algebraic-graphs-0.3: A library for algebraic graph construction and transformation

Algebra.Graph

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 data type Graph and associated algorithms. For graphs that are known to be non-empty at compile time, see Algebra.Graph.NonEmpty. Graph is an instance of type classes defined in modules Algebra.Graph.Class and Algebra.Graph.HigherKinded.Class, which can be used for polymorphic graph construction and manipulation.

Synopsis

# Algebraic data type for graphs

data Graph a Source #

The Graph data type is a deep embedding of the core graph construction primitives empty, vertex, overlay and connect. 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 Eq instance is currently implemented using the AdjacencyMap as the canonical graph representation and 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 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

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. For example, if g is a Graph then n, m and s can be computed as follows:

n == vertexCount g
m == edgeCount g
s == size g

Note that size counts all leaves of the expression:

vertexCount empty           == 0
size        empty           == 1
vertexCount (vertex x)      == 1
size        (vertex x)      == 1
vertexCount (empty + empty) == 0
size        (empty + empty) == 2

Converting a Graph to the corresponding AdjacencyMap takes O(s + m * log(m)) time and O(s + m) memory. This is also the complexity of the graph equality test, because it is currently implemented by converting graph expressions to canonical representations based on adjacency maps.

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 2
vertex 3 < edge 1 2
vertex 1 < edge 1 1
edge 1 1 < edge 1 2
edge 1 2 < edge 1 1 + edge 2 2
edge 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

Constructors

 Empty Vertex a Overlay (Graph a) (Graph a) Connect (Graph a) (Graph a)
Instances
isEmpty (removeEdge x y $edge x y) == False  size :: Graph a -> Int Source # The size of a graph, i.e. the number of leaves of the expression including empty leaves. Complexity: O(s) time. size empty == 1 size (vertex x) == 1 size (overlay x y) == size x + size y size (connect x y) == size x + size y size x >= 1 size x >= vertexCount x  hasVertex :: Eq a => a -> Graph a -> Bool Source # Check if a graph contains a given vertex. Complexity: O(s) time. hasVertex x empty == False hasVertex x (vertex x) == True hasVertex 1 (vertex 2) == False hasVertex x . removeVertex x == const False  hasEdge :: Eq a => a -> a -> Graph a -> Bool Source # Check if a graph contains a given edge. Complexity: O(s) time. hasEdge x y empty == False hasEdge x y (vertex z) == False hasEdge x y (edge x y) == True hasEdge x y . removeEdge x y == const False hasEdge x y == elem (x,y) . edgeList  vertexCount :: Ord a => Graph a -> Int Source # The number of vertices in a graph. Complexity: O(s * log(n)) time. vertexCount empty == 0 vertexCount (vertex x) == 1 vertexCount == length . vertexList vertexCount x < vertexCount y ==> x < y  edgeCount :: Ord a => Graph a -> Int Source # The number of edges in a graph. Complexity: O(s + m * log(m)) time. Note that the number of edges m of a graph can be quadratic with respect to the expression size s. edgeCount empty == 0 edgeCount (vertex x) == 0 edgeCount (edge x y) == 1 edgeCount == length . edgeList  vertexList :: Ord a => Graph a -> [a] Source # The sorted list of vertices of a given graph. Complexity: O(s * log(n)) time and O(n) memory. vertexList empty == [] vertexList (vertex x) == [x] vertexList . vertices == nub . sort  edgeList :: Ord a => Graph a -> [(a, a)] Source # The sorted list of edges of a graph. Complexity: O(s + m * log(m)) time and O(m) memory. Note that the number of edges m of a graph can be quadratic with respect to the expression size s. edgeList empty == [] edgeList (vertex x) == [] edgeList (edge x y) == [(x,y)] edgeList (star 2 [3,1]) == [(2,1), (2,3)] edgeList . edges == nub . sort edgeList . transpose == sort . map swap . edgeList  vertexSet :: Ord a => Graph a -> Set a Source # The set of vertices of a given graph. Complexity: O(s * log(n)) time and O(n) memory. vertexSet empty == Set.empty vertexSet . vertex == Set.singleton vertexSet . vertices == Set.fromList  edgeSet :: Ord a => Graph a -> Set (a, a) Source # The set of edges of a given graph. Complexity: O(s * log(m)) time and O(m) memory. edgeSet empty == Set.empty edgeSet (vertex x) == Set.empty edgeSet (edge x y) == Set.singleton (x,y) edgeSet . edges == Set.fromList  adjacencyList :: Ord a => Graph a -> [(a, [a])] Source # The sorted adjacency list of a graph. Complexity: O(n + m) time and O(m) memory. adjacencyList empty == [] adjacencyList (vertex x) == [(x, [])] adjacencyList (edge 1 2) == [(1, [2]), (2, [])] adjacencyList (star 2 [3,1]) == [(1, []), (2, [1,3]), (3, [])] stars . adjacencyList == id  # Standard families of graphs path :: [a] -> Graph a Source # 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 path . reverse == transpose . path  circuit :: [a] -> Graph a Source # 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)] circuit . reverse == transpose . circuit  clique :: [a] -> Graph a Source # 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) clique . reverse == transpose . clique  biclique :: [a] -> [a] -> Graph a Source # 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 :: a -> [a] -> Graph a Source # 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)  stars :: [(a, [a])] -> Graph a Source # The stars formed by overlaying a list of stars. An inverse of adjacencyList. Complexity: O(L) 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 == id overlay (stars xs) (stars ys) == stars (xs ++ ys)  tree :: Tree a -> Graph a 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 :: Forest a -> Graph a 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 . map tree  mesh :: [a] -> [b] -> Graph (a, b) Source # Construct a mesh graph from two lists of vertices. Complexity: O(L1 * L2) time, memory and size, where L1 and L2 are the lengths of the given lists. mesh xs [] == empty mesh [] ys == empty mesh [x] [y] == vertex (x, y) mesh xs ys == box (path xs) (path ys) mesh [1..3] "ab" == edges [ ((1,'a'),(1,'b')), ((1,'a'),(2,'a')), ((1,'b'),(2,'b')), ((2,'a'),(2,'b')) , ((2,'a'),(3,'a')), ((2,'b'),(3,'b')), ((3,'a'),(3,'b')) ]  torus :: [a] -> [b] -> Graph (a, b) Source # Construct a torus graph from two lists of vertices. Complexity: O(L1 * L2) time, memory and size, where L1 and L2 are the lengths of the given lists. torus xs [] == empty torus [] ys == empty torus [x] [y] == edge (x,y) (x,y) torus xs ys == box (circuit xs) (circuit ys) torus [1,2] "ab" == edges [ ((1,'a'),(1,'b')), ((1,'a'),(2,'a')), ((1,'b'),(1,'a')), ((1,'b'),(2,'b')) , ((2,'a'),(1,'a')), ((2,'a'),(2,'b')), ((2,'b'),(1,'b')), ((2,'b'),(2,'a')) ]  deBruijn :: Int -> [a] -> Graph [a] Source # Construct a De Bruijn graph of a given non-negative dimension using symbols from a given alphabet. Complexity: O(A^(D + 1)) time, memory and size, where A is the size of the alphabet and D is the dimension of the graph.  deBruijn 0 xs == edge [] [] n > 0 ==> deBruijn n [] == empty deBruijn 1 [0,1] == edges [ ([0],[0]), ([0],[1]), ([1],[0]), ([1],[1]) ] deBruijn 2 "0" == edge "00" "00" deBruijn 2 "01" == edges [ ("00","00"), ("00","01"), ("01","10"), ("01","11") , ("10","00"), ("10","01"), ("11","10"), ("11","11") ] transpose (deBruijn n xs) == fmap reverse$ deBruijn n xs
vertexCount (deBruijn n xs) == (length $nub xs)^n n > 0 ==> edgeCount (deBruijn n xs) == (length$ nub xs)^(n + 1)


