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

Copyright (c) Andrey Mokhov 2016-2019 MIT (see the file LICENSE) andrey.mokhov@gmail.com experimental None Haskell2010

Algebra.Graph.ToGraph

Contents

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 defines the type class ToGraph for capturing data types that can be converted to algebraic graphs. To make an instance of this class you need to define just a single method (toGraph or foldg), which gives you access to many other useful methods for free (although note that the default implementations may be suboptimal performance-wise).

This type class is similar to the standard type class Foldable defined for lists. Furthermore, one can define Foldable methods foldMap and toList using ToGraph.foldg:

foldMap f = foldg mempty f    (<>) (<>)
toList    = foldg []     pure (++) (++)


However, the resulting Foldable instance is problematic. For example, folding equivalent algebraic graphs 1 and 1 + 1 leads to different results:

toList (1    ) == [1]
toList (1 + 1) == [1, 1]


To avoid such cases, we do not provide Foldable instances for algebraic graph datatypes. Furthermore, we require that the four arguments passed to foldg satisfy the laws of the algebra of graphs. The above definitions of foldMap and toList violate this requirement, for example [1] ++ [1] /= [1], and are therefore disallowed.

Synopsis

# Type class

class ToGraph t where Source #

The ToGraph type class captures data types that can be converted to algebraic graphs. Instances of this type class should satisfy the laws specified by the default method definitions.

Minimal complete definition

Associated Types

type ToVertex t Source #

The type of vertices of the resulting graph.

Methods

toGraph :: t -> Graph (ToVertex t) Source #

Convert a value to the corresponding algebraic graph, see Algebra.Graph.

toGraph == foldg Empty Vertex Overlay Connect


foldg :: r -> (ToVertex t -> r) -> (r -> r -> r) -> (r -> r -> r) -> t -> r Source #

The method foldg is used for generalised graph folding. It collapses a given value by applying the provided graph construction primitives. The order of arguments is: empty, vertex, overlay and connect, and it is assumed that the arguments satisfy the axioms of the graph algebra.

foldg == Algebra.Graph.foldg . toGraph


isEmpty :: t -> Bool Source #

Check if a graph is empty.

isEmpty == foldg True (const False) (&&) (&&)


hasVertex :: Eq (ToVertex t) => ToVertex t -> t -> Bool Source #

Check if a graph contains a given vertex.

hasVertex x == foldg False (==x) (||) (||)


hasEdge :: Eq (ToVertex t) => ToVertex t -> ToVertex t -> t -> Bool Source #

Check if a graph contains a given edge.

hasEdge x y == Algebra.Graph.hasEdge x y . toGraph


vertexCount :: Ord (ToVertex t) => t -> Int Source #

The number of vertices in a graph.

vertexCount == Set.size . vertexSet


edgeCount :: Ord (ToVertex t) => t -> Int Source #

The number of edges in a graph.

edgeCount == Set.size . edgeSet


vertexList :: Ord (ToVertex t) => t -> [ToVertex t] Source #

The sorted list of vertices of a given graph.

vertexList == Set.toAscList . vertexSet


edgeList :: Ord (ToVertex t) => t -> [(ToVertex t, ToVertex t)] Source #

The sorted list of edges of a graph.

edgeList == Set.toAscList . edgeSet


vertexSet :: Ord (ToVertex t) => t -> Set (ToVertex t) Source #

The set of vertices of a graph.

vertexSet == foldg Set.empty Set.singleton Set.union Set.union


vertexIntSet :: ToVertex t ~ Int => t -> IntSet Source #

The set of vertices of a graph. Like vertexSet but specialised for graphs with vertices of type Int.

vertexIntSet == foldg IntSet.empty IntSet.singleton IntSet.union IntSet.union


edgeSet :: Ord (ToVertex t) => t -> Set (ToVertex t, ToVertex t) Source #

The set of edges of a graph.

edgeSet == Algebra.Graph.AdjacencyMap.edgeSet . toAdjacencyMap


preSet :: Ord (ToVertex t) => ToVertex t -> t -> Set (ToVertex t) Source #

The preset of a vertex is the set of its direct predecessors.

preSet x == Algebra.Graph.AdjacencyMap.preSet x . toAdjacencyMap


preIntSet :: ToVertex t ~ Int => Int -> t -> IntSet Source #

The preset (here preIntSet) of a vertex is the set of its direct predecessors. Like preSet but specialised for graphs with vertices of type Int.

preIntSet x == Algebra.Graph.AdjacencyIntMap.preIntSet x . toAdjacencyIntMap


postSet :: Ord (ToVertex t) => ToVertex t -> t -> Set (ToVertex t) Source #

The postset of a vertex is the set of its direct successors.

postSet x == Algebra.Graph.AdjacencyMap.postSet x . toAdjacencyMap


postIntSet :: ToVertex t ~ Int => Int -> t -> IntSet Source #

The postset (here postIntSet) of a vertex is the set of its direct successors. Like postSet but specialised for graphs with vertices of type Int.

