Copyright | (c) Andrey Mokhov 2016-2018 |
---|---|
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 Relation
data type, as well as associated
operations and algorithms. Relation
is an instance of the Graph
type
class, which can be used for polymorphic graph construction and manipulation.
- data Relation a
- domain :: Relation a -> Set a
- relation :: Relation a -> Set (a, a)
- empty :: Ord a => Relation a
- vertex :: Ord a => a -> Relation a
- edge :: Ord a => a -> a -> Relation a
- overlay :: Ord a => Relation a -> Relation a -> Relation a
- connect :: Ord a => Relation a -> Relation a -> Relation a
- vertices :: Ord a => [a] -> Relation a
- edges :: Ord a => [(a, a)] -> Relation a
- overlays :: Ord a => [Relation a] -> Relation a
- connects :: Ord a => [Relation a] -> Relation a
- fromAdjacencyList :: Ord a => [(a, [a])] -> Relation a
- isSubgraphOf :: Ord a => Relation a -> Relation a -> Bool
- isEmpty :: Relation a -> Bool
- hasVertex :: Ord a => a -> Relation a -> Bool
- hasEdge :: Ord a => a -> a -> Relation a -> Bool
- vertexCount :: Relation a -> Int
- edgeCount :: Relation a -> Int
- vertexList :: Relation a -> [a]
- edgeList :: Relation a -> [(a, a)]
- vertexSet :: Relation a -> Set a
- vertexIntSet :: Relation Int -> IntSet
- edgeSet :: Relation a -> Set (a, a)
- preSet :: Ord a => a -> Relation a -> Set a
- postSet :: Ord a => a -> Relation a -> Set a
- path :: Ord a => [a] -> Relation a
- circuit :: Ord a => [a] -> Relation a
- clique :: Ord a => [a] -> Relation a
- biclique :: Ord a => [a] -> [a] -> Relation a
- star :: Ord a => a -> [a] -> Relation a
- starTranspose :: Ord a => a -> [a] -> Relation a
- tree :: Ord a => Tree a -> Relation a
- forest :: Ord a => Forest a -> Relation a
- removeVertex :: Ord a => a -> Relation a -> Relation a
- removeEdge :: Ord a => a -> a -> Relation a -> Relation a
- replaceVertex :: Ord a => a -> a -> Relation a -> Relation a
- mergeVertices :: Ord a => (a -> Bool) -> a -> Relation a -> Relation a
- transpose :: Ord a => Relation a -> Relation a
- gmap :: Ord b => (a -> b) -> Relation a -> Relation b
- induce :: (a -> Bool) -> Relation a -> Relation a
- compose :: Ord a => Relation a -> Relation a -> Relation a
- reflexiveClosure :: Ord a => Relation a -> Relation a
- symmetricClosure :: Ord a => Relation a -> Relation a
- transitiveClosure :: Ord a => Relation a -> Relation a
- preorderClosure :: Ord a => Relation a -> Relation a
Data structure
The Relation
data type represents a graph as a binary relation. We
define a Num
instance as a convenient notation for working with graphs:
0 == vertex 0 1 + 2 == overlay (vertex 1) (vertex 2) 1 * 2 == connect (vertex 1) (vertex 2) 1 + 2 * 3 == overlay (vertex 1) (connect (vertex 2) (vertex 3)) 1 * (2 + 3) == connect (vertex 1) (overlay (vertex 2) (vertex 3))
The Show
instance is defined using basic graph construction primitives:
show (empty :: Relation Int) == "empty" show (1 :: Relation Int) == "vertex 1" show (1 + 2 :: Relation Int) == "vertices [1,2]" show (1 * 2 :: Relation Int) == "edge 1 2" show (1 * 2 * 3 :: Relation Int) == "edges [(1,2),(1,3),(2,3)]" show (1 * 2 + 3 :: Relation Int) == "overlay (vertex 3) (edge 1 2)"
The Eq
instance satisfies all axioms of algebraic graphs:
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
When specifying the time and memory complexity of graph algorithms, n and m will denote the number of vertices and edges in the graph, respectively.
relation :: Relation a -> Set (a, a) Source #
The set of pairs of elements that are related. It is guaranteed that each element belongs to the domain.