# Graph transformation

removeVertex :: Eq a => a -> Graph a -> Graph a Source #

Remove a vertex from a given graph. Complexity: O(s) time, memory and size.

removeVertex x (vertex x)       == empty
removeVertex 1 (vertex 2)       == vertex 2
removeVertex x (edge x x)       == empty
removeVertex 1 (edge 1 2)       == vertex 2
removeVertex x . removeVertex x == removeVertex x


removeEdge :: Eq a => a -> a -> Graph a -> Graph a Source #

Remove an edge from a given graph. Complexity: O(s) time, memory and size.

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
size (removeEdge x y z)         <= 3 * size z


replaceVertex :: Eq a => a -> a -> Graph a -> Graph a Source #

The function replaceVertex x y replaces vertex x with vertex y in a given Graph. If y already exists, x and y will be merged. Complexity: O(s) time, memory and size.

replaceVertex x x            == id
replaceVertex x y (vertex x) == vertex y
replaceVertex x y            == mergeVertices (== x) y


mergeVertices :: (a -> Bool) -> a -> Graph a -> Graph a Source #

Merge vertices satisfying a given predicate into a given vertex. Complexity: O(s) time, memory and size, 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


splitVertex :: Eq a => a -> [a] -> Graph a -> Graph a Source #

Split a vertex into a list of vertices with the same connectivity. Complexity: O(s + k * L) time, memory and size, where k is the number of occurrences of the vertex in the expression and L is the length of the given list.

splitVertex x []                  == removeVertex x
splitVertex x [x]                 == id
splitVertex x [y]                 == replaceVertex x y
splitVertex 1 [0,1] \$ 1 * (2 + 3) == (0 + 1) * (2 + 3)


transpose :: Graph a -> Graph a Source #

Transpose a given graph. Complexity: O(s) time, memory and size.

