Algebra.Graph.Label

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.

Semirings and dioids

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

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

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

Instances

Semiring Any

Defined in Algebra.Graph.Label

Methods

one :: Any

(<.>) :: Any -> Any -> Any

Semiring (Label a)

Defined in Algebra.Graph.Label

Methods

one :: Label a

(<.>) :: Label a -> Label a -> Label a

(Monoid a, Ord a) => Semiring (PowerSet a)

Defined in Algebra.Graph.Label

Methods

one :: PowerSet a

(<.>) :: PowerSet a -> PowerSet a -> PowerSet a

(Monoid a, Ord a) => Semiring (Minimum a)

Defined in Algebra.Graph.Label

Methods

one :: Minimum a

(<.>) :: Minimum a -> Minimum a -> Minimum a

(Num a, Ord a) => Semiring (Distance a)

Defined in Algebra.Graph.Label

Methods

one :: Distance a

(<.>) :: Distance a -> Distance a -> Distance a

(Num a, Ord a) => Semiring (Count a)

Defined in Algebra.Graph.Label

Methods

one :: Count a

(<.>) :: Count a -> Count a -> Count a

(Num a, Ord a) => Semiring (Capacity a)

Defined in Algebra.Graph.Label

Methods

one :: Capacity a

(<.>) :: Capacity a -> Capacity a -> Capacity a

(Eq o, Semiring a, Semiring o) => Semiring (Optimum o a)

Defined in Algebra.Graph.Label

Methods

one :: Optimum o a

(<.>) :: Optimum o a -> Optimum o a -> Optimum o a

zero :: Monoid a => a

An alias for mempty.

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

An alias for <>.

class Semiring a => StarSemiring a where

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

Instances

StarSemiring Any

Defined in Algebra.Graph.Label

Methods

star :: Any -> Any

StarSemiring (Label a)

Defined in Algebra.Graph.Label

Methods

star :: Label a -> Label a

(Num a, Ord a) => StarSemiring (Distance a)

Defined in Algebra.Graph.Label

Methods

star :: Distance a -> Distance a

(Num a, Ord a) => StarSemiring (Count a)

Defined in Algebra.Graph.Label

Methods

star :: Count a -> Count a

(Num a, Ord a) => StarSemiring (Capacity a)

Defined in Algebra.Graph.Label

Methods

star :: Capacity a -> Capacity a

(Eq o, StarSemiring a, StarSemiring o) => StarSemiring (Optimum o a)

Defined in Algebra.Graph.Label

Methods

star :: Optimum o a -> Optimum o a

class Semiring a => Dioid a

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

x <+> x == x

Instances

Dioid Any

Defined in Algebra.Graph.Label

(Monoid a, Ord a) => Dioid (PowerSet a)

Defined in Algebra.Graph.Label

(Monoid a, Ord a) => Dioid (Minimum a)

Defined in Algebra.Graph.Label

(Num a, Ord a) => Dioid (Distance a)

Defined in Algebra.Graph.Label

(Num a, Ord a) => Dioid (Capacity a)

Defined in Algebra.Graph.Label

(Eq o, Dioid a, Dioid o) => Dioid (Optimum o a)

Defined in Algebra.Graph.Label

Data types for edge labels

data NonNegative a

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

Monad NonNegative

Defined in Algebra.Graph.Label

Methods

(>>=) :: NonNegative a -> (a -> NonNegative b) -> NonNegative b

(>>) :: NonNegative a -> NonNegative b -> NonNegative b

return :: a -> NonNegative a

Functor NonNegative

Defined in Algebra.Graph.Label

Methods

fmap :: (a -> b) -> NonNegative a -> NonNegative b

(<$) :: a -> NonNegative b -> NonNegative a

Applicative NonNegative

Defined in Algebra.Graph.Label

Methods

pure :: a -> NonNegative a

(<*>) :: NonNegative (a -> b) -> NonNegative a -> NonNegative b

liftA2 :: (a -> b -> c) -> NonNegative a -> NonNegative b -> NonNegative c

(*>) :: NonNegative a -> NonNegative b -> NonNegative b

(<*) :: NonNegative a -> NonNegative b -> NonNegative a

Num a => Bounded (NonNegative a)

Defined in Algebra.Graph.Label

Methods

minBound :: NonNegative a

maxBound :: NonNegative a

Eq a => Eq (NonNegative a)

Defined in Algebra.Graph.Label

Methods

(==) :: NonNegative a -> NonNegative a -> Bool

(/=) :: NonNegative a -> NonNegative a -> Bool

(Num a, Ord a) => Num (NonNegative a)

