Algebra.Graph.NonEmpty

Non-empty algebraic graphs, which are constructed using three primitives: vertex, overlay and connect. See module Algebra.Graph for algebraic graphs that can be empty.

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 signum method of the type class Num cannot be implemented and will throw an error. Furthermore, 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 satisfies the following laws of non-empty algebraic graphs.

• overlay is commutative, associative and idempotent:

        x + y == y + x
  x + (y + z) == (x + y) + z
        x + x == x

• connect is associative:

  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

• connect satisfies absorption and saturation:

  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, defined as the number of vertex leaves (note that n <= s). If g is a Graph, the corresponding n, m and s can be computed as follows:

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

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 Ord 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

x     <= x + y
x + y <= x * y

Convert an algebraic graph (from Algebra.Graph) into a non-empty algebraic graph. Returns Nothing if the argument is empty. Complexity: O(s) time, memory and size.

toNonEmpty empty       == Nothing
toNonEmpty (toGraph x) == Just (x :: Graph a)

Construct the graph comprising a single isolated vertex. An alias for the constructor Vertex.

hasVertex x (vertex y) == (x == y)
vertexCount (vertex x) == 1
edgeCount   (vertex x) == 0
size        (vertex x) == 1

Construct the graph comprising a single edge.

edge x y               == connect (vertex x) (vertex y)
hasEdge x y (edge x y) == True
edgeCount   (edge x y) == 1
vertexCount (edge 1 1) == 1
vertexCount (edge 1 2) == 2

Overlay two graphs. An alias for the constructor Overlay. This is a commutative, associative and idempotent operation. Complexity: O(1) time and memory, O(s1 + s2) size.

hasVertex z (overlay x y) == hasVertex z x || hasVertex z y
vertexCount (overlay x y) >= vertexCount x
vertexCount (overlay x y) <= vertexCount x + vertexCount y
edgeCount   (overlay x y) >= edgeCount x
edgeCount   (overlay x y) <= edgeCount x   + edgeCount y
size        (overlay x y) == size x        + size y
vertexCount (overlay 1 2) == 2
edgeCount   (overlay 1 2) == 0

Overlay a possibly empty graph (from Algebra.Graph) with a non-empty graph. If the first argument is empty, the function returns the second argument; otherwise it is semantically the same as overlay. Complexity: O(s1) time and memory, and O(s1 + s2) size.

               overlay1 empty x == x
x /= empty ==> overlay1 x     y == overlay (fromJust $ toNonEmpty x) y

Connect two graphs. An alias for the constructor Connect. This is an associative operation, which distributes over overlay and obeys the decomposition axiom. Complexity: O(1) time and memory, O(s1 + s2) size. 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).

hasVertex z (connect x y) == hasVertex z x || hasVertex z y
vertexCount (connect x y) >= vertexCount x
vertexCount (connect x y) <= vertexCount x + vertexCount y
edgeCount   (connect x y) >= edgeCount x
edgeCount   (connect x y) >= edgeCount y
edgeCount   (connect x y) >= vertexCount x * vertexCount y
edgeCount   (connect x y) <= vertexCount x * vertexCount y + edgeCount x + edgeCount y
size        (connect x y) == size x        + size y
vertexCount (connect 1 2) == 2
edgeCount   (connect 1 2) == 1

Construct the graph comprising a given list of isolated vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

vertices1 [x]           == vertex x
hasVertex x . vertices1 == elem x
vertexCount . vertices1 == length . nub
vertexSet   . vertices1 == Set.fromList . toList

Construct the graph from a list of edges. Complexity: O(L) time, memory and size, where L is the length of the given list.

edges1 [(x,y)]     == edge x y
edges1             == overlays1 . fmap (uncurry edge)
edgeCount . edges1 == length . nub

Overlay a given list of graphs. Complexity: O(L) time and memory, and O(S) size, where L is the length of the given list, and S is the sum of sizes of the graphs in the list.

