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.Relation
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
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.
Synopsis
- data Relation a
- domain :: Relation a -> Set a
- relation :: Relation a -> Set (a, a)
- empty :: Relation a
- vertex :: 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
- 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)]
- adjacencyList :: Eq a => Relation a -> [(a, [a])]
- vertexSet :: Relation a -> Set a
- 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
- stars :: 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
- induceJust :: Ord a => Relation ( Maybe a) -> Relation a
- compose :: Ord a => Relation a -> Relation a -> Relation a
- closure :: Ord 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
- consistent :: Ord a => Relation a -> Bool
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))
Note:
the
Num
instance does not satisfy several "customary laws" of
Num
,
which dictate that
fromInteger
0
and
fromInteger
1
should act as
additive and multiplicative identities, and
negate
as additive inverse.
Nevertheless, overloading
fromInteger
,
+
and
*
is very convenient when
working with algebraic graphs; we hope that in future Haskell's Prelude will
provide a more fine-grained class hierarchy for algebraic structures, which we
would be able to utilise without violating any laws.
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.
The total order on graphs is defined using size-lexicographic comparison:
- Compare the number of vertices. In case of a tie, continue.
- Compare the sets of vertices. In case of a tie, continue.
- Compare the number of edges. In case of a tie, continue.
- Compare the sets of edges.
Here are a few examples:
vertex
1 <vertex
2vertex
3 <edge
1 2vertex
1 <edge
1 1edge
1 1 <edge
1 2edge
1 2 <edge
1 1 +edge
2 2edge
1 2 <edge
1 3
Note that the resulting order refines the
isSubgraphOf
relation and is compatible with
overlay
and
connect
operations:
isSubgraphOf
x y ==> x <= y
empty
<= x
x <= x + y
x + y <= x * y
Instances
domain :: Relation a -> Set a Source #
The domain of the relation. Complexity: O(1) time and memory.
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. Complexity: O(1) time and memory.
Basic graph construction primitives
Construct the empty graph .
isEmpty
empty == TruehasVertex
x empty == FalsevertexCount
empty == 0edgeCount
empty == 0
vertex :: a -> Relation a Source #
Construct the graph comprising a single isolated vertex .
isEmpty
(vertex x) == FalsehasVertex
x (vertex y) == (x == y)vertexCount
(vertex x) == 1edgeCount
(vertex x) == 0
edge :: Ord a => a -> a -> Relation a Source #
Construct the graph comprising a single edge .
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
x vertices ==overlays
. mapvertex
hasVertex
x . vertices ==elem
xvertexCount
. vertices ==length
.nub
vertexSet
. vertices == Set.fromList
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 isSubgraphOf x y ==> x <= y
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
y) == (x == y) 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
vertexCount x < vertexCount y ==> x < y
vertexList :: Relation a -> [a] Source #
adjacencyList :: Eq a => Relation a -> [(a, [a])] Source #
preSet :: Ord a => a -> Relation a -> Set a Source #
The
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
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
stars :: Ord a => [(a, [a])] -> Relation a Source #
The
stars
formed by overlaying a list of
star
s. An inverse of
adjacencyList
.
Complexity:
O(L * log(n))
time, memory and size, where
L
is the total
size of the input.
stars [] ==empty
stars [(x, [])] ==vertex
x stars [(x, [y])] ==edge
x y stars [(x, ys)] ==star
x ys stars ==overlays
.map
(uncurry
star
) stars .adjacencyList
== idoverlay
(stars xs) (stars ys) == stars (xs ++ 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 y
x
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 constant time.
mergeVertices (const
False) x == id mergeVertices (== x) y ==replaceVertex
x y mergeVerticeseven
1 (0 * 2) == 1 * 1 mergeVerticesodd
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(n + m) time, assuming that the predicate takes constant time.
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
induceJust :: Ord a => Relation ( Maybe a) -> Relation a Source #
Construct the
induced subgraph
of a given graph by removing the vertices
that are
Nothing
.
Complexity:
O(n + m)
time.
induceJust (vertex
Nothing
) ==empty
induceJust (edge
(Just
x)Nothing
) ==vertex
x induceJust .gmap
Just
==id
induceJust .gmap
(\x -> if p x thenJust
x elseNothing
) ==induce
p
Relational operations
compose :: Ord a => Relation a -> Relation a -> Relation a Source #
Left-to-right
relational composition
of graphs: vertices
x
and
z
are
connected in the resulting graph if there is a vertex
y
, such that
x
is
connected to
y
in the first graph, and
y
is connected to
z
in the
second graph. There are no isolated vertices in the result. This operation is
associative, has
empty
and single-
vertex
graphs as
annihilating zeroes
,
and distributes over
overlay
.
Complexity:
O(n * m * log(m))
time and
O(n + m)
memory.
composeempty
x ==empty
compose xempty
==empty
compose (vertex
x) y ==empty
compose x (vertex
y) ==empty
compose x (compose y z) == compose (compose x y) z compose x (overlay
y z) ==overlay
(compose x y) (compose x z) compose (overlay
x y) z ==overlay
(compose x z) (compose y z) compose (edge
x y) (edge
y z) ==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]
closure :: Ord a => Relation a -> Relation a Source #
Compute the reflexive and transitive closure of a graph. Complexity: O(n * m * log(n) * log(m)) time.
closureempty
==empty
closure (vertex
x) ==edge
x x closure (edge
x x) ==edge
x x closure (edge
x y) ==edges
[(x,x), (x,y), (y,y)] closure (path
$nub
xs) ==reflexiveClosure
(clique
$nub
xs) closure ==reflexiveClosure
.transitiveClosure
closure ==transitiveClosure
.reflexiveClosure
closure . closure == closurepostSet
x (closure y) == Set.fromList
(reachable
x y)
reflexiveClosure :: Ord a => Relation a -> Relation a Source #
Compute the reflexive closure of a graph. Complexity: O(n * log(m)) time.
reflexiveClosureempty
==empty
reflexiveClosure (vertex
x) ==edge
x x reflexiveClosure (edge
x x) ==edge
x x reflexiveClosure (edge
x y) ==edges
[(x,x), (x,y), (y,y)] reflexiveClosure . reflexiveClosure == reflexiveClosure
transitiveClosure :: Ord a => Relation a -> Relation a Source #
Compute the transitive closure of a graph. Complexity: O(n * m * log(n) * log(m)) time.
transitiveClosureempty
==empty
transitiveClosure (vertex
x) ==vertex
x transitiveClosure (edge
x y) ==edge
x y transitiveClosure (path
$nub
xs) ==clique
(nub
xs) transitiveClosure . transitiveClosure == transitiveClosure
Miscellaneous
consistent :: Ord a => Relation a -> Bool Source #
Check that the internal representation of a relation is consistent, i.e. if all
pairs of elements in the
relation
refer to existing elements in the
domain
.
It should be impossible to create an inconsistent
Relation
, and we use this
function in testing.
consistentempty
== True consistent (vertex
x) == True consistent (overlay
x y) == True consistent (connect
x y) == True consistent (edge
x y) == True consistent (edges
xs) == True consistent (stars
xs) == True