| Copyright | (c) Andrey Mokhov 2016-2021 |
|---|---|
| License | MIT (see the file LICENSE) |
| Maintainer | andrey.mokhov@gmail.com |
| Stability | experimental |
| Safe Haskell | None |
| Language | Haskell2010 |
Algebra.Graph.AdjacencyMap
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 AdjacencyMap data type and associated functions.
See Algebra.Graph.AdjacencyMap.Algorithm for basic graph algorithms.
AdjacencyMap is an instance of the Graph type class, which can be used
for polymorphic graph construction and manipulation.
Algebra.Graph.AdjacencyIntMap defines adjacency maps specialised to graphs
with Int vertices.
Synopsis
- data AdjacencyMap a
- adjacencyMap :: AdjacencyMap a -> Map a (Set a)
- empty :: AdjacencyMap a
- vertex :: 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
- 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 :: Eq a => AdjacencyMap a -> Set (a, a)
- preSet :: Ord a => a -> AdjacencyMap a -> Set 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
- stars :: Ord a => [(a, [a])] -> AdjacencyMap a
- fromAdjacencySets :: Ord a => [(a, Set 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 :: (a -> Bool) -> AdjacencyMap a -> AdjacencyMap a
- induceJust :: Ord a => AdjacencyMap (Maybe a) -> AdjacencyMap a
- compose :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a
- box :: (Ord a, Ord b) => AdjacencyMap a -> AdjacencyMap b -> AdjacencyMap (a, b)
- closure :: Ord a => AdjacencyMap a -> AdjacencyMap a
- reflexiveClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a
- symmetricClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a
- transitiveClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a
- consistent :: Ord a => AdjacencyMap a -> Bool
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 ==vertex0 1 + 2 ==overlay(vertex1) (vertex2) 1 * 2 ==connect(vertex1) (vertex2) 1 + 2 * 3 ==overlay(vertex1) (connect(vertex2) (vertex3)) 1 * (2 + 3) ==connect(vertex1) (overlay(vertex2) (vertex3))
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 :: 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:
overlayis commutative and associative:x + y == y + x x + (y + z) == (x + y) + z
connectis associative and hasemptyas the identity:x * empty == x empty * x == x x * (y * z) == (x * y) * z
connectdistributes overoverlay:x * (y + z) == x * y + x * z (x + y) * z == x * z + y * z
connectcan be decomposed:x * y * z == x * y + x * z + y * z
The following useful theorems can be proved from the above set of axioms.
overlayhasemptyas the identity and is idempotent:x + empty == x empty + x == x x + x == xAbsorption 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:
vertex1 <vertex2vertex3 <edge1 2vertex1 <edge1 1edge1 1 <edge1 2edge1 2 <edge1 1 +edge2 2edge1 2 <edge1 3
Note that the resulting order refines the isSubgraphOf relation and is
compatible with overlay and connect operations:
isSubgraphOf x y ==> x <= yempty <= x
x <= x + y
x + y <= x * yInstances
adjacencyMap :: AdjacencyMap a -> Map a (Set a) Source #
The adjacency map of a graph: each vertex is associated with a set of its direct successors. Complexity: O(1) time and memory.
adjacencyMapempty== Map.emptyadjacencyMap (vertexx) == Map.singletonx Set.emptyadjacencyMap (edge1 1) == Map.singleton1 (Set.singleton1) adjacencyMap (edge1 2) == Map.fromList[(1,Set.singleton2), (2,Set.empty)]
Basic graph construction primitives
empty :: AdjacencyMap a Source #
Construct the empty graph.
isEmptyempty == TruehasVertexx empty == FalsevertexCountempty == 0edgeCountempty == 0
vertex :: a -> AdjacencyMap a Source #
Construct the graph comprising a single isolated vertex.
isEmpty(vertex x) == FalsehasVertexx (vertex y) == (x == y)vertexCount(vertex x) == 1edgeCount(vertex x) == 0
edge :: Ord a => a -> a -> AdjacencyMap a Source #
Construct the graph comprising a single edge.
edge x y ==connect(vertexx) (vertexy)hasEdgex 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) ==isEmptyx &&isEmptyyhasVertexz (overlay x y) ==hasVertexz x ||hasVertexz yvertexCount(overlay x y) >=vertexCountxvertexCount(overlay x y) <=vertexCountx +vertexCountyedgeCount(overlay x y) >=edgeCountxedgeCount(overlay x y) <=edgeCountx +edgeCountyvertexCount(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) ==isEmptyx &&isEmptyyhasVertexz (connect x y) ==hasVertexz x ||hasVertexz yvertexCount(connect x y) >=vertexCountxvertexCount(connect x y) <=vertexCountx +vertexCountyedgeCount(connect x y) >=edgeCountxedgeCount(connect x y) >=edgeCountyedgeCount(connect x y) >=vertexCountx *vertexCountyedgeCount(connect x y) <=vertexCountx *vertexCounty +edgeCountx +edgeCountyvertexCount(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 [] ==emptyvertices [x] ==vertexx vertices ==overlays. mapvertexhasVertexx . vertices ==elemxvertexCount. vertices ==length.nubvertexSet. 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 #
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.
isSubgraphOfemptyx == True isSubgraphOf (vertexx)empty== False isSubgraphOf x (overlayx y) == True isSubgraphOf (overlayx y) (connectx y) == True isSubgraphOf (pathxs) (circuitxs) == True isSubgraphOf x y ==> x <= y
Graph properties
isEmpty :: AdjacencyMap a -> Bool Source #
Check if a graph is empty. Complexity: O(1) time.
