Copyright | (c) Andrey Mokhov 2016-2017 |
---|---|
License | MIT (see the file LICENSE) |
Maintainer | andrey.mokhov@gmail.com |
Stability | unstable |
Safe Haskell | None |
Language | Haskell2010 |
This module exposes the implementation of adjacency maps. The API is unstable and unsafe. Where possible use non-internal module Algebra.Graph.IntAdjacencyMap instead.
- newtype IntAdjacencyMap = IntAdjacencyMap {}
- consistent :: IntAdjacencyMap -> Bool
- empty :: IntAdjacencyMap
- vertex :: Int -> IntAdjacencyMap
- overlay :: IntAdjacencyMap -> IntAdjacencyMap -> IntAdjacencyMap
- connect :: IntAdjacencyMap -> IntAdjacencyMap -> IntAdjacencyMap
- vertices :: [Int] -> IntAdjacencyMap
- edges :: [(Int, Int)] -> IntAdjacencyMap
- fromAdjacencyList :: [(Int, [Int])] -> IntAdjacencyMap
- edgeList :: IntAdjacencyMap -> [(Int, Int)]
- adjacencyList :: IntAdjacencyMap -> [(Int, [Int])]
- removeVertex :: Int -> IntAdjacencyMap -> IntAdjacencyMap
- removeEdge :: Int -> Int -> IntAdjacencyMap -> IntAdjacencyMap
- gmap :: (Int -> Int) -> IntAdjacencyMap -> IntAdjacencyMap
- induce :: (Int -> Bool) -> IntAdjacencyMap -> IntAdjacencyMap
Adjacency map
newtype IntAdjacencyMap Source #
The IntAdjacencyMap
data type represents a graph by a map of vertices to
their adjacency sets. We define a law-abiding 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))
The Show
instance is defined using basic graph construction primitives:
show (empty
:: IntAdjacencyMap Int) == "empty"
show (1 :: IntAdjacencyMap Int) == "vertex 1"
show (1 + 2 :: IntAdjacencyMap Int) == "vertices [1,2]"
show (1 * 2 :: IntAdjacencyMap Int) == "edge 1 2"
show (1 * 2 * 3 :: IntAdjacencyMap Int) == "edges [(1,2),(1,3),(2,3)]"
show (1 * 2 + 3 :: IntAdjacencyMap Int) == "graph [1,2,3] [(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 hasempty
as the identity:x * empty == x empty * x == x x * (y * z) == (x * y) * z
connect
distributes overoverlay
: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
hasempty
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.
IntAdjacencyMap | |
|
consistent :: IntAdjacencyMap -> Bool Source #
Check if 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 (vertex
x) == True consistent (overlay
x y) == True consistent (connect
x y) == True consistent (edge
x y) == True consistent (edges
xs) == True consistent (graph
xs ys) == True consistent (fromAdjacencyList
xs) == True
Basic graph construction primitives
empty :: IntAdjacencyMap Source #
Construct the empty graph. Complexity: O(1) time and memory.
isEmpty
empty == TruehasVertex
x empty == FalsevertexCount
empty == 0edgeCount
empty == 0
vertex :: Int -> IntAdjacencyMap Source #
Construct the graph comprising a single isolated vertex. Complexity: O(1) time and memory.
isEmpty
(vertex x) == FalsehasVertex
x (vertex x) == TruehasVertex
1 (vertex 2) == FalsevertexCount
(vertex x) == 1edgeCount
(vertex x) == 0
overlay :: IntAdjacencyMap -> IntAdjacencyMap -> IntAdjacencyMap Source #
Overlay two graphs. This is an idempotent, commutative and associative
operation with the identity empty
.
Complexity: O((n + m) * log(n)) time and O(n + m) memory.
isEmpty
(overlay x y) ==isEmpty
x &&isEmpty
yhasVertex
z (overlay x y) ==hasVertex
z x ||hasVertex
z yvertexCount
(overlay x y) >=vertexCount
xvertexCount
(overlay x y) <=vertexCount
x +vertexCount
yedgeCount
(overlay x y) >=edgeCount
xedgeCount
(overlay x y) <=edgeCount
x +edgeCount
yvertexCount
(overlay 1 2) == 2edgeCount
(overlay 1 2) == 0
connect :: IntAdjacencyMap -> IntAdjacencyMap -> IntAdjacencyMap Source #
Connect two graphs. This is an associative operation with the identity
empty
, which distributes over the 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
yhasVertex
z (connect x y) ==hasVertex
z x ||hasVertex
z yvertexCount
(connect x y) >=vertexCount
xvertexCount
(connect x y) <=vertexCount
x +vertexCount
yedgeCount
(connect x y) >=edgeCount
xedgeCount
(connect x y) >=edgeCount
yedgeCount
(connect x y) >=vertexCount
x *vertexCount
yedgeCount
(connect x y) <=vertexCount
x *vertexCount
y +edgeCount
x +edgeCount
yvertexCount
(connect 1 2) == 2edgeCount
(connect 1 2) == 1
vertices :: [Int] -> IntAdjacencyMap 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 [] ==empty
vertices [x] ==vertex
xhasVertex
x . vertices ==elem
xvertexCount
. vertices ==length
.nub
vertexSet
. vertices == IntSet.fromList
fromAdjacencyList :: [(Int, [Int])] -> IntAdjacencyMap Source #
Construct a graph from an adjacency list. Complexity: O((n + m) * log(n)) time and O(n + m) memory.
fromAdjacencyList [] ==empty
fromAdjacencyList [(x, [])] ==vertex
x fromAdjacencyList [(x, [y])] ==edge
x y fromAdjacencyList .adjacencyList
== idoverlay
(fromAdjacencyList xs) (fromAdjacencyList ys) == fromAdjacencyList (xs ++ ys)
Graph properties
adjacencyList :: IntAdjacencyMap -> [(Int, [Int])] Source #
The sorted adjacency list of a graph. Complexity: O(n + m) time and O(m) memory.
adjacencyListempty
== [] adjacencyList (vertex
x) == [(x, [])] adjacencyList (edge
1 2) == [(1, [2]), (2, [])] adjacencyList (star
2 [3,1]) == [(1, []), (2, [1,3]), (3, [])]fromAdjacencyList
. adjacencyList == id
Graph transformation
removeVertex :: Int -> IntAdjacencyMap -> IntAdjacencyMap Source #
removeEdge :: Int -> Int -> IntAdjacencyMap -> IntAdjacencyMap Source #
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
gmap :: (Int -> Int) -> IntAdjacencyMap -> IntAdjacencyMap 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
IntAdjacencyMap
.
Complexity: O((n + m) * log(n)) time.
gmap fempty
==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)
induce :: (Int -> Bool) -> IntAdjacencyMap -> IntAdjacencyMap 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.
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