Basic graph construction primitives
empty :: Ord a => Relation a Source #
Construct the empty graph. Complexity: O(1) time and memory.
isEmpty
empty == TruehasVertex
x empty == FalsevertexCount
empty == 0edgeCount
empty == 0
vertex :: Ord a => a -> Relation a Source #
Construct the graph comprising a single isolated vertex. Complexity: O(1) time and memory.
isEmpty
(vertex x) == FalsehasVertex
x (vertex x) == TruevertexCount
(vertex x) == 1edgeCount
(vertex x) == 0
edge :: Ord a => a -> a -> Relation a Source #
Construct the graph comprising a single edge. Complexity: O(1) time, memory and size.
edge x y ==connect
(vertex
x) (vertex
y)hasEdge
x y (edge x y) == TrueedgeCount
(edge x y) == 1vertexCount
(edge 1 1) == 1vertexCount
(edge 1 2) == 2
overlay :: Ord a => Relation a -> Relation a -> Relation a Source #
Overlay two graphs. This is a commutative, associative and idempotent
operation with the identity empty
.
Complexity: O((n + m) * log(n)) time and O(n + m) memory.
isEmpty
(overlay x y) ==isEmpty
x &&isEmpty
yhasVertex
z (overlay x y) ==hasVertex
z x ||hasVertex
z yvertexCount
(overlay x y) >=vertexCount
xvertexCount
(overlay x y) <=vertexCount
x +vertexCount
yedgeCount
(overlay x y) >=edgeCount
xedgeCount
(overlay x y) <=edgeCount
x +edgeCount
yvertexCount
(overlay 1 2) == 2edgeCount
(overlay 1 2) == 0
connect :: Ord a => Relation a -> Relation a -> Relation a Source #
Connect two graphs. This is an associative operation with the identity
empty
, which distributes over overlay
and obeys the decomposition axiom.
Complexity: O((n + m) * log(n)) time and O(n + m) memory. Note that the
number of edges in the resulting graph is quadratic with respect to the number
of vertices of the arguments: m = O(m1 + m2 + n1 * n2).
isEmpty
(connect x y) ==isEmpty
x &&isEmpty
yhasVertex
z (connect x y) ==hasVertex
z x ||hasVertex
z yvertexCount
(connect x y) >=vertexCount
xvertexCount
(connect x y) <=vertexCount
x +vertexCount
yedgeCount
(connect x y) >=edgeCount
xedgeCount
(connect x y) >=edgeCount
yedgeCount
(connect x y) >=vertexCount
x *vertexCount
yedgeCount
(connect x y) <=vertexCount
x *vertexCount
y +edgeCount
x +edgeCount
yvertexCount
(connect 1 2) == 2edgeCount
(connect 1 2) == 1
vertices :: Ord a => [a] -> Relation a Source #
Construct the graph comprising a given list of isolated vertices. Complexity: O(L * log(L)) time and O(L) memory, where L is the length of the given list.
vertices [] ==empty
vertices [x] ==vertex
xhasVertex
x . vertices ==elem
xvertexCount
. vertices ==length
.nub
vertexSet
. vertices == Set.fromList
fromAdjacencyList :: Ord a => [(a, [a])] -> Relation a Source #
Relations on graphs
isSubgraphOf :: Ord a => Relation a -> Relation a -> Bool Source #
The isSubgraphOf
function takes two graphs and returns True
if the
first graph is a subgraph of the second.
Complexity: O((n + m) * log(n)) time.
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
Graph properties
isEmpty :: Relation a -> Bool Source #
Check if a relation is empty. Complexity: O(1) time.
isEmptyempty
== True isEmpty (overlay
empty
empty
) == True isEmpty (vertex
x) == False isEmpty (removeVertex
x $vertex
x) == True isEmpty (removeEdge
x y $edge
x y) == False
hasVertex :: Ord a => a -> Relation a -> Bool Source #
Check if a graph contains a given vertex. Complexity: O(log(n)) time.
hasVertex xempty
== False hasVertex x (vertex
x) == True hasVertex 1 (vertex
2) == False hasVertex x .removeVertex
x == const False
vertexCount :: Relation a -> Int Source #
The number of vertices in a graph. Complexity: O(1) time.
vertexCountempty
== 0 vertexCount (vertex
x) == 1 vertexCount ==length
.vertexList
vertexList :: Relation a -> [a] Source #
preSet :: Ord a => a -> Relation a -> Set a Source #
The preset (here preSet
) of an element x
is the set of elements that are related to
it on the left, i.e. preSet x == { a | aRx }
. In the context of directed
graphs, this corresponds to the set of direct predecessors of vertex x
.
