mconcat :: [ Capacity a] -> Capacity a Source #
( Num a, Ord a) => Dioid ( Capacity a) Source #
Instance details Defined in Algebra.Graph.Label
( Num a, Ord a) => StarSemiring ( Capacity a) Source #
Instance details Defined in Algebra.Graph.Label Methods star :: Capacity a -> Capacity a Source #
( Num a, Ord a) => Semiring ( Capacity a) Source #
Instance details Defined in Algebra.Graph.Label Methods one :: Capacity a Source # (<.>) :: Capacity a -> Capacity a -> Capacity a Source #

capacity :: NonNegative a -> Capacity a Source #

A non-negative capacity.

getCapacity :: Capacity a -> NonNegative a Source #

Get the value of a capacity.

data Count a Source #

A count is a non-negative value that can be finite or infinite . Counts form a Semiring as follows:

zero  = 0
one   = 1
(<+>) = (+)
(<.>) = (*)

Instances

Instances details

Num a => Bounded ( Count a) Source #
Instance details Defined in Algebra.Graph.Label Methods minBound :: Count a Source # maxBound :: Count a Source #
Eq a => Eq ( Count a) Source #
Instance details Defined in Algebra.Graph.Label Methods (==) :: Count a -> Count a -> Bool Source # (/=) :: Count a -> Count a -> Bool Source #
( Num a, Ord a) => Num ( Count a) Source #
Instance details Defined in Algebra.Graph.Label Methods (+) :: Count a -> Count a -> Count a Source # (-) :: Count a -> Count a -> Count a Source # (*) :: Count a -> Count a -> Count a Source # negate :: Count a -> Count a Source # abs :: Count a -> Count a Source # signum :: Count a -> Count a Source # fromInteger :: Integer -> Count a Source #
Ord a => Ord ( Count a) Source #
Instance details Defined in Algebra.Graph.Label Methods compare :: Count a -> Count a -> Ordering Source # (<) :: Count a -> Count a -> Bool Source # (<=) :: Count a -> Count a -> Bool Source # (>) :: Count a -> Count a -> Bool Source # (>=) :: Count a -> Count a -> Bool Source # max :: Count a -> Count a -> Count a Source # min :: Count a -> Count a -> Count a Source #
Show a => Show ( Count a) Source #
Instance details Defined in Algebra.Graph.Label Methods showsPrec :: Int -> Count a -> ShowS Source # show :: Count a -> String Source # showList :: [ Count a] -> ShowS Source #
( Num a, Ord a) => Semigroup ( Count a) Source #
Instance details Defined in Algebra.Graph.Label Methods (<>) :: Count a -> Count a -> Count a Source # sconcat :: NonEmpty ( Count a) -> Count a Source # stimes :: Integral b => b -> Count a -> Count a Source #
( Num a, Ord a) => Monoid ( Count a) Source #
Instance details Defined in Algebra.Graph.Label Methods mempty :: Count a Source # mappend :: Count a -> Count a -> Count a Source # mconcat :: [ Count a] -> Count a Source #
( Num a, Ord a) => StarSemiring ( Count a) Source #
Instance details Defined in Algebra.Graph.Label Methods star :: Count a -> Count a Source #
( Num a, Ord a) => Semiring ( Count a) Source #
Instance details Defined in Algebra.Graph.Label Methods one :: Count a Source # (<.>) :: Count a -> Count a -> Count a Source #

count :: NonNegative a -> Count a Source #

A non-negative count.

getCount :: Count a -> NonNegative a Source #

Get the value of a count.

newtype PowerSet a Source #

The power set over the underlying set of elements a . If a is a monoid, then the power set forms a Dioid as follows:

zero    = PowerSet Set.empty
one     = PowerSet $ Set.singleton mempty
x <+> y = PowerSet $ Set.union (getPowerSet x) (getPowerSet y)
x <.> y = PowerSet $ cartesianProductWith mappend (getPowerSet x) (getPowerSet y)

