algebraic-graphs-0.7: A library for algebraic graph construction and transformation

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.Relation.Transitive

Contents

- Data structure

Description

An abstract implementation of transitive binary relations. Use Algebra.Graph.Class for polymorphic construction and manipulation.

Synopsis

- data TransitiveRelation a
- fromRelation :: Relation a -> TransitiveRelation a
- toRelation :: Ord a => TransitiveRelation a -> Relation a

# Data structure

data TransitiveRelation a #

The `TransitiveRelation` data type represents a transitive binary relation over a set of elements. Transitive relations satisfy all laws of the `Transitive` type class and, in particular, the closure axiom:

    y /= empty ==> x * y + x * z + y * z == x * y + y * z

For example, the following holds:

    path xs == (clique xs :: TransitiveRelation Int)

The `Show` instance produces transitively closed expressions:

    show (1 * 2         :: TransitiveRelation Int) == "edge 1 2"
    show (1 * 2 + 2 * 3 :: TransitiveRelation Int) == "edges [(1,2),(1,3),(2,3)]"

#### Instances

Instances details

Ord a => Graph (TransitiveRelation a) # |

Instance details

Defined in Algebra.Graph.Relation.Transitive

Associated Types

type Vertex (TransitiveRelation a) |
Instance details Defined in Algebra.Graph.Relation.Transitive type Vertex (TransitiveRelation a) = a

Methods

empty :: TransitiveRelation a #

vertex :: Vertex (TransitiveRelation a) -> TransitiveRelation a #

overlay :: TransitiveRelation a -> TransitiveRelation a -> TransitiveRelation a #

connect :: TransitiveRelation a -> TransitiveRelation a -> TransitiveRelation a #

Ord a => Transitive (TransitiveRelation a) # |
Instance details Defined in Algebra.Graph.Relation.Transitive
NFData a => NFData (TransitiveRelation a) # |
Instance details Defined in Algebra.Graph.Relation.Transitive Methods rnf :: TransitiveRelation a -> () #
IsString a => IsString (TransitiveRelation a) # |
Instance details Defined in Algebra.Graph.Relation.Transitive Methods fromString :: String -> TransitiveRelation a #
(Ord a, Num a) => Num (TransitiveRelation a) # |

Instance details

Defined in Algebra.Graph.Relation.Transitive

Methods

(+) :: TransitiveRelation a -> TransitiveRelation a -> TransitiveRelation a #

(-) :: TransitiveRelation a -> TransitiveRelation a -> TransitiveRelation a #

(*) :: TransitiveRelation a -> TransitiveRelation a -> TransitiveRelation a #

negate :: TransitiveRelation a -> TransitiveRelation a #

abs :: TransitiveRelation a -> TransitiveRelation a #

signum :: TransitiveRelation a -> TransitiveRelation a #

fromInteger :: Integer -> TransitiveRelation a #

(Ord a, Show a) => Show (TransitiveRelation a) # |
Instance details Defined in Algebra.Graph.Relation.Transitive Methods showsPrec :: Int -> TransitiveRelation a -> ShowS # show :: TransitiveRelation a -> String # showList :: [TransitiveRelation a] -> ShowS #
Ord a => Eq (TransitiveRelation a) # |
Instance details Defined in Algebra.Graph.Relation.Transitive Methods (==) :: TransitiveRelation a -> TransitiveRelation a -> Bool # (/=) :: TransitiveRelation a -> TransitiveRelation a -> Bool #
Ord a => Ord (TransitiveRelation a) # |

Instance details

Defined in Algebra.Graph.Relation.Transitive

Methods

compare :: TransitiveRelation a -> TransitiveRelation a -> Ordering #

(<) :: TransitiveRelation a -> TransitiveRelation a -> Bool #

(<=) :: TransitiveRelation a -> TransitiveRelation a -> Bool #

(>) :: TransitiveRelation a -> TransitiveRelation a -> Bool #

(>=) :: TransitiveRelation a -> TransitiveRelation a -> Bool #

max :: TransitiveRelation a -> TransitiveRelation a -> TransitiveRelation a #

min :: TransitiveRelation a -> TransitiveRelation a -> TransitiveRelation a #

type Vertex (TransitiveRelation a) # |
Instance details Defined in Algebra.Graph.Relation.Transitive type Vertex (TransitiveRelation a) = a

fromRelation :: Relation a -> TransitiveRelation a #

Construct a transitive relation from a `Relation`. Complexity: O(1) time.

toRelation :: Ord a => TransitiveRelation a -> Relation a #

Extract the underlying relation. Complexity: O(n * m * log(m)) time.