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.Class

Contents

- The core type class
- Undirected graphs
- Reflexive graphs
- Transitive graphs
- Preorders
- Basic graph construction primitives
- Relations on graphs
- Standard families of graphs

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 core type class `Graph`, a few graph subclasses, and basic polymorphic graph construction primitives. Functions that cannot be implemented fully polymorphically and require the use of an intermediate data type are not included. For example, to compute the number of vertices in a `Graph` expression you will need to use a concrete data type, such as Algebra.Graph.Graph or Algebra.Graph.AdjacencyMap.

See Algebra.Graph.HigherKinded.Class for the higher-kinded version of the core graph type class.

Synopsis

- class Graph g where
  - type Vertex g
  - empty :: g
  - vertex :: Vertex g -> g
  - overlay :: g -> g -> g
  - connect :: g -> g -> g
- class Graph g => Undirected g
- class Graph g => Reflexive g
- class Graph g => Transitive g
- class (Reflexive g, Transitive g) => Preorder g
- edge :: Graph g => Vertex g -> Vertex g -> g
- vertices :: Graph g => [Vertex g] -> g
- overlays :: Graph g => [g] -> g
- connects :: Graph g => [g] -> g
- edges :: Graph g => [(Vertex g, Vertex g)] -> g
- isSubgraphOf :: (Graph g, Eq g) => g -> g -> Bool
- path :: Graph g => [Vertex g] -> g
- circuit :: Graph g => [Vertex g] -> g
- clique :: Graph g => [Vertex g] -> g
- biclique :: Graph g => [Vertex g] -> [Vertex g] -> g
- star :: Graph g => Vertex g -> [Vertex g] -> g
- tree :: Graph g => Tree (Vertex g) -> g
- forest :: Graph g => Forest (Vertex g) -> g

# The core type class

class Graph g where #

The core type class for constructing algebraic graphs, characterised by the following minimal set of axioms. In equations we use `+` and `*` as convenient shortcuts for `overlay` and `connect`, respectively.

- `overlay` is commutative and associative:

            x + y == y + x
      x + (y + z) == (x + y) + z

- `connect` is associative and has `empty` as the identity:

        x * empty == x
        empty * x == x
      x * (y * z) == (x * y) * z

- `connect` distributes over `overlay`:

      x * (y + z) == x * y + x * z
      (x + y) * z == x * z + y * z

- `connect` can be decomposed:

      x * y * z == x * y + x * z + y * z

The following useful theorems can be proved from the above set of axioms.

- `overlay` has `empty` as the identity and is idempotent:

        x + empty == x
        empty + x == x
            x + x == x

- Absorption and saturation of `connect`:

      x * y + x + y == x * y
          x * x * x == x * x

The core type class `Graph` corresponds to unlabelled directed graphs. `Undirected`, `Reflexive`, `Transitive` and `Preorder` graphs can be obtained by extending the minimal set of axioms.

When specifying the time and memory complexity of graph algorithms, n will denote the number of vertices in the graph, m will denote the number of edges in the graph, and s will denote the size of the corresponding `Graph` expression.

Associated Types

type Vertex g #

The type of graph vertices.

Methods

empty :: g #

Construct the empty graph.

vertex :: Vertex g -> g #

Construct the graph with a single vertex.

overlay :: g -> g -> g #

Overlay two graphs.

connect :: g -> g -> g #

Connect two graphs.

