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 AdjacencyMap
data type, as well as associated
operations and algorithms. AdjacencyMap
is an instance of the Graph
type
class, which can be used for polymorphic graph construction and manipulation.
Algebra.Graph.IntAdjacencyMap defines adjacency maps specialised to graphs
with Int
vertices.
- data AdjacencyMap a
- adjacencyMap :: AdjacencyMap a -> Map a (Set a)
- empty :: Ord a => AdjacencyMap a
- vertex :: Ord a => a -> AdjacencyMap a
- edge :: Ord a => a -> a -> AdjacencyMap a
- overlay :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a
- connect :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a
- vertices :: Ord a => [a] -> AdjacencyMap a
- edges :: Ord a => [(a, a)] -> AdjacencyMap a
- overlays :: Ord a => [AdjacencyMap a] -> AdjacencyMap a
- connects :: Ord a => [AdjacencyMap a] -> AdjacencyMap a
- fromAdjacencyList :: Ord a => [(a, [a])] -> AdjacencyMap a
- isSubgraphOf :: Ord a => AdjacencyMap a -> AdjacencyMap a -> Bool
- isEmpty :: AdjacencyMap a -> Bool
- hasVertex :: Ord a => a -> AdjacencyMap a -> Bool
- hasEdge :: Ord a => a -> a -> AdjacencyMap a -> Bool
- vertexCount :: AdjacencyMap a -> Int
- edgeCount :: AdjacencyMap a -> Int
- vertexList :: AdjacencyMap a -> [a]
- edgeList :: AdjacencyMap a -> [(a, a)]
- adjacencyList :: AdjacencyMap a -> [(a, [a])]
- vertexSet :: AdjacencyMap a -> Set a
- edgeSet :: Ord a => AdjacencyMap a -> Set (a, a)
- postSet :: Ord a => a -> AdjacencyMap a -> Set a
- path :: Ord a => [a] -> AdjacencyMap a
- circuit :: Ord a => [a] -> AdjacencyMap a
- clique :: Ord a => [a] -> AdjacencyMap a
- biclique :: Ord a => [a] -> [a] -> AdjacencyMap a
- star :: Ord a => a -> [a] -> AdjacencyMap a
- starTranspose :: Ord a => a -> [a] -> AdjacencyMap a
- tree :: Ord a => Tree a -> AdjacencyMap a
- forest :: Ord a => Forest a -> AdjacencyMap a
- removeVertex :: Ord a => a -> AdjacencyMap a -> AdjacencyMap a
- removeEdge :: Ord a => a -> a -> AdjacencyMap a -> AdjacencyMap a
- replaceVertex :: Ord a => a -> a -> AdjacencyMap a -> AdjacencyMap a
- mergeVertices :: Ord a => (a -> Bool) -> a -> AdjacencyMap a -> AdjacencyMap a
- transpose :: Ord a => AdjacencyMap a -> AdjacencyMap a
- gmap :: (Ord a, Ord b) => (a -> b) -> AdjacencyMap a -> AdjacencyMap b
- induce :: Ord a => (a -> Bool) -> AdjacencyMap a -> AdjacencyMap a
- dfsForest :: AdjacencyMap a -> Forest a
- dfsForestFrom :: [a] -> AdjacencyMap a -> Forest a
- dfs :: [a] -> AdjacencyMap a -> [a]
- topSort :: Ord a => AdjacencyMap a -> Maybe [a]
- isTopSort :: Ord a => [a] -> AdjacencyMap a -> Bool
- scc :: Ord a => AdjacencyMap a -> AdjacencyMap (Set a)
Data structure
data AdjacencyMap a Source #
The AdjacencyMap
data type represents a graph by a map of vertices to
their adjacency sets. 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 :: AdjacencyMap Int) == "empty" show (1 :: AdjacencyMap Int) == "vertex 1" show (1 + 2 :: AdjacencyMap Int) == "vertices [1,2]" show (1 * 2 :: AdjacencyMap Int) == "edge 1 2" show (1 * 2 * 3 :: AdjacencyMap Int) == "edges [(1,2),(1,3),(2,3)]" show (1 * 2 + 3 :: AdjacencyMap 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.
