algebraic-graphs-0.3: A library for algebraic graph construction and transformation

Copyright (c) Andrey Mokhov 2016-2018 MIT (see the file LICENSE) andrey.mokhov@gmail.com experimental None Haskell2010

Algebra.Graph.HigherKinded.Class

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 core type class Graph, a few graph subclasses, and basic polymorphic graph construction primitives. Functions that cannot be implemented fully polymorphically and require the use of an intermediate data type are not included. For example, to compute the size of a Graph expression you will need to use a concrete data type, such as Algebra.Graph.

See Algebra.Graph.Class for alternative definitions where the core type class is not higher-kinded and permits more instances.

Synopsis

# The core type class

class MonadPlus g => Graph g where Source #

The core type class for constructing algebraic graphs is defined by introducing the connect method to the standard MonadPlus class and reusing the following existing methods:

• The empty method comes from the Alternative class and corresponds to the empty graph. This module simply re-exports it.
• The vertex graph construction primitive is an alias for pure of the Applicative type class.
• Graph overlay is an alias for mplus of the MonadPlus type class.

The Graph type class is characterised by the following minimal set of axioms. In equations we use + and * as convenient shortcuts for overlay and connect, respectively.

• 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

The core type class Graph corresponds to unlabelled directed graphs. Undirected, Reflexive, Transitive and Preorder graphs can be obtained by extending the minimal set of axioms.

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.

Methods

connect :: g a -> g a -> g a Source #

Connect two graphs.

Instances
 Source # Instance detailsDefined in Algebra.Graph.HigherKinded.Class Methodsconnect :: Graph a -> Graph a -> Graph a Source # Source # Instance detailsDefined in Algebra.Graph.HigherKinded.Class Methodsconnect :: Fold a -> Fold a -> Fold a Source #

empty :: Alternative f => f a #

The identity of <|>

vertex :: Graph g => a -> g a Source #

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

overlay :: Graph g => g a -> g a -> g a Source #

Overlay two graphs. An alias for <|>.

# Undirected graphs

class Graph g => Undirected g Source #

The class of undirected graphs that satisfy the following additional axiom.

• connect is commutative:

x * y == y * x

# Reflexive graphs

class Graph g => Reflexive g Source #

The class of reflexive graphs that satisfy the following additional axiom.

• Each vertex has a self-loop:

vertex x == vertex x * vertex x

Or, alternatively, if we remember that vertex is an alias for pure:

pure x == pure x * pure x

Note that by applying the axiom in the reverse direction, one can always remove all self-loops resulting in an irreflexive graph. This type class can therefore be also used in the context of irreflexive graphs.

# Transitive graphs

class Graph g => Transitive g Source #

The class of transitive graphs that satisfy the following additional axiom.

• The closure axiom: graphs with equal transitive closures are equal.

y /= empty ==> x * y + x * z + y * z == x * y + y * z

By repeated application of the axiom one can turn any graph into its transitive closure or transitive reduction.

# Preorders

class (Reflexive g, Transitive g) => Preorder g Source #

The class of preorder graphs that are both reflexive and transitive.

# Basic graph construction primitives

edge :: Graph g => a -> a -> g a Source #

Construct the graph comprising a single edge. Complexity: O(1) time, memory and size.

edge x y               == connect (vertex x) (vertex y)
vertexCount (edge 1 1) == 1
vertexCount (edge 1 2) == 2


vertices :: Graph g => [a] -> g a Source #

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.

vertices []            == empty
vertices [x]           == vertex x
hasVertex x . vertices == elem x
vertexCount . vertices == length . nub
vertexSet   . vertices == Set.fromList


edges :: Graph g => [(a, a)] -> g a Source #

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

edges []      == empty
edges [(x,y)] == edge x y


overlays :: Graph g => [g a] -> g a Source #

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.

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


connects :: Graph g => [g a] -> g a Source #

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.

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


# Relations on graphs

isSubgraphOf :: (Graph g, Eq (g a)) => g a -> g a -> Bool Source #

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

isSubgraphOf x y = overlay x y == y


The complexity therefore depends on the complexity of equality testing of the specific graph instance.

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


# Graph properties

hasEdge :: (Eq (g a), Graph g, Ord a) => a -> a -> g a -> Bool Source #

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

hasEdge x y empty            == False
hasEdge x y (vertex z)       == False
hasEdge x y (edge x y)       == True
hasEdge x y                  == elem (x,y) . edgeList


# Standard families of graphs

path :: Graph g => [a] -> g a Source #

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

path []    == empty
path [x]   == vertex x
path [x,y] == edge x y


circuit :: Graph g => [a] -> g a Source #

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

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


clique :: Graph g => [a] -> g a Source #

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

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)


biclique :: Graph g => [a] -> [a] -> g a Source #

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.