#### Instances

Instances details

Graph AdjacencyIntMap # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex AdjacencyIntMap |
Instance details Defined in Algebra.Graph.Class type Vertex AdjacencyIntMap = Int
Methods empty :: AdjacencyIntMap # vertex :: Vertex AdjacencyIntMap -> AdjacencyIntMap # overlay :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap # connect :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap #
Graph () # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex () |
Instance details Defined in Algebra.Graph.Class type Vertex () = ()
Methods empty :: () # vertex :: Vertex () -> () # overlay :: () -> () -> () # connect :: () -> () -> () #
Graph (Graph a) # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex (Graph a) |
Instance details Defined in Algebra.Graph.Class type Vertex (Graph a) = a
Methods empty :: Graph a # vertex :: Vertex (Graph a) -> Graph a # overlay :: Graph a -> Graph a -> Graph a # connect :: Graph a -> Graph a -> Graph a #
Ord a => Graph (AdjacencyMap a) # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex (AdjacencyMap a) |
Instance details Defined in Algebra.Graph.Class type Vertex (AdjacencyMap a) = a
Methods empty :: AdjacencyMap a # vertex :: Vertex (AdjacencyMap a) -> AdjacencyMap a # overlay :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a # connect :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a #
Ord a => Graph (Todo a) # |
Instance details Defined in Algebra.Graph.Example.Todo Associated Types
type Vertex (Todo a) |
Instance details Defined in Algebra.Graph.Example.Todo type Vertex (Todo a) = a
Methods empty :: Todo a # vertex :: Vertex (Todo a) -> Todo a # overlay :: Todo a -> Todo a -> Todo a # connect :: Todo a -> Todo a -> Todo a #
Ord a => Graph (Relation a) # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex (Relation a) |
Instance details Defined in Algebra.Graph.Class type Vertex (Relation a) = a
Methods empty :: Relation a # vertex :: Vertex (Relation a) -> Relation a # overlay :: Relation a -> Relation a -> Relation a # connect :: Relation a -> Relation a -> Relation a #
Ord a => Graph (PreorderRelation a) # |
Instance details Defined in Algebra.Graph.Relation.Preorder Associated Types
type Vertex (PreorderRelation a) |
Instance details Defined in Algebra.Graph.Relation.Preorder type Vertex (PreorderRelation a) = a
Methods empty :: PreorderRelation a # vertex :: Vertex (PreorderRelation a) -> PreorderRelation a # overlay :: PreorderRelation a -> PreorderRelation a -> PreorderRelation a # connect :: PreorderRelation a -> PreorderRelation a -> PreorderRelation a #
Ord a => Graph (ReflexiveRelation a) # |
Instance details Defined in Algebra.Graph.Relation.Reflexive Associated Types
type Vertex (ReflexiveRelation a) |
Instance details Defined in Algebra.Graph.Relation.Reflexive type Vertex (ReflexiveRelation a) = a
Methods empty :: ReflexiveRelation a # vertex :: Vertex (ReflexiveRelation a) -> ReflexiveRelation a # overlay :: ReflexiveRelation a -> ReflexiveRelation a -> ReflexiveRelation a # connect :: ReflexiveRelation a -> ReflexiveRelation a -> ReflexiveRelation a #
Ord a => Graph (Relation a) # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex (Relation a) |
Instance details Defined in Algebra.Graph.Class type Vertex (Relation a) = a
Methods empty :: Relation a # vertex :: Vertex (Relation a) -> Relation a # overlay :: Relation a -> Relation a -> Relation a # connect :: Relation a -> Relation a -> Relation a #
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 #
Graph (Graph a) # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex (Graph a) |
Instance details Defined in Algebra.Graph.Class type Vertex (Graph a) = a
Methods empty :: Graph a # vertex :: Vertex (Graph a) -> Graph a # overlay :: Graph a -> Graph a -> Graph a # connect :: Graph a -> Graph a -> Graph a #
Graph g => Graph (Maybe g) # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex (Maybe g) |
Instance details Defined in Algebra.Graph.Class type Vertex (Maybe g) = Vertex g
Methods empty :: Maybe g # vertex :: Vertex (Maybe g) -> Maybe g # overlay :: Maybe g -> Maybe g -> Maybe g # connect :: Maybe g -> Maybe g -> Maybe g #
Dioid e => Graph (Graph e a) # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex (Graph e a) |
Instance details Defined in Algebra.Graph.Class type Vertex (Graph e a) = a
Methods empty :: Graph e a # vertex :: Vertex (Graph e a) -> Graph e a # overlay :: Graph e a -> Graph e a -> Graph e a # connect :: Graph e a -> Graph e a -> Graph e a #
(Dioid e, Eq e, Ord a) => Graph (AdjacencyMap e a) # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex (AdjacencyMap e a) |
Instance details Defined in Algebra.Graph.Class type Vertex (AdjacencyMap e a) = a
Methods empty :: AdjacencyMap e a # vertex :: Vertex (AdjacencyMap e a) -> AdjacencyMap e a # overlay :: AdjacencyMap e a -> AdjacencyMap e a -> AdjacencyMap e a # connect :: AdjacencyMap e a -> AdjacencyMap e a -> AdjacencyMap e a #
(Graph g, Graph h) => Graph (g, h) # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex (g, h) |
Instance details Defined in Algebra.Graph.Class type Vertex (g, h) = (Vertex g, Vertex h)
Methods empty :: (g, h) # vertex :: Vertex (g, h) -> (g, h) # overlay :: (g, h) -> (g, h) -> (g, h) # connect :: (g, h) -> (g, h) -> (g, h) #
Graph g => Graph (a -> g) # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex (a -> g) |
Instance details Defined in Algebra.Graph.Class type Vertex (a -> g) = Vertex g
Methods empty :: a -> g # vertex :: Vertex (a -> g) -> a -> g # overlay :: (a -> g) -> (a -> g) -> a -> g # connect :: (a -> g) -> (a -> g) -> a -> g #
(Graph g, Graph h, Graph i) => Graph (g, h, i) # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex (g, h, i) |
Instance details Defined in Algebra.Graph.Class type Vertex (g, h, i) = (Vertex g, Vertex h, Vertex i)
Methods empty :: (g, h, i) # vertex :: Vertex (g, h, i) -> (g, h, i) # overlay :: (g, h, i) -> (g, h, i) -> (g, h, i) # connect :: (g, h, i) -> (g, h, i) -> (g, h, i) #

