algebraic-graphs-0.5: A library for algebraic graph construction and transformation

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 AdjacencyIntMap data type and associated functions. See Algebra.Graph.AdjacencyIntMap.Algorithm for implementations of basic graph algorithms. AdjacencyIntMap is an instance of the Graph type class, which can be used for polymorphic graph construction and manipulation. See Algebra.Graph.AdjacencyMap for graphs with non-Int vertices.

Synopsis

# Data structure

The AdjacencyIntMap 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))

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     :: AdjacencyIntMap Int) == "empty"
show (1         :: AdjacencyIntMap Int) == "vertex 1"
show (1 + 2     :: AdjacencyIntMap Int) == "vertices [1,2]"
show (1 * 2     :: AdjacencyIntMap Int) == "edge 1 2"
show (1 * 2 * 3 :: AdjacencyIntMap Int) == "edges [(1,2),(1,3),(2,3)]"
show (1 * 2 + 3 :: AdjacencyIntMap 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 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 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 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
Instances
 Source # Instance detailsDefined in Algebra.Graph.AdjacencyIntMap Methods Source # Note: this does not satisfy the usual ring laws; see AdjacencyIntMap for more details. Instance detailsDefined in Algebra.Graph.AdjacencyIntMap Methods Source # Instance detailsDefined in Algebra.Graph.AdjacencyIntMap Methods Source # Instance detailsDefined in Algebra.Graph.AdjacencyIntMap MethodsshowList :: [AdjacencyIntMap] -> ShowS # Source # Instance detailsDefined in Algebra.Graph.AdjacencyIntMap Associated Typestype Rep AdjacencyIntMap :: Type -> Type # Methods Source # Instance detailsDefined in Algebra.Graph.AdjacencyIntMap Methodsrnf :: AdjacencyIntMap -> () # Source # Instance detailsDefined in Algebra.Graph.ToGraph Associated Types Methodsfoldg :: r -> (ToVertex AdjacencyIntMap -> r) -> (r -> r -> r) -> (r -> r -> r) -> AdjacencyIntMap -> r Source # Source # Instance detailsDefined in Algebra.Graph.Class Associated Types Methods Source # Instance detailsDefined in Algebra.Graph.AdjacencyIntMap type Rep AdjacencyIntMap = D1 (MetaData "AdjacencyIntMap" "Algebra.Graph.AdjacencyIntMap" "algebraic-graphs-0.5-4dnrALfehjHELqhQlGFoFS" True) (C1 (MetaCons "AM" PrefixI True) (S1 (MetaSel (Just "adjacencyIntMap") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (IntMap IntSet)))) Source # Instance detailsDefined in Algebra.Graph.ToGraph Source # Instance detailsDefined in Algebra.Graph.Class

The adjacency map of a graph: each vertex is associated with a set of its direct successors. Complexity: O(1) time and memory.

adjacencyIntMap empty      == IntMap.empty
adjacencyIntMap (vertex x) == IntMap.singleton x IntSet.empty
adjacencyIntMap (edge 1 1) == IntMap.singleton 1 (IntSet.singleton 1)
adjacencyIntMap (edge 1 2) == IntMap.fromList [(1,IntSet.singleton 2), (2,IntSet.empty)]


Construct an AdjacencyIntMap from an AdjacencyMap with vertices of type Int. Complexity: O(n + m) time and memory.

fromAdjacencyMap == stars . AdjacencyMap.adjacencyList


# Basic graph construction primitives

Construct the empty graph. Complexity: O(1) time and memory.

isEmpty     empty == True
hasVertex x empty == False
vertexCount empty == 0
edgeCount   empty == 0


Construct the graph comprising a single isolated vertex. Complexity: O(1) time and memory.

isEmpty     (vertex x) == False
hasVertex x (vertex y) == (x == y)
vertexCount (vertex x) == 1
edgeCount   (vertex x) == 0


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) == True
edgeCount   (edge x y) == 1
vertexCount (edge 1 1) == 1
vertexCount (edge 1 2) == 2


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   y
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
vertexCount (overlay 1 2) == 2
edgeCount   (overlay 1 2) == 0


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   y
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
vertexCount (connect 1 2) == 2
edgeCount   (connect 1 2) == 1


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
hasVertex x  . vertices == elem x
vertexCount  . vertices == length . nub
vertexIntSet . vertices == IntSet.fromList


edges :: [(Int, Int)] -> AdjacencyIntMap Source #

Construct the graph from a list of edges. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

edges []          == empty
edges [(x,y)]     == edge x y
edges             == overlays . map (uncurry edge)
edgeCount . edges == length . nub
edgeList . edges  == nub . sort


Overlay a given list of graphs. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

overlays []        == empty
overlays [x]       == x
overlays [x,y]     == overlay x y
overlays           == foldr overlay empty
isEmpty . overlays == all isEmpty


Connect a given list of graphs. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

connects []        == empty
connects [x]       == x
connects [x,y]     == connect x y
connects           == foldr connect empty
isEmpty . connects == all isEmpty


# Relations on graphs

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.

isSubgraphOf empty         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

Check if a graph is empty. Complexity: O(1) time.

isEmpty empty                       == True
isEmpty (overlay empty empty)       == True
isEmpty (vertex x)                  == False
isEmpty (removeVertex x $vertex x) == True isEmpty (removeEdge x y$ edge x y) == False


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

hasVertex x empty            == False
hasVertex x (vertex y)       == (x == y)
hasVertex x . removeVertex x == const False


Check if a graph contains a given edge. Complexity: O(log(n)) 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


The number of vertices in a graph. Complexity: O(1) time.

vertexCount empty             ==  0
vertexCount (vertex x)        ==  1
vertexCount                   ==  length . vertexList
vertexCount x < vertexCount y ==> x < y


The number of edges in a graph. Complexity: O(n) time.