Defined in Algebra.Graph.Label

Methods

(+) :: NonNegative a -> NonNegative a -> NonNegative a

(-) :: NonNegative a -> NonNegative a -> NonNegative a

(*) :: NonNegative a -> NonNegative a -> NonNegative a

negate :: NonNegative a -> NonNegative a

abs :: NonNegative a -> NonNegative a

signum :: NonNegative a -> NonNegative a

fromInteger :: Integer -> NonNegative a

Ord a => Ord (NonNegative a)

Defined in Algebra.Graph.Label

Methods

compare :: NonNegative a -> NonNegative a -> Ordering

(<) :: NonNegative a -> NonNegative a -> Bool

(<=) :: NonNegative a -> NonNegative a -> Bool

(>) :: NonNegative a -> NonNegative a -> Bool

(>=) :: NonNegative a -> NonNegative a -> Bool

max :: NonNegative a -> NonNegative a -> NonNegative a

min :: NonNegative a -> NonNegative a -> NonNegative a

(Num a, Show a) => Show (NonNegative a)

Defined in Algebra.Graph.Label

Methods

showsPrec :: Int -> NonNegative a -> ShowS

show :: NonNegative a -> String

showList :: [NonNegative a] -> ShowS

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

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

finiteWord :: Word -> NonNegative Word

A finite Word.

unsafeFinite :: a -> NonNegative a

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

The (non-negative) infinite value.

getFinite :: NonNegative a -> Maybe a

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

data Distance a

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

Num a => Bounded (Distance a)

Defined in Algebra.Graph.Label

Methods

minBound :: Distance a

maxBound :: Distance a

Eq a => Eq (Distance a)

Defined in Algebra.Graph.Label

Methods

(==) :: Distance a -> Distance a -> Bool

(/=) :: Distance a -> Distance a -> Bool

(Num a, Ord a) => Num (Distance a)

Defined in Algebra.Graph.Label

Methods

(+) :: Distance a -> Distance a -> Distance a

(-) :: Distance a -> Distance a -> Distance a

(*) :: Distance a -> Distance a -> Distance a

negate :: Distance a -> Distance a

abs :: Distance a -> Distance a

signum :: Distance a -> Distance a

fromInteger :: Integer -> Distance a

Ord a => Ord (Distance a)

Defined in Algebra.Graph.Label

Methods

compare :: Distance a -> Distance a -> Ordering

(<) :: Distance a -> Distance a -> Bool

(<=) :: Distance a -> Distance a -> Bool

(>) :: Distance a -> Distance a -> Bool

(>=) :: Distance a -> Distance a -> Bool

max :: Distance a -> Distance a -> Distance a

min :: Distance a -> Distance a -> Distance a

Show a => Show (Distance a)

Defined in Algebra.Graph.Label

Methods

showsPrec :: Int -> Distance a -> ShowS

show :: Distance a -> String

showList :: [Distance a] -> ShowS

Ord a => Semigroup (Distance a)

Defined in Algebra.Graph.Label

Methods

(<>) :: Distance a -> Distance a -> Distance a

sconcat :: NonEmpty (Distance a) -> Distance a

stimes :: Integral b => b -> Distance a -> Distance a

(Ord a, Num a) => Monoid (Distance a)

Defined in Algebra.Graph.Label

Methods

mempty :: Distance a

mappend :: Distance a -> Distance a -> Distance a

mconcat :: [Distance a] -> Distance a

(Num a, Ord a) => Dioid (Distance a)

Defined in Algebra.Graph.Label

(Num a, Ord a) => StarSemiring (Distance a)

Defined in Algebra.Graph.Label

Methods

star :: Distance a -> Distance a

(Num a, Ord a) => Semiring (Distance a)

Defined in Algebra.Graph.Label

Methods

one :: Distance a

(<.>) :: Distance a -> Distance a -> Distance a

distance :: NonNegative a -> Distance a

A non-negative distance.

getDistance :: Distance a -> NonNegative a

Get the value of a distance.

data Capacity a

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

Num a => Bounded (Capacity a)

Defined in Algebra.Graph.Label

Methods

minBound :: Capacity a

maxBound :: Capacity a

Eq a => Eq (Capacity a)

Defined in Algebra.Graph.Label

Methods

(==) :: Capacity a -> Capacity a -> Bool

(/=) :: Capacity a -> Capacity a -> Bool

(Num a, Ord a) => Num (Capacity a)