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 :: Graph g => a -> [a] -> g a Source #

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] == edges [(x,y), (x,z)]
star x ys    == connect (vertex x) (vertices ys)


stars :: Graph g => [(a, [a])] -> g a Source #

The stars formed by overlaying a list of stars. An inverse of adjacencyList. Complexity: O(L) 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)


tree :: Graph g => Tree a -> g a Source #

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 []]])                       == 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)]


forest :: Graph g => Forest a -> g a Source #

The forest graph constructed from a given Forest data structure. Complexity: O(F) time, memory and size, where F is the size of the given forest (i.e. the number of vertices in the forest).

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


mesh :: Graph g => [a] -> [b] -> g (a, b) Source #

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.

mesh xs     []   == empty
mesh []     ys   == empty
mesh [x]    [y]  == vertex (x, y)
mesh xs     ys   == box (path xs) (path ys)
mesh [1..3] "ab" == edges [ ((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')) ]


torus :: Graph g => [a] -> [b] -> g (a, b) Source #

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.

torus xs    []   == empty
torus []    ys   == empty
torus [x]   [y]  == edge (x,y) (x,y)
torus xs    ys   == box (circuit xs) (circuit ys)
torus [1,2] "ab" == edges [ ((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')) ]


deBruijn :: Graph g => Int -> [a] -> g [a] Source #

Construct a De Bruijn graph of a given non-negative dimension using symbols from a given alphabet. Complexity: O(A^(D + 1)) time, memory and size, where A is the size of the alphabet and D is the dimension of the graph.

          deBruijn 0 xs               == edge [] []
n > 0 ==> deBruijn n []               == empty
deBruijn 1 [0,1]            == edges [ (,), (,), (,), (,) ]
deBruijn 2 "0"              == edge "00" "00"
deBruijn 2 "01"             == edges [ ("00","00"), ("00","01"), ("01","10"), ("01","11")
, ("10","00"), ("10","01"), ("11","10"), ("11","11") ]
transpose   (deBruijn n xs) == fmap reverse $deBruijn n xs vertexCount (deBruijn n xs) == (length$ nub xs)^n
n > 0 ==> edgeCount   (deBruijn n xs) == (length $nub xs)^(n + 1)  # Graph transformation removeVertex :: (Eq a, Graph g) => a -> g a -> g a Source # Remove a vertex from a given graph. Complexity: O(s) time, memory and size. removeVertex x (vertex x) == empty removeVertex 1 (vertex 2) == vertex 2 removeVertex x (edge x x) == empty removeVertex 1 (edge 1 2) == vertex 2 removeVertex x . removeVertex x == removeVertex x  replaceVertex :: (Eq a, Graph g) => a -> a -> g a -> g a Source # The function replaceVertex x y replaces vertex x with vertex y in a given Graph. If y already exists, x and y will be merged. Complexity: O(s) time, memory and size. replaceVertex x x == id replaceVertex x y (vertex x) == vertex y replaceVertex x y == mergeVertices (== x) y  mergeVertices :: Graph g => (a -> Bool) -> a -> g a -> g a Source # Merge vertices satisfying a given predicate into a given vertex. Complexity: O(s) time, memory and size, assuming that the predicate takes O(1) to be evaluated. mergeVertices (const False) x == id mergeVertices (== x) y == replaceVertex x y mergeVertices even 1 (0 * 2) == 1 * 1 mergeVertices odd 1 (3 + 4 * 5) == 4 * 1  splitVertex :: (Eq a, Graph g) => a -> [a] -> g a -> g a Source # Split a vertex into a list of vertices with the same connectivity. Complexity: O(s + k * L) time, memory and size, where k is the number of occurrences of the vertex in the expression and L is the length of the given list. splitVertex x [] == removeVertex x splitVertex x [x] == id splitVertex x [y] == replaceVertex x y splitVertex 1 [0,1]$ 1 * (2 + 3) == (0 + 1) * (2 + 3)


induce :: Graph g => (a -> Bool) -> g a -> g a Source #

Construct the induced subgraph of a given graph by removing the vertices that do not satisfy a given predicate. Complexity: O(s) time, memory and size, 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