Copyright | (c) Andrey Mokhov 2016-2017 |
---|---|
License | MIT (see the file LICENSE) |
Maintainer | andrey.mokhov@gmail.com |
Stability | experimental |
Safe Haskell | None |
Language | Haskell2010 |
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.Fold. Other useful Graph
instances are defined in
Algebra.Graph, Algebra.Graph.AdjacencyMap and Algebra.Graph.Relation.
See Algebra.Graph.HigherKinded.Class for the higher-kinded version of the core graph type class.
- class Graph g where
- type Vertex 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
- graph :: Graph g => [Vertex 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
- class ToGraph t where
- type ToVertex t
The core type class
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 hasempty
as the identity:x * empty == x empty * x == x x * (y * z) == (x * y) * z
connect
distributes overoverlay
: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
hasempty
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.
Construct the empty graph.
vertex :: Vertex g -> g Source #
Construct the graph with a single vertex.
overlay :: g -> g -> g Source #
Overlay two graphs.
connect :: g -> g -> g Source #
Connect two graphs.
Graph () Source # | |
Graph IntAdjacencyMap Source # | |
Graph g => Graph (Maybe g) Source # | |
Ord a => Graph (Relation a) Source # | |
Ord a => Graph (PreorderRelation a) Source # | |
Ord a => Graph (TransitiveRelation a) Source # | |
Ord a => Graph (SymmetricRelation a) Source # | |
Ord a => Graph (ReflexiveRelation a) Source # | |
Ord a => Graph (AdjacencyMap a) Source # | |
Graph (Fold a) Source # | |
Graph (Graph a) Source # | |
Graph g => Graph (a -> g) Source # | |
(Graph g, Graph h) => Graph (g, h) Source # | |
(Graph g, Graph h, Graph i) => Graph (g, h, i) Source # | |
Undirected graphs
class Graph g => Undirected g Source #
The class of undirected graphs that satisfy the following additional axiom.
connect
is commutative:x * y == y * x
Undirected () Source # | |
Undirected g => Undirected (Maybe g) Source # | |
Ord a => Undirected (SymmetricRelation a) Source # | |
Undirected g => Undirected (a -> g) Source # | |
(Undirected g, Undirected h) => Undirected (g, h) Source # | |
(Undirected g, Undirected h, Undirected i) => Undirected (g, h, i) Source # | |
Reflexive graphs
class Graph g => Reflexive g Source #
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.
Reflexive () Source # | |
Reflexive g => Reflexive (Maybe g) Source # | |
Ord a => Reflexive (PreorderRelation a) Source # | |
Ord a => Reflexive (ReflexiveRelation a) Source # | |
Reflexive g => Reflexive (a -> g) Source # | |
(Reflexive g, Reflexive h) => Reflexive (g, h) Source # | |
(Reflexive g, Reflexive h, Reflexive i) => Reflexive (g, h, i) Source # | |
Transitive graphs
class Graph g => Transitive g Source #
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.
Transitive () Source # | |
Transitive g => Transitive (Maybe g) Source # | |
Ord a => Transitive (PreorderRelation a) Source # | |
Ord a => Transitive (TransitiveRelation a) Source # | |
Transitive g => Transitive (a -> g) Source # | |
(Transitive g, Transitive h) => Transitive (g, h) Source # | |
(Transitive g, Transitive h, Transitive i) => Transitive (g, h, i) Source # | |
Preorders
class (Reflexive g, Transitive g) => Preorder g Source #
The class of preorder graphs that are both reflexive and transitive.
Basic graph construction primitives
graph :: Graph g => [Vertex g] -> [(Vertex g, Vertex g)] -> g Source #
Construct the graph from given lists of vertices V and edges E. The resulting graph contains the vertices V as well as all the vertices referred to by the edges E. Complexity: O(|V| + |E|) time, memory and size.
graph [] [] ==empty
graph [x] [] ==vertex
x graph [] [(x,y)] ==edge
x y graph vs es ==overlay
(vertices
vs) (edges
es)
Relations on graphs
isSubgraphOf :: (Graph g, Eq g) => g -> g -> Bool Source #
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.
isSubgraphOfempty
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
biclique :: Graph g => [Vertex g] -> [Vertex g] -> g Source #
The biclique on a list 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)
tree :: Graph g => Tree (Vertex g) -> g Source #
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 Source #
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
. maptree
Conversion between graph data types
class ToGraph t where Source #
The ToGraph
type class captures data types that can be converted to
polymorphic graph expressions. The conversion method toGraph
semantically
acts as the identity on graph data structures, but allows to convert graphs
between different data representations.
toGraph (g ::Graph
a ) ::Graph
a == gshow
(toGraph (1 * 2 ::Graph
Int) ::Relation
Int) == "edge 1 2"