Constructors

PowerSet
Fields getPowerSet :: Set a

Instances

Instances details

Eq a => Eq ( PowerSet a) Source #
Instance details Defined in Algebra.Graph.Label Methods (==) :: PowerSet a -> PowerSet a -> Bool Source # (/=) :: PowerSet a -> PowerSet a -> Bool Source #
Ord a => Ord ( PowerSet a) Source #
Instance details Defined in Algebra.Graph.Label Methods compare :: PowerSet a -> PowerSet a -> Ordering Source # (<) :: PowerSet a -> PowerSet a -> Bool Source # (<=) :: PowerSet a -> PowerSet a -> Bool Source # (>) :: PowerSet a -> PowerSet a -> Bool Source # (>=) :: PowerSet a -> PowerSet a -> Bool Source # max :: PowerSet a -> PowerSet a -> PowerSet a Source # min :: PowerSet a -> PowerSet a -> PowerSet a Source #
Show a => Show ( PowerSet a) Source #
Instance details Defined in Algebra.Graph.Label Methods showsPrec :: Int -> PowerSet a -> ShowS Source # show :: PowerSet a -> String Source # showList :: [ PowerSet a] -> ShowS Source #
Ord a => Semigroup ( PowerSet a) Source #
Instance details Defined in Algebra.Graph.Label Methods (<>) :: PowerSet a -> PowerSet a -> PowerSet a Source # sconcat :: NonEmpty ( PowerSet a) -> PowerSet a Source # stimes :: Integral b => b -> PowerSet a -> PowerSet a Source #
Ord a => Monoid ( PowerSet a) Source #
Instance details Defined in Algebra.Graph.Label Methods mempty :: PowerSet a Source # mappend :: PowerSet a -> PowerSet a -> PowerSet a Source # mconcat :: [ PowerSet a] -> PowerSet a Source #
( Monoid a, Ord a) => Dioid ( PowerSet a) Source #
Instance details Defined in Algebra.Graph.Label
( Monoid a, Ord a) => Semiring ( PowerSet a) Source #
Instance details Defined in Algebra.Graph.Label Methods one :: PowerSet a Source # (<.>) :: PowerSet a -> PowerSet a -> PowerSet a Source #

data Minimum a Source #

If a is a monoid, Minimum a forms the following Dioid :

zero  = noMinimum
one   = pure mempty
(<+>) = liftA2 min
(<.>) = liftA2 mappend

To create a singleton value of type Minimum a use the pure function. For example:

getMinimum (pure "Hello, " <+> pure "World!") == Just "Hello, "
getMinimum (pure "Hello, " <.> pure "World!") == Just "Hello, World!"