overlays1 [x]   == x
overlays1 [x,y] == overlay x y

Connect a given list of graphs. Complexity: O(L) time and memory, and O(S) size, where L is the length of the given list, and S is the sum of sizes of the graphs in the list.

connects1 [x]   == x
connects1 [x,y] == connect x y

Generalised graph folding: recursively collapse a Graph by applying the provided functions to the leaves and internal nodes of the expression. The order of arguments is: vertex, overlay and connect. Complexity: O(s) applications of the given functions. As an example, the complexity of size is O(s), since const and + have constant costs.

foldg1 vertex    overlay connect        == id
foldg1 vertex    overlay (flip connect) == transpose
foldg1 (const 1) (+)     (+)            == size
foldg1 (== x)    (||)    (||)           == hasVertex x

The isSubgraphOf function takes two graphs and returns True if the first graph is a subgraph of the second. 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.

isSubgraphOf x             (overlay x y) ==  True
isSubgraphOf (overlay x y) (connect x y) ==  True
isSubgraphOf (path1 xs)    (circuit1 xs) ==  True
isSubgraphOf x y                         ==> x <= y

Structural equality on graph expressions. Complexity: O(s) time.

    x === x     == True
x + y === x + y == True
1 + 2 === 2 + 1 == False
x + y === x * y == False

The size of a graph, i.e. the number of leaves of the expression. Complexity: O(s) time.

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

Check if a graph contains a given vertex. Complexity: O(s) time.

hasVertex x (vertex y) == (x == y)

Check if a graph contains a given edge. Complexity: O(s) time.

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

The number of vertices in a graph. Complexity: O(s * log(n)) time.

vertexCount (vertex x)        ==  1
vertexCount                   ==  length . vertexList
vertexCount x < vertexCount y ==> x < y

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 (vertex x) == 0
edgeCount (edge x y) == 1
edgeCount            == length . edgeList

The sorted list of vertices of a given graph. Complexity: O(s * log(n)) time and O(n) memory.

vertexList1 (vertex x)  == [x]
vertexList1 . vertices1 == nub . sort

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 (vertex x)     == []
edgeList (edge x y)     == [(x,y)]
edgeList (star 2 [3,1]) == [(2,1), (2,3)]
edgeList . edges1       == nub . sort . toList
edgeList . transpose    == sort . map swap . edgeList

The set of vertices of a given graph. Complexity: O(s * log(n)) time and O(n) memory.

vertexSet . vertex    == Set.singleton
vertexSet . vertices1 == Set.fromList . toList
vertexSet . clique1   == Set.fromList . toList

The set of edges of a given graph. Complexity: O(s * log(m)) time and O(m) memory.

edgeSet (vertex x) == Set.empty
edgeSet (edge x y) == Set.singleton (x,y)
edgeSet . edges1   == Set.fromList . toList

The path on a list of vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

path1 [x]       == vertex x
path1 [x,y]     == edge x y
path1 . reverse == transpose . path1

The circuit on a list of vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

circuit1 [x]       == edge x x
circuit1 [x,y]     == edges1 [(x,y), (y,x)]
circuit1 . reverse == transpose . circuit1

The clique on a list of vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

clique1 [x]        == vertex x
clique1 [x,y]      == edge x y
clique1 [x,y,z]    == edges1 [(x,y), (x,z), (y,z)]
clique1 (xs <> ys) == connect (clique1 xs) (clique1 ys)
clique1 . reverse  == transpose . clique1

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.

biclique1 [x1,x2] [y1,y2] == edges1 [(x1,y1), (x1,y2), (x2,y1), (x2,y2)]
biclique1 xs      ys      == connect (vertices1 xs) (vertices1 ys)

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] == edges1 [(x,y), (x,z)]

The stars formed by overlaying a non-empty list of stars. Complexity: O(L) time, memory and size, where L is the total size of the input.