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


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

vertexList empty      == []
vertexList (vertex x) == [x]
vertexList . vertices == nub . sort


edgeList :: AdjacencyIntMap -> [(Int, Int)] Source #

The sorted list of edges of a graph. Complexity: O(n + m) time and O(m) memory.

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


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


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

vertexIntSet empty      == IntSet.empty
vertexIntSet . vertex   == IntSet.singleton
vertexIntSet . vertices == IntSet.fromList
vertexIntSet . clique   == IntSet.fromList


The set of edges of a given graph. Complexity: O((n + m) * 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


The preset (here preIntSet) of an element x is the set of its direct predecessors. Complexity: O(n * log(n)) time and O(n) memory.

preIntSet x empty      == Set.empty
preIntSet x (vertex x) == Set.empty
preIntSet 1 (edge 1 2) == Set.empty
preIntSet y (edge x y) == Set.fromList [x]


The postset (here postIntSet) of a vertex is the set of its direct successors.

postIntSet x empty      == IntSet.empty
postIntSet x (vertex x) == IntSet.empty
postIntSet x (edge x y) == IntSet.fromList [y]
postIntSet 2 (edge 1 2) == IntSet.empty


# Standard families of graphs

path :: [Int] -> AdjacencyIntMap Source #

The path on a list of vertices. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

path []        == empty
path [x]       == vertex x
path [x,y]     == edge x y
path . reverse == transpose . path


circuit :: [Int] -> AdjacencyIntMap Source #

The circuit on a list of vertices. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

circuit []        == empty
circuit [x]       == edge x x
circuit [x,y]     == edges [(x,y), (y,x)]
circuit . reverse == transpose . circuit


clique :: [Int] -> AdjacencyIntMap Source #

The clique on a list of vertices. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

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 :: [Int] -> [Int] -> AdjacencyIntMap Source #

The biclique on two lists of vertices. Complexity: O(n * log(n) + m) time and O(n + m) memory.

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 :: Int -> [Int] -> AdjacencyIntMap Source #

The star formed by a centre vertex connected to a list of leaves. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

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 :: [(Int, [Int])] -> AdjacencyIntMap Source #

The stars formed by overlaying a list of stars. 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         == id
overlay (stars xs) (stars ys) == stars (xs ++ ys)


Construct a graph from a list of adjacency sets; a variation of stars. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

fromAdjacencyIntSets []                                     == empty
fromAdjacencyIntSets [(x, IntSet.empty)]                    == vertex x
fromAdjacencyIntSets [(x, IntSet.singleton y)]              == edge x y
fromAdjacencyIntSets . map (fmap IntSet.fromList)           == stars
overlay (fromAdjacencyIntSets xs) (fromAdjacencyIntSets ys) == fromAdjacencyIntSets (xs ++ ys)


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)]


The forest graph constructed from a given Forest data structure. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

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


# Graph transformation

Remove a vertex from a given graph. Complexity: O(n*log(n)) time.

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


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


The function replaceVertex x y replaces vertex x with vertex y in a given AdjacencyIntMap. 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


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 a given graph. Complexity: O(m * log(n)) time, O(n + m) memory.

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


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 AdjacencyIntMap. Complexity: O((n + m) * log(n)) time.

gmap f empty      == 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)


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 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


# Relational operations

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 and O(n + m) memory.

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]


Compute the reflexive and transitive closure of a graph. Complexity: O(n * m * log(n)^2) time.

closure empty            == 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        == closure
postIntSet x (closure y) == IntSet.fromList (reachable x y)


Compute the reflexive closure of a graph by adding a self-loop to every vertex. Complexity: O(n * log(n)) time.

reflexiveClosure empty              == 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


Compute the symmetric closure of a graph by overlaying it with its own transpose. Complexity: O((n + m) * log(n)) time.

symmetricClosure empty              == empty
symmetricClosure (vertex x)         == vertex x
symmetricClosure (edge x y)         == edges [(x,y), (y,x)]
symmetricClosure x                  == overlay x (transpose x)
symmetricClosure . symmetricClosure == symmetricClosure


Compute the transitive closure of a graph. Complexity: O(n * m * log(n)^2) time.

transitiveClosure empty               == empty
transitiveClosure (vertex x)          == vertex x
transitiveClosure (edge x y)          == edge x y
transitiveClosure (path \$ nub xs)     == clique (nub xs)
transitiveClosure . transitiveClosure == transitiveClosure


# Miscellaneous

Check that the internal graph representation is consistent, i.e. that all edges refer to existing vertices. It should be impossible to create an inconsistent adjacency map, and we use this function in testing.

consistent empty         == 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