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Language	Haskell98

Math.Combinatorics.Digraph

Description

A module for working with directed graphs (digraphs). Some of the functions are specifically for working with directed acyclic graphs (DAGs), that is, directed graphs containing no cycles.

Synopsis

data Digraph v = DG [v] [(v, v)]
nf :: Ord v => Digraph v -> Digraph v
vertices :: Digraph v -> [v]
edges :: Digraph v -> [(v, v)]
predecessors :: Eq a => Digraph a -> a -> [a]
successors :: Eq a => Digraph a -> a -> [a]
adjLists :: Ord a => Digraph a -> (Map a [a], Map a [a])
digraphIsos1 :: (Eq a1, Eq a2) => Digraph a1 -> Digraph a2 -> [[(a1, a2)]]
digraphIsos2 :: (Ord k2, Ord k1) => Digraph k2 -> Digraph k1 -> [[(k2, k1)]]
heightPartitionDAG :: Ord k => Digraph k -> [[k]]
isDAG :: Ord a => Digraph a -> Bool
dagIsos :: (Ord a1, Ord a2) => Digraph a2 -> Digraph a1 -> [[(a2, a1)]]
isDagIso :: (Ord a, Ord b) => Digraph a -> Digraph b -> Bool
perms :: [a] -> [[a]]
isoRepDAG1 :: Ord k => Digraph k -> Digraph Int
isoRepDAG2 :: (Ord b, Num b, Enum b, Ord a) => Digraph a -> [(a, b)]
isoRepDAG3 :: Ord v => Digraph v -> Digraph Int
isoRepDAG :: Ord a => Digraph a -> Digraph Int

Documentation

data Digraph v Source #

A digraph is represented as DG vs es, where vs is the list of vertices, and es is the list of edges. Edges are directed: an edge (u,v) means an edge from u to v. A digraph is considered to be in normal form if both es and vs are in ascending order. This is the preferred form, and some functions will only work for digraphs in normal form.

Constructors

DG [v] [(v, v)]

Instances

Functor Digraph Source #
Instance details Defined in Math.Combinatorics.Digraph Methods fmap :: (a -> b) -> Digraph a -> Digraph b # (<$) :: a -> Digraph b -> Digraph a #
Eq v => Eq (Digraph v) Source #
Instance details Defined in Math.Combinatorics.Digraph Methods (==) :: Digraph v -> Digraph v -> Bool # (/=) :: Digraph v -> Digraph v -> Bool #
Ord v => Ord (Digraph v) Source #
Instance details Defined in Math.Combinatorics.Digraph Methods compare :: Digraph v -> Digraph v -> Ordering # (<) :: Digraph v -> Digraph v -> Bool # (<=) :: Digraph v -> Digraph v -> Bool # (>) :: Digraph v -> Digraph v -> Bool # (>=) :: Digraph v -> Digraph v -> Bool # max :: Digraph v -> Digraph v -> Digraph v # min :: Digraph v -> Digraph v -> Digraph v #
Show v => Show (Digraph v) Source #
Instance details Defined in Math.Combinatorics.Digraph Methods showsPrec :: Int -> Digraph v -> ShowS # show :: Digraph v -> String # showList :: [Digraph v] -> ShowS #

nf :: Ord v => Digraph v -> Digraph v Source #

vertices :: Digraph v -> [v] Source #

edges :: Digraph v -> [(v, v)] Source #

predecessors :: Eq a => Digraph a -> a -> [a] Source #

successors :: Eq a => Digraph a -> a -> [a] Source #

adjLists :: Ord a => Digraph a -> (Map a [a], Map a [a]) Source #

digraphIsos1 :: (Eq a1, Eq a2) => Digraph a1 -> Digraph a2 -> [[(a1, a2)]] Source #

digraphIsos2 :: (Ord k2, Ord k1) => Digraph k2 -> Digraph k1 -> [[(k2, k1)]] Source #

heightPartitionDAG :: Ord k => Digraph k -> [[k]] Source #

isDAG :: Ord a => Digraph a -> Bool Source #

dagIsos :: (Ord a1, Ord a2) => Digraph a2 -> Digraph a1 -> [[(a2, a1)]] Source #

isDagIso :: (Ord a, Ord b) => Digraph a -> Digraph b -> Bool Source #

Are the two DAGs isomorphic?

perms :: [a] -> [[a]] Source #

isoRepDAG1 :: Ord k => Digraph k -> Digraph Int Source #

isoRepDAG2 :: (Ord b, Num b, Enum b, Ord a) => Digraph a -> [(a, b)] Source #

isoRepDAG3 :: Ord v => Digraph v -> Digraph Int Source #

isoRepDAG :: Ord a => Digraph a -> Digraph Int Source #

Given a directed acyclic graph (DAG), return a canonical representative for its isomorphism class. isoRepDAG dag is isomorphic to dag. It follows that if isoRepDAG dagA == isoRepDAG dagB then dagA is isomorphic to dagB. Conversely, isoRepDAG dag is the minimal element in the isomorphism class, subject to some constraints. It follows that if dagA is isomorphic to dagB, then isoRepDAG dagA == isoRepDAG dagB.

The algorithm of course is faster on some DAGs than others: roughly speaking, it prefers "tall" DAGs (long chains) to "wide" DAGs (long antichains), and it prefers asymmetric DAGs (ie those with smaller automorphism groups).