# Undirected graphs

class Graph g => Undirected g #

The class of undirected graphs that satisfy the following additional axiom.

- `connect` is commutative:

      x * y == y * x

#### Instances

Instances details

Undirected () # |
Instance details Defined in Algebra.Graph.Class
Ord a => Undirected (Relation a) # |
Instance details Defined in Algebra.Graph.Class
Undirected (Graph a) # |
Instance details Defined in Algebra.Graph.Class
Undirected g => Undirected (Maybe g) # |
Instance details Defined in Algebra.Graph.Class
(Undirected g, Undirected h) => Undirected (g, h) # |
Instance details Defined in Algebra.Graph.Class
Undirected g => Undirected (a -> g) # |
Instance details Defined in Algebra.Graph.Class
(Undirected g, Undirected h, Undirected i) => Undirected (g, h, i) # |
Instance details Defined in Algebra.Graph.Class

# Reflexive graphs

class Graph g => Reflexive g #

The class of reflexive graphs that satisfy the following additional axiom.

- Each vertex has a self-loop:

      vertex x == vertex x * vertex x

Note that by applying the axiom in the reverse direction, one can always remove all self-loops resulting in an irreflexive graph. This type class can therefore be also used in the context of irreflexive graphs.

#### Instances

Instances details

Reflexive () # |
Instance details Defined in Algebra.Graph.Class
Ord a => Reflexive (PreorderRelation a) # |
Instance details Defined in Algebra.Graph.Relation.Preorder
Ord a => Reflexive (ReflexiveRelation a) # |
Instance details Defined in Algebra.Graph.Relation.Reflexive
Reflexive g => Reflexive (Maybe g) # |
Instance details Defined in Algebra.Graph.Class
(Reflexive g, Reflexive h) => Reflexive (g, h) # |
Instance details Defined in Algebra.Graph.Class
Reflexive g => Reflexive (a -> g) # |
Instance details Defined in Algebra.Graph.Class
(Reflexive g, Reflexive h, Reflexive i) => Reflexive (g, h, i) # |
Instance details Defined in Algebra.Graph.Class

# Transitive graphs

class Graph g => Transitive g #

The class of transitive graphs that satisfy the following additional axiom.

- The closure axiom: graphs with equal transitive closures are equal.

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

By repeated application of the axiom one can turn any graph into its transitive closure or transitive reduction.

#### Instances

Instances details

Transitive () # |
Instance details Defined in Algebra.Graph.Class
Ord a => Transitive (PreorderRelation a) # |
Instance details Defined in Algebra.Graph.Relation.Preorder
Ord a => Transitive (TransitiveRelation a) # |
Instance details Defined in Algebra.Graph.Relation.Transitive
Transitive g => Transitive (Maybe g) # |
Instance details Defined in Algebra.Graph.Class
(Transitive g, Transitive h) => Transitive (g, h) # |
Instance details Defined in Algebra.Graph.Class
Transitive g => Transitive (a -> g) # |
Instance details Defined in Algebra.Graph.Class
(Transitive g, Transitive h, Transitive i) => Transitive (g, h, i) # |
Instance details Defined in Algebra.Graph.Class

# Preorders

class (Reflexive g, Transitive g) => Preorder g #

The class of preorder graphs that are both reflexive and transitive.