Defined in Algebra.Graph.Label

Methods

(+) :: Capacity a -> Capacity a -> Capacity a

(-) :: Capacity a -> Capacity a -> Capacity a

(*) :: Capacity a -> Capacity a -> Capacity a

negate :: Capacity a -> Capacity a

abs :: Capacity a -> Capacity a

signum :: Capacity a -> Capacity a

fromInteger :: Integer -> Capacity a

Ord a => Ord (Capacity a)

Defined in Algebra.Graph.Label

Methods

compare :: Capacity a -> Capacity a -> Ordering

(<) :: Capacity a -> Capacity a -> Bool

(<=) :: Capacity a -> Capacity a -> Bool

(>) :: Capacity a -> Capacity a -> Bool

(>=) :: Capacity a -> Capacity a -> Bool

max :: Capacity a -> Capacity a -> Capacity a

min :: Capacity a -> Capacity a -> Capacity a

Show a => Show (Capacity a)

Defined in Algebra.Graph.Label

Methods

showsPrec :: Int -> Capacity a -> ShowS

show :: Capacity a -> String

showList :: [Capacity a] -> ShowS

Ord a => Semigroup (Capacity a)

Defined in Algebra.Graph.Label

Methods

(<>) :: Capacity a -> Capacity a -> Capacity a

sconcat :: NonEmpty (Capacity a) -> Capacity a

stimes :: Integral b => b -> Capacity a -> Capacity a

(Ord a, Num a) => Monoid (Capacity a)

Defined in Algebra.Graph.Label

Methods

mempty :: Capacity a

mappend :: Capacity a -> Capacity a -> Capacity a

mconcat :: [Capacity a] -> Capacity a

(Num a, Ord a) => Dioid (Capacity a)

Defined in Algebra.Graph.Label

(Num a, Ord a) => StarSemiring (Capacity a)

Defined in Algebra.Graph.Label

Methods

star :: Capacity a -> Capacity a

(Num a, Ord a) => Semiring (Capacity a)

Defined in Algebra.Graph.Label

Methods

one :: Capacity a

(<.>) :: Capacity a -> Capacity a -> Capacity a

capacity :: NonNegative a -> Capacity a

A non-negative capacity.

getCapacity :: Capacity a -> NonNegative a

Get the value of a capacity.

data Count a

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

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

Instances

Num a => Bounded (Count a)

Defined in Algebra.Graph.Label

Methods

minBound :: Count a

maxBound :: Count a

Eq a => Eq (Count a)

Defined in Algebra.Graph.Label

Methods

(==) :: Count a -> Count a -> Bool

(/=) :: Count a -> Count a -> Bool

(Num a, Ord a) => Num (Count a)

Defined in Algebra.Graph.Label

Methods

(+) :: Count a -> Count a -> Count a

(-) :: Count a -> Count a -> Count a

(*) :: Count a -> Count a -> Count a

negate :: Count a -> Count a

abs :: Count a -> Count a

signum :: Count a -> Count a

fromInteger :: Integer -> Count a

Ord a => Ord (Count a)

Defined in Algebra.Graph.Label

Methods

compare :: Count a -> Count a -> Ordering

(<) :: Count a -> Count a -> Bool

(<=) :: Count a -> Count a -> Bool

(>) :: Count a -> Count a -> Bool

(>=) :: Count a -> Count a -> Bool

max :: Count a -> Count a -> Count a

min :: Count a -> Count a -> Count a

Show a => Show (Count a)

Defined in Algebra.Graph.Label

Methods

showsPrec :: Int -> Count a -> ShowS

show :: Count a -> String

showList :: [Count a] -> ShowS

(Num a, Ord a) => Semigroup (Count a)

Defined in Algebra.Graph.Label

Methods

(<>) :: Count a -> Count a -> Count a

sconcat :: NonEmpty (Count a) -> Count a

stimes :: Integral b => b -> Count a -> Count a

(Num a, Ord a) => Monoid (Count a)

Defined in Algebra.Graph.Label

Methods

mempty :: Count a

mappend :: Count a -> Count a -> Count a

mconcat :: [Count a] -> Count a

(Num a, Ord a) => StarSemiring (Count a)

Defined in Algebra.Graph.Label

Methods

star :: Count a -> Count a

(Num a, Ord a) => Semiring (Count a)

Defined in Algebra.Graph.Label

Methods

one :: Count a

(<.>) :: Count a -> Count a -> Count a

count :: NonNegative a -> Count a

A non-negative count.

getCount :: Count a -> NonNegative a

Get the value of a count.

newtype PowerSet a

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

Eq a => Eq (PowerSet a)