Eq a => Eq (AdjacencyMap a) Source # | |
(Ord a, Num a) => Num (AdjacencyMap a) Source # | |
(Ord a, Show a) => Show (AdjacencyMap a) Source # | |
ToGraph (AdjacencyMap a) Source # | |
Ord a => Graph (AdjacencyMap a) Source # | |
type ToVertex (AdjacencyMap a) Source # | |
type Vertex (AdjacencyMap a) Source # | |
adjacencyMap :: AdjacencyMap a -> Map a (Set a) Source #
The adjacency map of the graph: each vertex is associated with a set of its direct successors.
Basic graph construction primitives
empty :: Ord a => AdjacencyMap 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 -> AdjacencyMap 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 -> AdjacencyMap a Source #
Construct the graph comprising a single edge. Complexity: O(1) time, memory.
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 => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap 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 => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap 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] -> AdjacencyMap 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
edges :: Ord a => [(a, a)] -> AdjacencyMap a Source #
overlays :: Ord a => [AdjacencyMap a] -> AdjacencyMap a Source #
connects :: Ord a => [AdjacencyMap a] -> AdjacencyMap a Source #
fromAdjacencyList :: Ord a => [(a, [a])] -> AdjacencyMap a Source #
Construct a graph from an adjacency list. Complexity: O((n + m) * log(n)) time and O(n + m) memory.
fromAdjacencyList [] ==empty
fromAdjacencyList [(x, [])] ==vertex
x fromAdjacencyList [(x, [y])] ==edge
x y fromAdjacencyList .adjacencyList
== idoverlay
(fromAdjacencyList xs) (fromAdjacencyList ys) == fromAdjacencyList (xs ++ ys)
Relations on graphs
isSubgraphOf :: Ord a => AdjacencyMap a -> AdjacencyMap 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 :: AdjacencyMap a -> Bool Source #
Check if a graph 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 -> AdjacencyMap 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 :: AdjacencyMap a -> Int Source #
The number of vertices in a graph. Complexity: O(1) time.
vertexCountempty
== 0 vertexCount (vertex
x) == 1 vertexCount ==length
.vertexList
edgeCount :: AdjacencyMap a -> Int Source #
vertexList :: AdjacencyMap a -> [a] Source #
edgeList :: AdjacencyMap a -> [(a, a)] Source #
adjacencyList :: AdjacencyMap a -> [(a, [a])] Source #
The sorted adjacency list of a graph. Complexity: O(n + m) time and O(m) memory.
adjacencyListempty
== [] adjacencyList (vertex
x) == [(x, [])] adjacencyList (edge
1 2) == [(1, [2]), (2, [])] adjacencyList (star
2 [3,1]) == [(1, []), (2, [1,3]), (3, [])]fromAdjacencyList
. adjacencyList == id
vertexSet :: AdjacencyMap a -> Set a Source #
Standard families of graphs
path :: Ord a => [a] -> AdjacencyMap a Source #
circuit :: Ord a => [a] -> AdjacencyMap a Source #
clique :: Ord a => [a] -> AdjacencyMap a Source #
biclique :: Ord a => [a] -> [a] -> AdjacencyMap a Source #
star :: Ord a => a -> [a] -> AdjacencyMap a Source #
starTranspose :: Ord a => a -> [a] -> AdjacencyMap 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 -> AdjacencyMap 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
removeVertex :: Ord a => a -> AdjacencyMap a -> AdjacencyMap a Source #
removeEdge :: Ord a => a -> a -> AdjacencyMap a -> AdjacencyMap a Source #
Remove an edge from a given graph. Complexity: O(log(n)) 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 -> AdjacencyMap a -> AdjacencyMap 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 -> AdjacencyMap a -> AdjacencyMap 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
transpose :: Ord a => AdjacencyMap a -> AdjacencyMap a Source #
gmap :: (Ord a, Ord b) => (a -> b) -> AdjacencyMap a -> AdjacencyMap 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
AdjacencyMap
.