#### Instances

Instances details

Preorder () # |
Instance details Defined in Algebra.Graph.Class
Ord a => Preorder (PreorderRelation a) # |
Instance details Defined in Algebra.Graph.Relation.Preorder
Preorder g => Preorder (Maybe g) # |
Instance details Defined in Algebra.Graph.Class
(Preorder g, Preorder h) => Preorder (g, h) # |
Instance details Defined in Algebra.Graph.Class
Preorder g => Preorder (a -> g) # |
Instance details Defined in Algebra.Graph.Class
(Preorder g, Preorder h, Preorder i) => Preorder (g, h, i) # |
Instance details Defined in Algebra.Graph.Class

# Basic graph construction primitives

edge :: Graph g => Vertex g -> Vertex g -> g #

Construct the graph comprising a single edge.

    edge x y == connect (vertex x) (vertex y)

vertices :: Graph g => [Vertex g] -> g #

Construct the graph comprising a given list of isolated vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

    vertices []  == empty
    vertices [x] == vertex x
    vertices     == overlays . map vertex

overlays :: Graph g => [g] -> g #

Overlay a given list of graphs. Complexity: O(L) time and memory, and O(S) size, where L is the length of the given list, and S is the sum of sizes of the graphs in the list.

    overlays []    == empty
    overlays [x]   == x
    overlays [x,y] == overlay x y
    overlays       == foldr overlay empty

connects :: Graph g => [g] -> g #

Connect a given list of graphs. Complexity: O(L) time and memory, and O(S) size, where L is the length of the given list, and S is the sum of sizes of the graphs in the list.

    connects []    == empty
    connects [x]   == x
    connects [x,y] == connect x y
    connects       == foldr connect empty

edges :: Graph g => [(Vertex g, Vertex g)] -> g #

Construct the graph from a list of edges. Complexity: O(L) time, memory and size, where L is the length of the given list.

    edges []      == empty
    edges [(x,y)] == edge x y

# Relations on graphs

isSubgraphOf :: (Graph g, Eq g) => g -> g -> Bool #

The `isSubgraphOf` function takes two graphs and returns `True` if the first graph is a subgraph of the second. Here is the current implementation:

    isSubgraphOf x y = overlay x y == y

The complexity therefore depends on the complexity of equality testing of the specific graph instance.

    isSubgraphOf empty         x             == True
    isSubgraphOf (vertex x)    empty         == False
    isSubgraphOf x             (overlay x y) == True
    isSubgraphOf (overlay x y) (connect x y) == True
    isSubgraphOf (path xs)     (circuit xs)  == True

# Standard families of graphs

path :: Graph g => [Vertex g] -> g #

The path on a list of vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

    path []    == empty
    path [x]   == vertex x
    path [x,y] == edge x y

circuit :: Graph g => [Vertex g] -> g #

The circuit on a list of vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

    circuit []    == empty
    circuit [x]   == edge x x
    circuit [x,y] == edges [(x,y), (y,x)]

clique :: Graph g => [Vertex g] -> g #

The clique on a list of vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

    clique []         == empty
    clique [x]        == vertex x
    clique [x,y]      == edge x y
    clique [x,y,z]    == edges [(x,y), (x,z), (y,z)]
    clique (xs ++ ys) == connect (clique xs) (clique ys)

biclique :: Graph g => [Vertex g] -> [Vertex g] -> g #

The biclique on two lists of vertices. Complexity: O(L1 + L2) time, memory and size, where L1 and L2 are the lengths of the given lists.

    biclique []      []      == empty
    biclique [x]     []      == vertex x
    biclique []      [y]     == vertex y
    biclique [x1,x2] [y1,y2] == edges [(x1,y1), (x1,y2), (x2,y1), (x2,y2)]
    biclique xs      ys      == connect (vertices xs) (vertices ys)

star :: Graph g => Vertex g -> [Vertex g] -> g #

The star formed by a centre vertex connected to a list of leaves. Complexity: O(L) time, memory and size, where L is the length of the given list.

    star x []    == vertex x
    star x [y]   == edge x y
    star x [y,z] == edges [(x,y), (x,z)]
    star x ys    == connect (vertex x) (vertices ys)

tree :: Graph g => Tree (Vertex g) -> g #

The tree graph constructed from a given `Tree` data structure. Complexity: O(T) time, memory and size, where T is the size of the given tree (i.e. the number of vertices in the tree).

    tree (Node x [])                                         == vertex x
    tree (Node x [Node y [Node z []]])                       == path [x,y,z]
    tree (Node x [Node y [], Node z []])                     == star x [y,z]
    tree (Node 1 [Node 2 [], Node 3 [Node 4 [], Node 5 []]]) == edges [(1,2), (1,3), (3,4), (3,5)]

forest :: Graph g => Forest (Vertex g) -> g #

The forest graph constructed from a given `Forest` data structure. Complexity: O(F) time, memory and size, where F is the size of the given forest (i.e. the number of vertices in the forest).

    forest []                                                  == empty
    forest [x]                                                 == tree x
    forest [Node 1 [Node 2 [], Node 3 []], Node 4 [Node 5 []]] == edges [(1,2), (1,3), (4,5)]
    forest                                                     == overlays . map tree