Defined in Algebra.Graph.Label

Methods

(==) :: PowerSet a -> PowerSet a -> Bool

(/=) :: PowerSet a -> PowerSet a -> Bool

Ord a => Ord (PowerSet a)

Defined in Algebra.Graph.Label

Methods

compare :: PowerSet a -> PowerSet a -> Ordering

(<) :: PowerSet a -> PowerSet a -> Bool

(<=) :: PowerSet a -> PowerSet a -> Bool

(>) :: PowerSet a -> PowerSet a -> Bool

(>=) :: PowerSet a -> PowerSet a -> Bool

max :: PowerSet a -> PowerSet a -> PowerSet a

min :: PowerSet a -> PowerSet a -> PowerSet a

Show a => Show (PowerSet a)

Defined in Algebra.Graph.Label

Methods

showsPrec :: Int -> PowerSet a -> ShowS

show :: PowerSet a -> String

showList :: [PowerSet a] -> ShowS

Ord a => Semigroup (PowerSet a)

Defined in Algebra.Graph.Label

Methods

(<>) :: PowerSet a -> PowerSet a -> PowerSet a

sconcat :: NonEmpty (PowerSet a) -> PowerSet a

stimes :: Integral b => b -> PowerSet a -> PowerSet a

Ord a => Monoid (PowerSet a)

Defined in Algebra.Graph.Label

Methods

mempty :: PowerSet a

mappend :: PowerSet a -> PowerSet a -> PowerSet a

mconcat :: [PowerSet a] -> PowerSet a

(Monoid a, Ord a) => Dioid (PowerSet a)

Defined in Algebra.Graph.Label

(Monoid a, Ord a) => Semiring (PowerSet a)

Defined in Algebra.Graph.Label

Methods

one :: PowerSet a

(<.>) :: PowerSet a -> PowerSet a -> PowerSet a

data Minimum a

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

Monad Minimum

Defined in Algebra.Graph.Label

Methods

(>>=) :: Minimum a -> (a -> Minimum b) -> Minimum b

(>>) :: Minimum a -> Minimum b -> Minimum b

return :: a -> Minimum a

Functor Minimum

Defined in Algebra.Graph.Label

Methods

fmap :: (a -> b) -> Minimum a -> Minimum b

(<$) :: a -> Minimum b -> Minimum a

Applicative Minimum

Defined in Algebra.Graph.Label

Methods

pure :: a -> Minimum a

(<*>) :: Minimum (a -> b) -> Minimum a -> Minimum b

liftA2 :: (a -> b -> c) -> Minimum a -> Minimum b -> Minimum c

(*>) :: Minimum a -> Minimum b -> Minimum b

(<*) :: Minimum a -> Minimum b -> Minimum a

IsList a => IsList (Minimum a)

Defined in Algebra.Graph.Label

Associated Types

type Item (Minimum a)

Methods

fromList :: [Item (Minimum a)] -> Minimum a

fromListN :: Int -> [Item (Minimum a)] -> Minimum a

toList :: Minimum a -> [Item (Minimum a)]

Eq a => Eq (Minimum a)

Defined in Algebra.Graph.Label

Methods

(==) :: Minimum a -> Minimum a -> Bool

(/=) :: Minimum a -> Minimum a -> Bool

Ord a => Ord (Minimum a)

Defined in Algebra.Graph.Label

Methods

compare :: Minimum a -> Minimum a -> Ordering

(<) :: Minimum a -> Minimum a -> Bool

(<=) :: Minimum a -> Minimum a -> Bool

(>) :: Minimum a -> Minimum a -> Bool

(>=) :: Minimum a -> Minimum a -> Bool

max :: Minimum a -> Minimum a -> Minimum a

min :: Minimum a -> Minimum a -> Minimum a

Show a => Show (Minimum a)

Defined in Algebra.Graph.Label

Methods

showsPrec :: Int -> Minimum a -> ShowS

show :: Minimum a -> String

showList :: [Minimum a] -> ShowS

Ord a => Semigroup (Minimum a)

Defined in Algebra.Graph.Label

Methods

(<>) :: Minimum a -> Minimum a -> Minimum a

sconcat :: NonEmpty (Minimum a) -> Minimum a

stimes :: Integral b => b -> Minimum a -> Minimum a

(Monoid a, Ord a) => Monoid (Minimum a)

Defined in Algebra.Graph.Label

Methods

mempty :: Minimum a

mappend :: Minimum a -> Minimum a -> Minimum a

mconcat :: [Minimum a] -> Minimum a

(Monoid a, Ord a) => Dioid (Minimum a)