stars1 [(x, [] )]               == vertex x
stars1 [(x, [y])]               == edge x y
stars1 [(x, ys )]               == star x ys
stars1                          == overlays1 . fmap (uncurry star)
overlay (stars1 xs) (stars1 ys) == stars1 (xs <> ys)

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 []]])                       == path1 [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 []]]) == edges1 [(1,2), (1,3), (3,4), (3,5)]

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.

mesh1 [x]     [y]        == vertex (x, y)
mesh1 xs      ys         == box (path1 xs) (path1 ys)
mesh1 [1,2,3] ['a', 'b'] == edges1 [ ((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')) ]

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.

torus1 [x]   [y]        == edge (x,y) (x,y)
torus1 xs    ys         == box (circuit1 xs) (circuit1 ys)
torus1 [1,2] ['a', 'b'] == edges1 [ ((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')) ]

Remove a vertex from a given graph. Returns Nothing if the resulting graph is empty. Complexity: O(s) time, memory and size.

removeVertex1 x (vertex x)          == Nothing
removeVertex1 1 (vertex 2)          == Just (vertex 2)
removeVertex1 x (edge x x)          == Nothing
removeVertex1 1 (edge 1 2)          == Just (vertex 2)
removeVertex1 x >=> removeVertex1 x == removeVertex1 x

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

removeEdge x y (edge x y)       == vertices1 [x,y]
removeEdge x y . removeEdge x y == removeEdge x y
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

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

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

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

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.

splitVertex1 x [x]                 == id
splitVertex1 x [y]                 == replaceVertex x y
splitVertex1 1 [0,1] $ 1 * (2 + 3) == (0 + 1) * (2 + 3)

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

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

Construct the induced subgraph of a given graph by removing the vertices that do not satisfy a given predicate. Returns Nothing if the resulting graph is empty. Complexity: O(s) time, memory and size, assuming that the predicate takes constant time.

induce1 (const True ) x == Just x
induce1 (const False) x == Nothing
induce1 (/= x)          == removeVertex1 x
induce1 p >=> induce1 q == induce1 (\x -> p x && q x)

Construct the induced subgraph of a given graph by removing the vertices that are Nothing. Returns Nothing if the resulting graph is empty. Complexity: O(s) time, memory and size.

induceJust1 (vertex Nothing)                               == Nothing
induceJust1 (edge (Just x) Nothing)                        == Just (vertex x)
induceJust1 . fmap Just                                    == Just
induceJust1 . fmap (\x -> if p x then Just x else Nothing) == induce1 p

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 1           === 1
simplify (1 + 1)     === 1
simplify (1 + 2 + 1) === 1 + 2
simplify (1 * 1 * 1) === 1 * 1

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

Sparsify a graph whose vertices are integers in the range [1..n], where n is the first argument of the function, producing an array-based graph representation from Data.Graph (introduced by King and Launchbury, hence the name of the function). In the resulting graph, vertices [1..n] correspond to the original vertices, and all vertices greater than n are introduced by the sparsification procedure.

Complexity: O(s) time and memory. Note that thanks to sparsification, the resulting graph has a linear number of edges with respect to the size of the original algebraic representation even though the latter can potentially contain a quadratic O(s^2) number of edges.

sort . reachable x                 == sort . filter (<= n) . reachable (sparsifyKL n x)
length (vertices $ sparsifyKL n x) <= vertexCount x + size x + 1
length (edges    $ sparsifyKL n x) <= 3 * size x

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 (path1 [0,1]) (path1 ['a','b']) == edges1 [ ((0,'a'), (0,'b'))
                                              , ((0,'a'), (1,'a'))
                                              , ((0,'b'), (1,'b'))
                                              , ((1,'a'), (1,'b')) ]

Up to isomorphism between the resulting vertex types, this operation is commutative, associative, distributes over overlay, and has singleton graphs as identities. Below ~~ stands for 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
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