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 data type AdjacencyMap for graphs that are known to be non-empty at compile time. To avoid name clashes with Algebra.Graph.AdjacencyMap, this module can be imported qualified:

import qualified Algebra.Graph.NonEmpty.AdjacencyMap as NonEmpty


The naming convention generally follows that of Data.List.NonEmpty: we use suffix 1 to indicate the functions whose interface must be changed compared to Algebra.Graph.AdjacencyMap, e.g. vertices1.

Synopsis

# Data structure

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))

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 Show instance is defined using basic graph construction primitives:

show (1         :: AdjacencyMap Int) == "vertex 1"
show (1 + 2     :: AdjacencyMap Int) == "vertices1 [1,2]"
show (1 * 2     :: AdjacencyMap Int) == "edge 1 2"
show (1 * 2 * 3 :: AdjacencyMap Int) == "edges1 [(1,2),(1,3),(2,3)]"
show (1 * 2 + 3 :: AdjacencyMap Int) == "overlay (vertex 3) (edge 1 2)"

The Eq instance satisfies the following laws of 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 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
x     <= x + y
x + y <= x * y
Instances
 Eq a => Eq (AdjacencyMap a) Source # Instance detailsDefined in Algebra.Graph.NonEmpty.AdjacencyMap Methods(==) :: AdjacencyMap a -> AdjacencyMap a -> Bool #(/=) :: AdjacencyMap a -> AdjacencyMap a -> Bool # (Ord a, Num a) => Num (AdjacencyMap a) Source # Note: this does not satisfy the usual ring laws; see AdjacencyMap for more details. Instance detailsDefined in Algebra.Graph.NonEmpty.AdjacencyMap Methods(+) :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a #(-) :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a #(*) :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a #abs :: AdjacencyMap a -> AdjacencyMap a # Ord a => Ord (AdjacencyMap a) Source # Instance detailsDefined in Algebra.Graph.NonEmpty.AdjacencyMap Methodscompare :: AdjacencyMap a -> AdjacencyMap a -> Ordering #(<) :: AdjacencyMap a -> AdjacencyMap a -> Bool #(<=) :: AdjacencyMap a -> AdjacencyMap a -> Bool #(>) :: AdjacencyMap a -> AdjacencyMap a -> Bool #(>=) :: AdjacencyMap a -> AdjacencyMap a -> Bool #max :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a #min :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a # (Ord a, Show a) => Show (AdjacencyMap a) Source # Instance detailsDefined in Algebra.Graph.NonEmpty.AdjacencyMap MethodsshowsPrec :: Int -> AdjacencyMap a -> ShowS #show :: AdjacencyMap a -> String #showList :: [AdjacencyMap a] -> ShowS # Source # Instance detailsDefined in Algebra.Graph.NonEmpty.AdjacencyMap Associated Typestype Rep (AdjacencyMap a) :: Type -> Type # Methodsfrom :: AdjacencyMap a -> Rep (AdjacencyMap a) x #to :: Rep (AdjacencyMap a) x -> AdjacencyMap a # NFData a => NFData (AdjacencyMap a) Source # Instance detailsDefined in Algebra.Graph.NonEmpty.AdjacencyMap Methodsrnf :: AdjacencyMap a -> () # Ord a => ToGraph (AdjacencyMap a) Source # Instance detailsDefined in Algebra.Graph.ToGraph Associated Typestype ToVertex (AdjacencyMap a) :: Type Source # Methodsfoldg :: r -> (ToVertex (AdjacencyMap a) -> r) -> (r -> r -> r) -> (r -> r -> r) -> AdjacencyMap a -> r Source #hasEdge :: ToVertex (AdjacencyMap a) -> ToVertex (AdjacencyMap a) -> AdjacencyMap a -> Bool Source #edgeList :: AdjacencyMap a -> [(ToVertex (AdjacencyMap a), ToVertex (AdjacencyMap a))] Source #preSet :: ToVertex (AdjacencyMap a) -> AdjacencyMap a -> Set (ToVertex (AdjacencyMap a)) Source #adjacencyList :: AdjacencyMap a -> [(ToVertex (AdjacencyMap a), [ToVertex (AdjacencyMap a)])] Source #dfsForestFrom :: [ToVertex (AdjacencyMap a)] -> AdjacencyMap a -> Forest (ToVertex (AdjacencyMap a)) Source #dfs :: [ToVertex (AdjacencyMap a)] -> AdjacencyMap a -> [ToVertex (AdjacencyMap a)] Source #reachable :: ToVertex (AdjacencyMap a) -> AdjacencyMap a -> [ToVertex (AdjacencyMap a)] Source #isTopSortOf :: [ToVertex (AdjacencyMap a)] -> AdjacencyMap a -> Bool Source # type Rep (AdjacencyMap a) Source # Instance detailsDefined in Algebra.Graph.NonEmpty.AdjacencyMap type Rep (AdjacencyMap a) = D1 (MetaData "AdjacencyMap" "Algebra.Graph.NonEmpty.AdjacencyMap" "algebraic-graphs-0.5-4dnrALfehjHELqhQlGFoFS" True) (C1 (MetaCons "NAM" PrefixI True) (S1 (MetaSel (Just "am") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (AdjacencyMap a)))) type ToVertex (AdjacencyMap a) Source # Instance detailsDefined in Algebra.Graph.ToGraph type ToVertex (AdjacencyMap a) = a