Defined in Algebra.Graph.Label

(Monoid a, Ord a) => Semiring (Minimum a)

Defined in Algebra.Graph.Label

Methods

one :: Minimum a

(<.>) :: Minimum a -> Minimum a -> Minimum a

type Item (Minimum a)

Defined in Algebra.Graph.Label

type Item (Minimum a) = Item a

getMinimum :: Minimum a -> Maybe a

Extract the minimum or Nothing if it does not exist.

noMinimum :: Minimum a

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

type Path a = [(a, a)]

A path is a list of edges.

data Label a

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

Instances

Functor Label

Defined in Algebra.Graph.Label

Methods

fmap :: (a -> b) -> Label a -> Label b

(<$) :: a -> Label b -> Label a

IsList (Label a)

Defined in Algebra.Graph.Label

Associated Types

type Item (Label a)

Methods

fromList :: [Item (Label a)] -> Label a

fromListN :: Int -> [Item (Label a)] -> Label a

toList :: Label a -> [Item (Label a)]

Show a => Show (Label a)

Defined in Algebra.Graph.Label

Methods

showsPrec :: Int -> Label a -> ShowS

show :: Label a -> String

showList :: [Label a] -> ShowS

Semigroup (Label a)

Defined in Algebra.Graph.Label

Methods

(<>) :: Label a -> Label a -> Label a

sconcat :: NonEmpty (Label a) -> Label a

stimes :: Integral b => b -> Label a -> Label a

Monoid (Label a)

Defined in Algebra.Graph.Label

Methods

mempty :: Label a

mappend :: Label a -> Label a -> Label a

mconcat :: [Label a] -> Label a

StarSemiring (Label a)

Defined in Algebra.Graph.Label

Methods

star :: Label a -> Label a

Semiring (Label a)

Defined in Algebra.Graph.Label

Methods

one :: Label a

(<.>) :: Label a -> Label a -> Label a

type Item (Label a)

Defined in Algebra.Graph.Label

type Item (Label a) = a

isZero :: Label a -> Bool

Check if a Label is zero.

type RegularExpression a = Label a

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

Combining edge labels

data Optimum o a

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

(Eq o, Eq a) => Eq (Optimum o a)

Defined in Algebra.Graph.Label

Methods

(==) :: Optimum o a -> Optimum o a -> Bool

(/=) :: Optimum o a -> Optimum o a -> Bool

(Ord o, Ord a) => Ord (Optimum o a)

Defined in Algebra.Graph.Label

Methods

compare :: Optimum o a -> Optimum o a -> Ordering

(<) :: Optimum o a -> Optimum o a -> Bool

(<=) :: Optimum o a -> Optimum o a -> Bool

(>) :: Optimum o a -> Optimum o a -> Bool

(>=) :: Optimum o a -> Optimum o a -> Bool

max :: Optimum o a -> Optimum o a -> Optimum o a

min :: Optimum o a -> Optimum o a -> Optimum o a

(Show o, Show a) => Show (Optimum o a)

Defined in Algebra.Graph.Label

Methods

showsPrec :: Int -> Optimum o a -> ShowS

show :: Optimum o a -> String

showList :: [Optimum o a] -> ShowS

(Eq o, Monoid a, Monoid o) => Semigroup (Optimum o a)

Defined in Algebra.Graph.Label

Methods

(<>) :: Optimum o a -> Optimum o a -> Optimum o a

sconcat :: NonEmpty (Optimum o a) -> Optimum o a

stimes :: Integral b => b -> Optimum o a -> Optimum o a

(Eq o, Monoid a, Monoid o) => Monoid (Optimum o a)

Defined in Algebra.Graph.Label

Methods

mempty :: Optimum o a

mappend :: Optimum o a -> Optimum o a -> Optimum o a

mconcat :: [Optimum o a] -> Optimum o a

(Eq o, Dioid a, Dioid o) => Dioid (Optimum o a)

Defined in Algebra.Graph.Label

(Eq o, StarSemiring a, StarSemiring o) => StarSemiring (Optimum o a)

Defined in Algebra.Graph.Label

Methods

star :: Optimum o a -> Optimum o a

(Eq o, Semiring a, Semiring o) => Semiring (Optimum o a)

Defined in Algebra.Graph.Label

Methods

one :: Optimum o a

(<.>) :: Optimum o a -> Optimum o a -> Optimum o a

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

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

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

The Optimum semiring specialised to finding all shortest paths.

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

The Optimum semiring specialised to counting all shortest paths.

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

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