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 :: Ord a => (a -> Bool) -> AdjacencyMap a -> AdjacencyMap 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
Algorithms
dfsForest :: AdjacencyMap a -> Forest a Source #
Compute the depth-first search forest of a graph.
forest
(dfsForest $edge
1 1) ==vertex
1forest
(dfsForest $edge
1 2) ==edge
1 2forest
(dfsForest $edge
2 1) ==vertices
[1, 2]isSubgraphOf
(forest
$ dfsForest x) x == True dfsForest .forest
. dfsForest == dfsForest dfsForest (vertices
vs) == map (\v -> Node v []) (nub
$sort
vs)dfsForestFrom
(vertexList
x) x == dfsForest x dfsForest $ 3 * (1 + 4) * (1 + 5) == [ Node { rootLabel = 1 , subForest = [ Node { rootLabel = 5 , subForest = [] }]} , Node { rootLabel = 3 , subForest = [ Node { rootLabel = 4 , subForest = [] }]}]
dfsForestFrom :: [a] -> AdjacencyMap a -> Forest a 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.
forest
(dfsForestFrom [1] $edge
1 1) ==vertex
1forest
(dfsForestFrom [1] $edge
1 2) ==edge
1 2forest
(dfsForestFrom [2] $edge
1 2) ==vertex
2forest
(dfsForestFrom [3] $edge
1 2) ==empty
forest
(dfsForestFrom [2, 1] $edge
1 2) ==vertices
[1, 2]isSubgraphOf
(forest
$ dfsForestFrom vs x) x == True dfsForestFrom (vertexList
x) x ==dfsForest
x dfsForestFrom vs (vertices
vs) == map (\v -> Node v []) (nub
vs) dfsForestFrom [] x == [] dfsForestFrom [1, 4] $ 3 * (1 + 4) * (1 + 5) == [ Node { rootLabel = 1 , subForest = [ Node { rootLabel = 5 , subForest = [] } , Node { rootLabel = 4 , subForest = [] }]
dfs :: [a] -> AdjacencyMap a -> [a] 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 [1] $edge
1 1 == [1] dfs [1] $edge
1 2 == [1, 2] dfs [2] $edge
1 2 == [2] dfs [3] $edge
1 2 == [] dfs [1, 2] $edge
1 2 == [1, 2] dfs [2, 1] $edge
1 2 == [2, 1] dfs [] $ x == [] dfs [1, 4] $ 3 * (1 + 4) * (1 + 5) == [1, 5, 4]isSubgraphOf
(vertices
$ dfs vs x) x == True
topSort :: Ord a => AdjacencyMap a -> Maybe [a] Source #
Compute the topological sort of a graph or return Nothing
if the graph
is cyclic.
topSort (1 * 2 + 3 * 1) == Just [3,1,2]
topSort (1 * 2 + 2 * 1) == Nothing
fmap (flip isTopSort
x) (topSort x) /= Just False
scc :: Ord a => AdjacencyMap a -> AdjacencyMap (Set a) Source #
Compute the condensation of a graph, where each vertex corresponds to a strongly-connected component of the original graph.
sccempty
==empty
scc (vertex
x) ==vertex
(Set.singleton
x) scc (edge
x y) ==edge
(Set.singleton
x) (Set.singleton
y) scc (circuit
(1:xs)) ==edge
(Set.fromList
(1:xs)) (Set.fromList
(1:xs)) scc (3 * 1 * 4 * 1 * 5) ==edges
[ (Set.fromList
[1,4], Set.fromList
[1,4]) , (Set.fromList
[1,4], Set.fromList
[5] ) , (Set.fromList
[3] , Set.fromList
[1,4]) , (Set.fromList
[3] , Set.fromList
[5] )]