isEmptyempty== True isEmpty (overlayemptyempty) == True isEmpty (vertexx) == False isEmpty (removeVertexx $vertexx) == True isEmpty (removeEdgex y $edgex 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 (vertexy) == (x == y) hasVertex x .removeVertexx ==constFalse
vertexCount :: AdjacencyMap a -> Int Source #
The number of vertices in a graph. Complexity: O(1) time.
vertexCountempty== 0 vertexCount (vertexx) == 1 vertexCount ==length.vertexListvertexCount x < vertexCount y ==> x < y
edgeCount :: AdjacencyMap a -> Int Source #
vertexList :: AdjacencyMap a -> [a] Source #
edgeList :: AdjacencyMap a -> [(a, a)] Source #
adjacencyList :: AdjacencyMap a -> [(a, [a])] Source #
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 #
stars :: Ord a => [(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.
stars [] ==emptystars [(x, [])] ==vertexx stars [(x, [y])] ==edgex y stars [(x, ys)] ==starx ys stars ==overlays.map(uncurrystar) stars .adjacencyList== idoverlay(stars xs) (stars ys) == stars (xs++ys)
fromAdjacencySets :: Ord a => [(a, Set a)] -> AdjacencyMap a Source #
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.
fromAdjacencySets [] ==emptyfromAdjacencySets [(x, Set.empty)] ==vertexx fromAdjacencySets [(x, Set.singletony)] ==edgex y fromAdjacencySets .map(fmapSet.fromList) ==starsoverlay(fromAdjacencySets xs) (fromAdjacencySets ys) == fromAdjacencySets (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 []) ==vertexx tree (Node x [Node y [Node z []]]) ==path[x,y,z] tree (Node x [Node y [], Node z []]) ==starx [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 (edgex y) ==vertices[x,y] removeEdge x y . removeEdge x y == removeEdge x y removeEdge x y .removeVertexx ==removeVertexx 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 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 (vertexx) ==vertexy 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 constant time.
mergeVertices (constFalse) x == id mergeVertices (== x) y ==replaceVertexx y mergeVerticeseven1 (0 * 2) == 1 * 1 mergeVerticesodd1 (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==emptygmap f (vertexx) ==vertex(f x) gmap f (edgex y) ==edge(f x) (f y) gmapid==idgmap f . gmap g == gmap (f . g)
induce :: (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(n + m) time, assuming that the predicate takes constant time.
induce (constTrue ) x == x induce (constFalse) x ==emptyinduce (/= x) ==removeVertexx induce p . induce q == induce (\x -> p x && q x)isSubgraphOf(induce p x) x == True
induceJust :: Ord a => AdjacencyMap (Maybe a) -> AdjacencyMap a Source #
Graph composition
compose :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap 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 and O(n + m) memory.
composeemptyx ==emptycompose xempty==emptycompose (vertexx) y ==emptycompose x (vertexy) ==emptycompose x (compose y z) == compose (compose x y) z compose x (overlayy z) ==overlay(compose x y) (compose x z) compose (overlayx y) z ==overlay(compose x z) (compose y z) compose (edgex y) (edgey z) ==edgex 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]
box :: (Ord a, Ord b) => AdjacencyMap a -> AdjacencyMap b -> AdjacencyMap (a, b) Source #
Compute the Cartesian product of graphs. Complexity: O((n + m) * log(n)) time and O(n + m) memory.
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 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 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 (overlayy z) ==overlay(box x y) (box x z) box x (vertex()) ~~ x box xempty~~emptytranspose(box x y) == box (transposex) (transposey)vertexCount(box x y) ==vertexCountx *vertexCountyedgeCount(box x y) <=vertexCountx *edgeCounty +edgeCountx *vertexCounty
Relational operations
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.
closureempty==emptyclosure (vertexx) ==edgex x closure (edgex x) ==edgex x closure (edgex y) ==edges[(x,x), (x,y), (y,y)] closure (path$nubxs) ==reflexiveClosure(clique$nubxs) closure ==reflexiveClosure.transitiveClosureclosure ==transitiveClosure.reflexiveClosureclosure . closure == closurepostSetx (closure y) == Set.fromList(reachablex y)
reflexiveClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a Source #
Compute the reflexive closure of a graph by adding a self-loop to every vertex. Complexity: O(n * log(n)) time.
reflexiveClosureempty==emptyreflexiveClosure (vertexx) ==edgex x reflexiveClosure (edgex x) ==edgex x reflexiveClosure (edgex y) ==edges[(x,x), (x,y), (y,y)] reflexiveClosure . reflexiveClosure == reflexiveClosure
symmetricClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a Source #
Compute the symmetric closure of a graph by overlaying it with its own transpose. Complexity: O((n + m) * log(n)) time.
symmetricClosureempty==emptysymmetricClosure (vertexx) ==vertexx symmetricClosure (edgex y) ==edges[(x,y), (y,x)] symmetricClosure x ==overlayx (transposex) symmetricClosure . symmetricClosure == symmetricClosure
transitiveClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a Source #
Compute the transitive closure of a graph. Complexity: O(n * m * log(n)^2) time.
transitiveClosureempty==emptytransitiveClosure (vertexx) ==vertexx transitiveClosure (edgex y) ==edgex y transitiveClosure (path$nubxs) ==clique(nubxs) 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. It should be impossible to create an inconsistent adjacency map, and we use this function in testing.
consistentempty== True consistent (vertexx) == True consistent (overlayx y) == True consistent (connectx y) == True consistent (edgex y) == True consistent (edgesxs) == True consistent (starsxs) == True