Instances

Instances details

Monad Minimum Source #
Instance details Defined in Algebra.Graph.Label Methods (>>=) :: Minimum a -> (a -> Minimum b) -> Minimum b Source # (>>) :: Minimum a -> Minimum b -> Minimum b Source # return :: a -> Minimum a Source #
Functor Minimum Source #
Instance details Defined in Algebra.Graph.Label Methods fmap :: (a -> b) -> Minimum a -> Minimum b Source # (<$) :: a -> Minimum b -> Minimum a Source #
Applicative Minimum Source #
Instance details Defined in Algebra.Graph.Label Methods pure :: a -> Minimum a Source # (<>) :: Minimum (a -> b) -> Minimum a -> Minimum b Source # liftA2 :: (a -> b -> c) -> Minimum a -> Minimum b -> Minimum c Source # (>) :: Minimum a -> Minimum b -> Minimum b Source # (<*) :: Minimum a -> Minimum b -> Minimum a Source #
IsList a => IsList ( Minimum a) Source #
Instance details Defined in Algebra.Graph.Label Associated Types type Item ( Minimum a) Source # Methods fromList :: [ Item ( Minimum a)] -> Minimum a Source # fromListN :: Int -> [ Item ( Minimum a)] -> Minimum a Source # toList :: Minimum a -> [ Item ( Minimum a)] Source #
Eq a => Eq ( Minimum a) Source #
Instance details Defined in Algebra.Graph.Label Methods (==) :: Minimum a -> Minimum a -> Bool Source # (/=) :: Minimum a -> Minimum a -> Bool Source #
Ord a => Ord ( Minimum a) Source #
Instance details Defined in Algebra.Graph.Label Methods compare :: Minimum a -> Minimum a -> Ordering Source # (<) :: Minimum a -> Minimum a -> Bool Source # (<=) :: Minimum a -> Minimum a -> Bool Source # (>) :: Minimum a -> Minimum a -> Bool Source # (>=) :: Minimum a -> Minimum a -> Bool Source # max :: Minimum a -> Minimum a -> Minimum a Source # min :: Minimum a -> Minimum a -> Minimum a Source #
Show a => Show ( Minimum a) Source #
Instance details Defined in Algebra.Graph.Label Methods showsPrec :: Int -> Minimum a -> ShowS Source # show :: Minimum a -> String Source # showList :: [ Minimum a] -> ShowS Source #
Ord a => Semigroup ( Minimum a) Source #
Instance details Defined in Algebra.Graph.Label Methods (<>) :: Minimum a -> Minimum a -> Minimum a Source # sconcat :: NonEmpty ( Minimum a) -> Minimum a Source # stimes :: Integral b => b -> Minimum a -> Minimum a Source #
( Monoid a, Ord a) => Monoid ( Minimum a) Source #
Instance details Defined in Algebra.Graph.Label Methods mempty :: Minimum a Source # mappend :: Minimum a -> Minimum a -> Minimum a Source # mconcat :: [ Minimum a] -> Minimum a Source #
( Monoid a, Ord a) => Dioid ( Minimum a) Source #
Instance details Defined in Algebra.Graph.Label
( Monoid a, Ord a) => Semiring ( Minimum a) Source #
Instance details Defined in Algebra.Graph.Label Methods one :: Minimum a Source # (<.>) :: Minimum a -> Minimum a -> Minimum a Source #
type Item ( Minimum a) Source #
Instance details Defined in Algebra.Graph.Label type Item ( Minimum a) = Item a

getMinimum :: Minimum a -> Maybe a Source #

Extract the minimum or Nothing if it does not exist.

noMinimum :: Minimum a Source #

The value corresponding to the lack of minimum, e.g. the minimum of the empty set.

type Path a = [(a, a)] Source #

A path is a list of edges.

data Label a Source #

The type of free labels over the underlying set of symbols a . This data type is an instance of classes StarSemiring and Dioid .

Instances

Instances details

Functor Label Source #
Instance details Defined in Algebra.Graph.Label Methods fmap :: (a -> b) -> Label a -> Label b Source # (<$) :: a -> Label b -> Label a Source #
IsList ( Label a) Source #
Instance details Defined in Algebra.Graph.Label Associated Types type Item ( Label a) Source # Methods fromList :: [ Item ( Label a)] -> Label a Source # fromListN :: Int -> [ Item ( Label a)] -> Label a Source # toList :: Label a -> [ Item ( Label a)] Source #
Show a => Show ( Label a) Source #
Instance details Defined in Algebra.Graph.Label Methods showsPrec :: Int -> Label a -> ShowS Source # show :: Label a -> String Source # showList :: [ Label a] -> ShowS Source #
Semigroup ( Label a) Source #
Instance details Defined in Algebra.Graph.Label Methods (<>) :: Label a -> Label a -> Label a Source # sconcat :: NonEmpty ( Label a) -> Label a Source # stimes :: Integral b => b -> Label a -> Label a Source #
Monoid ( Label a) Source #
Instance details Defined in Algebra.Graph.Label Methods mempty :: Label a Source # mappend :: Label a -> Label a -> Label a Source # mconcat :: [ Label a] -> Label a Source #
StarSemiring ( Label a) Source #
Instance details Defined in Algebra.Graph.Label Methods star :: Label a -> Label a Source #
Semiring ( Label a) Source #
Instance details Defined in Algebra.Graph.Label Methods one :: Label a Source # (<.>) :: Label a -> Label a -> Label a Source #
type Item ( Label a) Source #
Instance details Defined in Algebra.Graph.Label type Item ( Label a) = a

isZero :: Label a -> Bool Source #

Check if a Label is zero .

type RegularExpression a = Label a Source #

A type synonym for regular expressions , built on top of free labels .

