algebraic-graphs-0.5: A library for algebraic graph construction and transformation

Algebra.Graph.Relation.Preorder

Contents

Description

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

Synopsis

# Data structure

The PreorderRelation data type represents a binary relation that is both reflexive and transitive. Preorders satisfy all laws of the Preorder type class and, in particular, the self-loop axiom:

vertex x == vertex x * vertex x

and the closure axiom:

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

For example, the following holds:

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

The Show instance produces reflexively and transitively closed expressions:

show (1             :: PreorderRelation Int) == "edge 1 1"
show (1 * 2         :: PreorderRelation Int) == "edges [(1,1),(1,2),(2,2)]"
show (1 * 2 + 2 * 3 :: PreorderRelation Int) == "edges [(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)]"
Instances
 Ord a => Eq (PreorderRelation a) Source # Instance detailsDefined in Algebra.Graph.Relation.Preorder Methods (Ord a, Num a) => Num (PreorderRelation a) Source # Instance detailsDefined in Algebra.Graph.Relation.Preorder Methods Ord a => Ord (PreorderRelation a) Source # Instance detailsDefined in Algebra.Graph.Relation.Preorder Methods(<) :: PreorderRelation a -> PreorderRelation a -> Bool #(>) :: PreorderRelation a -> PreorderRelation a -> Bool # (Ord a, Show a) => Show (PreorderRelation a) Source # Instance detailsDefined in Algebra.Graph.Relation.Preorder MethodsshowList :: [PreorderRelation a] -> ShowS # NFData a => NFData (PreorderRelation a) Source # Instance detailsDefined in Algebra.Graph.Relation.Preorder Methodsrnf :: PreorderRelation a -> () # Ord a => Preorder (PreorderRelation a) Source # Instance detailsDefined in Algebra.Graph.Relation.Preorder Source # Instance detailsDefined in Algebra.Graph.Relation.Preorder Ord a => Reflexive (PreorderRelation a) Source # Instance detailsDefined in Algebra.Graph.Relation.Preorder Ord a => Graph (PreorderRelation a) Source # Instance detailsDefined in Algebra.Graph.Relation.Preorder Associated Typestype Vertex (PreorderRelation a) :: Type Source # Methods type Vertex (PreorderRelation a) Source # Instance detailsDefined in Algebra.Graph.Relation.Preorder type Vertex (PreorderRelation a) = a

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

toRelation :: Ord a => PreorderRelation a -> Relation a Source #

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