transpose empty       == empty
transpose (vertex x)  == vertex x
transpose (edge x y)  == edge y x
transpose . transpose == id
transpose (box x y)   == box (transpose x) (transpose y)
edgeList . transpose  == sort . map swap . edgeList


induce :: (a -> Bool) -> Graph a -> Graph a Source #

Construct the induced subgraph of a given graph by removing the vertices that do not satisfy a given predicate. Complexity: O(s) time, memory and size, 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


simplify :: Ord a => Graph a -> Graph a Source #

Simplify a graph expression. Semantically, this is the identity function, but it simplifies a given expression according to the laws of the algebra. The function does not compute the simplest possible expression, but uses heuristics to obtain useful simplifications in reasonable time. Complexity: the function performs O(s) graph comparisons. It is guaranteed that the size of the result does not exceed the size of the given expression.

simplify              == id
size (simplify x)     <= size x
simplify empty       === empty
simplify 1           === 1
simplify (1 + 1)     === 1
simplify (1 + 2 + 1) === 1 + 2
simplify (1 * 1 * 1) === 1 * 1


sparsify :: Graph a -> Graph (Either Int a) Source #

Sparsify a graph by adding intermediate Left Int vertices between the original vertices (wrapping the latter in Right) such that the resulting graph is sparse, i.e. contains only O(s) edges, but preserves the reachability relation between the original vertices. Sparsification is useful when working with dense graphs, as it can reduce the number of edges from O(n^2) down to O(n) by replacing cliques, bicliques and similar densely connected structures by sparse subgraphs built out of intermediate vertices. Complexity: O(s) time, memory and size.

sort . reachable x       == sort . rights . reachable (Right x) . sparsify
vertexCount (sparsify x) <= vertexCount x + size x + 1
edgeCount   (sparsify x) <= 3 * size x
size        (sparsify x) <= 3 * size x


# Graph composition

compose :: Ord a => Graph a -> Graph a -> Graph 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(n)) time, O(n + m) memory, and O(m1 + m2) size, where n and m stand for the number of vertices and edges in the resulting graph, while m1 and m2 are the number of edges in the original graphs. Note that the number of edges in the resulting graph may be quadratic, i.e. m = O(m1 * m2), but the algebraic representation requires only O(m1 + m2) operations to list them.

compose empty            x                == empty
compose x                empty            == 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]
size (compose x y)                        <= edgeCount x + edgeCount y + 1


box :: Graph a -> Graph b -> Graph (a, b) Source #

Compute the Cartesian product of graphs. Complexity: O(s1 * s2) time, memory and size, where s1 and s2 are the sizes of the given graphs.

box (path [0,1]) (path "ab") == edges [ ((0,'a'), (0,'b'))
, ((0,'a'), (1,'a'))
, ((0,'b'), (1,'b'))
, ((1,'a'), (1,'b')) ]


Up to an isomorphism between the resulting vertex types, this operation is commutative, associative, distributes over overlay, has singleton graphs as identities and empty as the annihilating zero. Below ~~ stands for the equality up to an isomorphism, e.g. (x, ()) ~~ x.

box x y               ~~ box y x
box x (box y z)       ~~ box (box x y) z
box x (overlay y z)   == overlay (box x y) (box x z)
box x (vertex ())     ~~ x
box x empty           ~~ empty
transpose   (box x y) == box (transpose x) (transpose y)
vertexCount (box x y) == vertexCount x * vertexCount y
edgeCount   (box x y) <= vertexCount x * edgeCount y + edgeCount x * vertexCount y


# Context

data Context a Source #

The Context of a subgraph comprises its inputs and outputs, i.e. all the vertices that are connected to the subgraph's vertices. Note that inputs and outputs can belong to the subgraph itself. In general, there are no guarantees on the order of vertices in inputs and outputs; furthermore, there may be repetitions.

Constructors

 Context Fieldsinputs :: [a] outputs :: [a]
Instances
 Eq a => Eq (Context a) Source # Instance detailsDefined in Algebra.Graph Methods(==) :: Context a -> Context a -> Bool #(/=) :: Context a -> Context a -> Bool # Show a => Show (Context a) Source # Instance detailsDefined in Algebra.Graph MethodsshowsPrec :: Int -> Context a -> ShowS #show :: Context a -> String #showList :: [Context a] -> ShowS #

context :: (a -> Bool) -> Graph a -> Maybe (Context a) Source #

Extract the Context of a subgraph specified by a given predicate. Returns Nothing if the specified subgraph is empty.

context (const False) x                   == Nothing
context (== 1)        (edge 1 2)          == Just (Context [   ] [2  ])
context (== 2)        (edge 1 2)          == Just (Context [1  ] [   ])
context (const True ) (edge 1 2)          == Just (Context [1  ] [2  ])
context (== 4)        (3 * 1 * 4 * 1 * 5) == Just (Context [3,1] [1,5])