| 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.NonEmpty.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 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 AdjacencyMap a
- toNonEmpty :: AdjacencyMap a -> Maybe (AdjacencyMap a)
- fromNonEmpty :: AdjacencyMap a -> 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
- vertices1 :: Ord a => NonEmpty a -> AdjacencyMap a
- edges1 :: Ord a => NonEmpty (a, a) -> AdjacencyMap a
- overlays1 :: Ord a => NonEmpty (AdjacencyMap a) -> AdjacencyMap a
- connects1 :: Ord a => NonEmpty (AdjacencyMap a) -> AdjacencyMap a
- isSubgraphOf :: Ord a => AdjacencyMap a -> 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
- vertexList1 :: AdjacencyMap a -> NonEmpty a
- edgeList :: AdjacencyMap a -> [(a, a)]
- vertexSet :: AdjacencyMap a -> Set a
- edgeSet :: Ord a => AdjacencyMap a -> Set (a, a)
- preSet :: Ord a => a -> AdjacencyMap a -> Set a
- postSet :: Ord a => a -> AdjacencyMap a -> Set a
- path1 :: Ord a => NonEmpty a -> AdjacencyMap a
- circuit1 :: Ord a => NonEmpty a -> AdjacencyMap a
- clique1 :: Ord a => NonEmpty a -> AdjacencyMap a
- biclique1 :: Ord a => NonEmpty a -> NonEmpty a -> AdjacencyMap a
- star :: Ord a => a -> [a] -> AdjacencyMap a
- stars1 :: Ord a => NonEmpty (a, [a]) -> AdjacencyMap a
- tree :: Ord a => Tree a -> AdjacencyMap a
- removeVertex1 :: Ord a => a -> AdjacencyMap a -> Maybe (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
- induce1 :: (a -> Bool) -> AdjacencyMap a -> Maybe (AdjacencyMap a)
- induceJust1 :: Ord a => AdjacencyMap (Maybe a) -> Maybe (AdjacencyMap a)
- 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 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:
overlayis commutative, associative and idempotent:x + y == y + x x + (y + z) == (x + y) + z x + x == xconnectis associative: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
connectsatisfies 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:
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 <= yx <= x + y x + y <= x * y
Instances
toNonEmpty :: AdjacencyMap a -> Maybe (AdjacencyMap a) Source #
Convert a possibly empty AdjacencyMap into NonEmpty.AdjacencyMap.
Returns Nothing if the argument is empty.
Complexity: O(1) time, memory and size.
toNonEmptyempty==NothingtoNonEmpty .fromNonEmpty==Just
fromNonEmpty :: AdjacencyMap a -> AdjacencyMap a Source #
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 ==constFalsetoNonEmpty. fromNonEmpty ==Just
Basic graph construction primitives
vertex :: a -> AdjacencyMap a Source #
Construct the graph comprising a single isolated vertex.
hasVertexx (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.
hasVertexz (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).
hasVertexz (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
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] ==vertexxhasVertexx . vertices1 ==elemxvertexCount. vertices1 ==length.nubvertexSet. vertices1 == Set.fromList.toList
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 (overlayx y) == True isSubgraphOf (overlayx y) (connectx y) == True isSubgraphOf (path1xs) (circuit1xs) == 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)
vertexCount :: AdjacencyMap a -> Int Source #
edgeCount :: AdjacencyMap a -> Int Source #
vertexList1 :: AdjacencyMap a -> NonEmpty a Source #
edgeList :: AdjacencyMap a -> [(a, a)] Source #
vertexSet :: AdjacencyMap a -> Set a Source #
Standard families of graphs
star :: Ord a => a -> [a] -> AdjacencyMap a Source #
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, [] )] ==vertexx stars1 [(x, [y])] ==edgex y stars1 [(x, ys )] ==starx ys stars1 ==overlays1.fmap(uncurrystar)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 []) ==vertexx tree (Node x [Node y [Node z []]]) ==path1[x,y,z] tree (Node x [Node y [], Node z []]) ==starx [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 #
removeEdge :: Ord a => a -> a -> AdjacencyMap a -> AdjacencyMap a Source #
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 f (vertexx) ==vertex(f x) gmap f (edgex 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 constant time.
induce1 (constTrue ) x == Just x induce1 (constFalse) x == Nothing induce1 (/= x) ==removeVertex1x 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 (vertexNothing) ==NothinginduceJust1 (edge(Justx)Nothing) ==Just(vertexx) induceJust1 .gmapJust==JustinduceJust1 .gmap(\x -> if p x thenJustx elseNothing) ==induce1p
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 (vertexx) ==edgex x closure (edgex x) ==edgex x closure (edgex y) ==edges1[(x,x), (x,y), (y,y)] closure (path1$nubxs) ==reflexiveClosure(clique1$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.
reflexiveClosure (vertexx) ==edgex x reflexiveClosure (edgex x) ==edgex x reflexiveClosure (edgex y) ==edges1[(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.
symmetricClosure (vertexx) ==vertexx symmetricClosure (edgex y) ==edges1[(x,y), (y,x)] symmetricClosure x ==overlayx (transposex) symmetricClosure . symmetricClosure == symmetricClosure
transitiveClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a Source #
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 (vertexx) == True consistent (overlayx y) == True consistent (connectx y) == True consistent (edgex y) == True consistent (edgesxs) == True consistent (starsxs) == True