postIntSet x == Algebra.Graph.AdjacencyIntMap.postIntSet x . toAdjacencyIntMap


adjacencyList :: Ord (ToVertex t) => t -> [(ToVertex t, [ToVertex t])] Source #

The sorted adjacency list of a graph.

adjacencyList == Algebra.Graph.AdjacencyMap.adjacencyList . toAdjacencyMap


dfsForest :: Ord (ToVertex t) => t -> Forest (ToVertex t) Source #

Compute the depth-first search forest of a graph that corresponds to searching from each of the graph vertices in the Ord a order.

dfsForest == Algebra.Graph.AdjacencyMap.dfsForest . toAdjacencyMap


dfsForestFrom :: Ord (ToVertex t) => [ToVertex t] -> t -> Forest (ToVertex t) Source #

Compute the depth-first search forest of a graph, searching from each of the given vertices in order. Note that the resulting forest does not necessarily span the whole graph, as some vertices may be unreachable.

dfsForestFrom vs == Algebra.Graph.AdjacencyMap.dfsForestFrom vs . toAdjacencyMap


dfs :: Ord (ToVertex t) => [ToVertex t] -> t -> [ToVertex t] Source #

Compute the list of vertices visited by the depth-first search in a graph, when searching from each of the given vertices in order.

dfs vs == Algebra.Graph.AdjacencyMap.dfs vs . toAdjacencyMap


reachable :: Ord (ToVertex t) => ToVertex t -> t -> [ToVertex t] Source #

Compute the list of vertices that are reachable from a given source vertex in a graph. The vertices in the resulting list appear in the depth-first order.

reachable x == Algebra.Graph.AdjacencyMap.reachable x . toAdjacencyMap


topSort :: Ord (ToVertex t) => t -> Either (Cycle (ToVertex t)) [ToVertex t] Source #

Compute the topological sort of a graph or a AM.Cycle if the graph is cyclic.

topSort == Algebra.Graph.AdjacencyMap.topSort . toAdjacencyMap


isAcyclic :: Ord (ToVertex t) => t -> Bool Source #

Check if a given graph is acyclic.

isAcyclic == Algebra.Graph.AdjacencyMap.isAcyclic . toAdjacencyMap


toAdjacencyMap :: Ord (ToVertex t) => t -> AdjacencyMap (ToVertex t) Source #

Convert a value to the corresponding AdjacencyMap.

toAdjacencyMap == foldg empty vertex overlay connect


toAdjacencyMapTranspose :: Ord (ToVertex t) => t -> AdjacencyMap (ToVertex t) Source #

Convert a value to the corresponding AdjacencyMap and transpose the result.

toAdjacencyMapTranspose == foldg empty vertex overlay (flip connect)


Convert a value to the corresponding AdjacencyIntMap.

toAdjacencyIntMap == foldg empty vertex overlay connect


Convert a value to the corresponding AdjacencyIntMap and transpose the result.

toAdjacencyIntMapTranspose == foldg empty vertex overlay (flip connect)


isDfsForestOf :: Ord (ToVertex t) => Forest (ToVertex t) -> t -> Bool Source #

Check if a given forest is a valid depth-first search forest of a graph.

isDfsForestOf f == Algebra.Graph.AdjacencyMap.isDfsForestOf f . toAdjacencyMap


isTopSortOf :: Ord (ToVertex t) => [ToVertex t] -> t -> Bool Source #

Check if a given list of vertices is a valid topological sort of a graph.

isTopSortOf vs == Algebra.Graph.AdjacencyMap.isTopSortOf vs . toAdjacencyMap

Instances

# Derived functions

adjacencyMap :: ToGraph t => Ord (ToVertex t) => t -> Map (ToVertex t) (Set (ToVertex t)) Source #

The adjacency map of a graph: each vertex is associated with a set of its direct successors.

adjacencyMap == Algebra.Graph.AdjacencyMap.adjacencyMap . toAdjacencyMap


adjacencyIntMap :: (ToGraph t, ToVertex t ~ Int) => t -> IntMap IntSet Source #

The adjacency map of a graph: each vertex is associated with a set of its direct successors. Like adjacencyMap but specialised for graphs with vertices of type Int.

adjacencyIntMap == Algebra.Graph.AdjacencyIntMap.adjacencyIntMap . toAdjacencyIntMap


adjacencyMapTranspose :: (ToGraph t, Ord (ToVertex t)) => t -> Map (ToVertex t) (Set (ToVertex t)) Source #

The transposed adjacency map of a graph: each vertex is associated with a set of its direct predecessors.

adjacencyMapTranspose == Algebra.Graph.AdjacencyMap.adjacencyMap . toAdjacencyMapTranspose


adjacencyIntMapTranspose :: (ToGraph t, ToVertex t ~ Int) => t -> IntMap IntSet Source #

The transposed adjacency map of a graph: each vertex is associated with a set of its direct predecessors. Like adjacencyMapTranspose but specialised for graphs with vertices of type Int.

adjacencyIntMapTranspose == Algebra.Graph.AdjacencyIntMap.adjacencyIntMap . toAdjacencyIntMapTranspose