Combining edge labels

data Optimum o a Source #

An optimum semiring obtained by combining a semiring o that defines an optimisation criterion , and a semiring a that describes the arguments of an optimisation problem. For example, by choosing o = Distance Int and and a = Minimum ( Path String) , we obtain the shortest path semiring for computing the shortest path in an Int -labelled graph with String vertices.

We assume that the semiring o is selective i.e. for all x and y :

x <+> y == x || x <+> y == y

In words, the operation <+> always simply selects one of its arguments. For example, the Capacity and Distance semirings are selective, whereas the the Count semiring is not.

Constructors

Optimum
Fields getOptimum :: o getArgument :: a

Instances

Instances details

( Eq o, Eq a) => Eq ( Optimum o a) Source #
Instance details Defined in Algebra.Graph.Label Methods (==) :: Optimum o a -> Optimum o a -> Bool Source # (/=) :: Optimum o a -> Optimum o a -> Bool Source #
( Ord o, Ord a) => Ord ( Optimum o a) Source #
Instance details Defined in Algebra.Graph.Label Methods compare :: Optimum o a -> Optimum o a -> Ordering Source # (<) :: Optimum o a -> Optimum o a -> Bool Source # (<=) :: Optimum o a -> Optimum o a -> Bool Source # (>) :: Optimum o a -> Optimum o a -> Bool Source # (>=) :: Optimum o a -> Optimum o a -> Bool Source # max :: Optimum o a -> Optimum o a -> Optimum o a Source # min :: Optimum o a -> Optimum o a -> Optimum o a Source #
( Show o, Show a) => Show ( Optimum o a) Source #
Instance details Defined in Algebra.Graph.Label Methods showsPrec :: Int -> Optimum o a -> ShowS Source # show :: Optimum o a -> String Source # showList :: [ Optimum o a] -> ShowS Source #
( Eq o, Monoid a, Monoid o) => Semigroup ( Optimum o a) Source #
Instance details Defined in Algebra.Graph.Label Methods (<>) :: Optimum o a -> Optimum o a -> Optimum o a Source # sconcat :: NonEmpty ( Optimum o a) -> Optimum o a Source # stimes :: Integral b => b -> Optimum o a -> Optimum o a Source #
( Eq o, Monoid a, Monoid o) => Monoid ( Optimum o a) Source #
Instance details Defined in Algebra.Graph.Label Methods mempty :: Optimum o a Source # mappend :: Optimum o a -> Optimum o a -> Optimum o a Source # mconcat :: [ Optimum o a] -> Optimum o a Source #
( Eq o, Dioid a, Dioid o) => Dioid ( Optimum o a) Source #
Instance details Defined in Algebra.Graph.Label
( Eq o, StarSemiring a, StarSemiring o) => StarSemiring ( Optimum o a) Source #
Instance details Defined in Algebra.Graph.Label Methods star :: Optimum o a -> Optimum o a Source #
( Eq o, Semiring a, Semiring o) => Semiring ( Optimum o a) Source #
Instance details Defined in Algebra.Graph.Label Methods one :: Optimum o a Source # (<.>) :: Optimum o a -> Optimum o a -> Optimum o a Source #

type ShortestPath e a = Optimum ( Distance e) ( Minimum ( Path a)) Source #

The Optimum semiring specialised to finding the lexicographically smallest shortest path .

type AllShortestPaths e a = Optimum ( Distance e) ( PowerSet ( Path a)) Source #

The Optimum semiring specialised to finding all shortest paths .

type CountShortestPaths e = Optimum ( Distance e) ( Count Integer ) Source #

The Optimum semiring specialised to counting all shortest paths .

type WidestPath e a = Optimum ( Capacity e) ( Minimum ( Path a)) Source #

The Optimum semiring specialised to finding the lexicographically smallest widest path .