Complexity: O(n + m) time and O(n) memory.
preSet xempty
== Set.empty
preSet x (vertex
x) == Set.empty
preSet 1 (edge
1 2) == Set.empty
preSet y (edge
x y) == Set.fromList
[x]
postSet :: Ord a => a -> Relation a -> Set a Source #
The postset (here postSet
) of an element x
is the set of elements that are related to
it on the right, i.e. postSet x == { a | xRa }
. In the context of directed
graphs, this corresponds to the set of direct successors of vertex x
.
Complexity: O(n + m) time and O(n) memory.
postSet xempty
== Set.empty
postSet x (vertex
x) == Set.empty
postSet x (edge
x y) == Set.fromList
[y] postSet 2 (edge
1 2) == Set.empty
Standard families of graphs
starTranspose :: Ord a => a -> [a] -> Relation a Source #
The star transpose formed by a list of leaves connected to a centre vertex. Complexity: O(L) time, memory and size, where L is the length of the given list.
starTranspose x [] ==vertex
x starTranspose x [y] ==edge
y x starTranspose x [y,z] ==edges
[(y,x), (z,x)] starTranspose x ys ==connect
(vertices
ys) (vertex
x) starTranspose x ys ==transpose
(star
x ys)
tree :: Ord a => Tree a -> Relation a Source #
The tree graph constructed from a given Tree
data structure.
Complexity: O((n + m) * log(n)) time and O(n + m) memory.
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)]
Graph transformation
removeEdge :: Ord a => a -> a -> Relation a -> Relation a Source #
Remove an edge from a given graph. Complexity: O(log(m)) time.
removeEdge x y (edge
x y) ==vertices
[x, y] removeEdge x y . removeEdge x y == removeEdge x y removeEdge x y .removeVertex
x ==removeVertex
x removeEdge 1 1 (1 * 1 * 2 * 2) == 1 * 2 * 2 removeEdge 1 2 (1 * 1 * 2 * 2) == 1 * 1 + 2 * 2
replaceVertex :: Ord a => a -> a -> Relation a -> Relation a Source #
The function
replaces vertex replaceVertex
x yx
with vertex y
in a
given AdjacencyMap
. If y
already exists, x
and y
will be merged.
Complexity: O((n + m) * log(n)) time.
replaceVertex x x == id replaceVertex x y (vertex
x) ==vertex
y replaceVertex x y ==mergeVertices
(== x) y
mergeVertices :: Ord a => (a -> Bool) -> a -> Relation a -> Relation a Source #
Merge vertices satisfying a given predicate into a given vertex. Complexity: O((n + m) * log(n)) time, assuming that the predicate takes O(1) to be evaluated.
mergeVertices (const False) x == id
mergeVertices (== x) y == replaceVertex
x y
mergeVertices even 1 (0 * 2) == 1 * 1
mergeVertices odd 1 (3 + 4 * 5) == 4 * 1
gmap :: Ord b => (a -> b) -> Relation a -> Relation b Source #
Transform a graph by applying a function to each of its vertices. This is
similar to Functor
's fmap
but can be used with non-fully-parametric
Relation
.
Complexity: O((n + m) * log(n)) time.
gmap fempty
==empty
gmap f (vertex
x) ==vertex
(f x) gmap f (edge
x y) ==edge
(f x) (f y) gmap id == id gmap f . gmap g == gmap (f . g)
induce :: (a -> Bool) -> Relation a -> Relation a Source #
Construct the induced subgraph of a given graph by removing the vertices that do not satisfy a given predicate. Complexity: O(m) time, assuming that the predicate takes O(1) to be evaluated.
induce (const True ) x == x induce (const False) x ==empty
induce (/= x) ==removeVertex
x induce p . induce q == induce (\x -> p x && q x)isSubgraphOf
(induce p x) x == True
Operations on binary relations
compose :: Ord a => Relation a -> Relation a -> Relation a Source #
Compose two relations: R =
. Two elements compose
Q Px
and y
are
related in the resulting relation, i.e. xRy
, if there exists an element z
,
such that xPz
and zQy
. This is an associative operation which has empty
as the annihilating zero.
Complexity: O(n * m * log(m)) time and O(n + m) memory.
composeempty
x ==empty
compose xempty
==empty
compose x (compose y z) == compose (compose x y) z compose (edge
y z) (edge
x y) ==edge
x z compose (path
[1..5]) (path
[1..5]) ==edges
[(1,3),(2,4),(3,5)] compose (circuit
[1..5]) (circuit
[1..5]) ==circuit
[1,3,5,2,4]