Convert a possibly empty AdjacencyMap into NonEmpty.AdjacencyMap. Returns Nothing if the argument is empty. Complexity: O(1) time, memory and size.

toNonEmpty empty          == Nothing
toNonEmpty . fromNonEmpty == Just


Convert a NonEmpty.AdjacencyMap into an AdjacencyMap. The resulting graph is guaranteed to be non-empty. Complexity: O(1) time, memory and size.

isEmpty . fromNonEmpty    == const False
toNonEmpty . fromNonEmpty == Just


# Basic graph construction primitives

vertex :: a -> AdjacencyMap a Source #

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

hasVertex x (vertex y) == (x == y)
vertexCount (vertex x) == 1
edgeCount   (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) == 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.

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).

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


vertices1 :: Ord a => NonEmpty 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.

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


edges1 :: Ord a => NonEmpty (a, a) -> AdjacencyMap a Source #

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

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


overlays1 :: Ord a => NonEmpty (AdjacencyMap a) -> AdjacencyMap a Source #

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

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


connects1 :: Ord a => NonEmpty (AdjacencyMap a) -> AdjacencyMap a Source #

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

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


# 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.

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


# Graph properties

hasVertex :: Ord a => a -> AdjacencyMap a -> Bool Source #

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

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


hasEdge :: Ord a => a -> a -> AdjacencyMap a -> Bool Source #

Check if a graph contains a given edge. Complexity: O(log(n)) 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(1) time.

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 (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.

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


edgeList :: AdjacencyMap a -> [(a, a)] Source #

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

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

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


edgeSet :: Ord a => AdjacencyMap a -> Set (a, a) Source #

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

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


preSet :: Ord a => a -> AdjacencyMap a -> Set a Source #

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

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 -> AdjacencyMap a -> Set a Source #

The postset of a vertex is the set of its direct successors. Complexity: O(log(n)) time and O(1) memory.

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

path1 :: Ord a => NonEmpty a -> AdjacencyMap a Source #

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

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


circuit1 :: Ord a => NonEmpty a -> AdjacencyMap a Source #

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

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


clique1 :: Ord a => NonEmpty a -> AdjacencyMap a Source #

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

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


biclique1 :: Ord a => NonEmpty a -> NonEmpty a -> AdjacencyMap a Source #

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

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


star :: Ord a => a -> [a] -> AdjacencyMap a 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] == edges1 [(x,y), (x,z)]


stars1 :: Ord a => NonEmpty (a, [a]) -> AdjacencyMap a 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.

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)


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 []]])                       == 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)]


# Graph transformation

removeVertex1 :: Ord a => a -> AdjacencyMap a -> Maybe (AdjacencyMap a) Source #

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

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


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)       == 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


replaceVertex :: Ord a => a -> a -> AdjacencyMap a -> AdjacencyMap a Source #

The function replaceVertex x y replaces vertex 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 -> 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 #

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

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


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 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)


induce1 :: (a -> Bool) -> AdjacencyMap a -> Maybe (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.

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)


induceJust1 :: Ord a => AdjacencyMap (Maybe a) -> Maybe (AdjacencyMap a) Source #

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(n + m) time.

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


# Graph closure

closure :: Ord a => AdjacencyMap a -> AdjacencyMap a Source #

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

closure (vertex x)       == edge x x
closure (edge x x)       == edge x x
closure (edge x y)       == edges1 [(x,x), (x,y), (y,y)]
closure (path1 $nub xs) == reflexiveClosure (clique1$ nub xs)
closure                  == reflexiveClosure . transitiveClosure
closure                  == transitiveClosure . reflexiveClosure
closure . closure        == closure
postSet x (closure y)    == Set.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 (vertex x)         == edge x x
reflexiveClosure (edge x x)         == edge x x
reflexiveClosure (edge x y)         == edges1 [(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 (vertex x)         == vertex x
symmetricClosure (edge x y)         == edges1 [(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 (vertex x)          == vertex x
transitiveClosure (edge x y)          == edge x y
transitiveClosure (path1 \$ nub xs)    == clique1 (nub xs)
transitiveClosure . transitiveClosure == transitiveClosure


# Miscellaneous

consistent :: Ord a => AdjacencyMap a -> Bool Source #

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

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