Copyright | (c) Andrey Mokhov 2016-2018 |
---|---|
License | MIT (see the file LICENSE) |
Maintainer | andrey.mokhov@gmail.com |
Stability | experimental |
Safe Haskell | None |
Language | Haskell2010 |
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 AdjacencyIntMap
data type, as well as associated
operations and algorithms. AdjacencyIntMap
is an instance of the Graph
type class, which can be used for polymorphic graph construction
and manipulation. See Algebra.Graph.AdjacencyMap for graphs with
non-Int
vertices.
Synopsis
- data AdjacencyIntMap
- adjacencyIntMap :: AdjacencyIntMap -> IntMap IntSet
- empty :: AdjacencyIntMap
- vertex :: Int -> AdjacencyIntMap
- edge :: Int -> Int -> AdjacencyIntMap
- overlay :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap
- connect :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap
- vertices :: [Int] -> AdjacencyIntMap
- edges :: [(Int, Int)] -> AdjacencyIntMap
- overlays :: [AdjacencyIntMap] -> AdjacencyIntMap
- connects :: [AdjacencyIntMap] -> AdjacencyIntMap
- isSubgraphOf :: AdjacencyIntMap -> AdjacencyIntMap -> Bool
- isEmpty :: AdjacencyIntMap -> Bool
- hasVertex :: Int -> AdjacencyIntMap -> Bool
- hasEdge :: Int -> Int -> AdjacencyIntMap -> Bool
- vertexCount :: AdjacencyIntMap -> Int
- edgeCount :: AdjacencyIntMap -> Int
- vertexList :: AdjacencyIntMap -> [Int]
- edgeList :: AdjacencyIntMap -> [(Int, Int)]
- adjacencyList :: AdjacencyIntMap -> [(Int, [Int])]
- vertexIntSet :: AdjacencyIntMap -> IntSet
- edgeSet :: AdjacencyIntMap -> Set (Int, Int)
- preIntSet :: Int -> AdjacencyIntMap -> IntSet
- postIntSet :: Int -> AdjacencyIntMap -> IntSet
- path :: [Int] -> AdjacencyIntMap
- circuit :: [Int] -> AdjacencyIntMap
- clique :: [Int] -> AdjacencyIntMap
- biclique :: [Int] -> [Int] -> AdjacencyIntMap
- star :: Int -> [Int] -> AdjacencyIntMap
- stars :: [(Int, [Int])] -> AdjacencyIntMap
- tree :: Tree Int -> AdjacencyIntMap
- forest :: Forest Int -> AdjacencyIntMap
- removeVertex :: Int -> AdjacencyIntMap -> AdjacencyIntMap
- removeEdge :: Int -> Int -> AdjacencyIntMap -> AdjacencyIntMap
- replaceVertex :: Int -> Int -> AdjacencyIntMap -> AdjacencyIntMap
- mergeVertices :: (Int -> Bool) -> Int -> AdjacencyIntMap -> AdjacencyIntMap
- transpose :: AdjacencyIntMap -> AdjacencyIntMap
- gmap :: (Int -> Int) -> AdjacencyIntMap -> AdjacencyIntMap
- induce :: (Int -> Bool) -> AdjacencyIntMap -> AdjacencyIntMap
- dfsForest :: AdjacencyIntMap -> Forest Int
- dfsForestFrom :: [Int] -> AdjacencyIntMap -> Forest Int
- dfs :: [Int] -> AdjacencyIntMap -> [Int]
- reachable :: Int -> AdjacencyIntMap -> [Int]
- topSort :: AdjacencyIntMap -> Maybe [Int]
- isAcyclic :: AdjacencyIntMap -> Bool
- isDfsForestOf :: Forest Int -> AdjacencyIntMap -> Bool
- isTopSortOf :: [Int] -> AdjacencyIntMap -> Bool
Data structure
data AdjacencyIntMap Source #
The AdjacencyIntMap
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))
The Show
instance is defined using basic graph construction primitives:
show (empty :: AdjacencyIntMap Int) == "empty" show (1 :: AdjacencyIntMap Int) == "vertex 1" show (1 + 2 :: AdjacencyIntMap Int) == "vertices [1,2]" show (1 * 2 :: AdjacencyIntMap Int) == "edge 1 2" show (1 * 2 * 3 :: AdjacencyIntMap Int) == "edges [(1,2),(1,3),(2,3)]" show (1 * 2 + 3 :: AdjacencyIntMap Int) == "overlay (vertex 3) (edge 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.
Instances
adjacencyIntMap :: AdjacencyIntMap -> IntMap IntSet Source #
The adjacency map of the graph: each vertex is associated with a set of its direct successors. Complexity: O(1) time and memory.
adjacencyIntMapempty
== IntMap.empty
adjacencyIntMap (vertex
x) == IntMap.singleton
x IntSet.empty
adjacencyIntMap (edge
1 1) == IntMap.singleton
1 (IntSet.singleton
1) adjacencyIntMap (edge
1 2) == IntMap.fromList
[(1,IntSet.singleton
2), (2,IntSet.empty
)]
Basic graph construction primitives
empty :: AdjacencyIntMap Source #
Construct the empty graph. Complexity: O(1) time and memory.
isEmpty
empty == TruehasVertex
x empty == FalsevertexCount
empty == 0edgeCount
empty == 0
vertex :: Int -> AdjacencyIntMap Source #
Construct the graph comprising a single isolated vertex. Complexity: O(1) time and memory.
isEmpty
(vertex x) == FalsehasVertex
x (vertex x) == TruevertexCount
(vertex x) == 1edgeCount
(vertex x) == 0
edge :: Int -> Int -> AdjacencyIntMap 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) == TrueedgeCount
(edge x y) == 1vertexCount
(edge 1 1) == 1vertexCount
(edge 1 2) == 2
overlay :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap 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) ==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 :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap 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) ==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] -> AdjacencyIntMap 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
vertexIntSet
. vertices == IntSet.fromList
overlays :: [AdjacencyIntMap] -> AdjacencyIntMap Source #
connects :: [AdjacencyIntMap] -> AdjacencyIntMap Source #
Relations on graphs
isSubgraphOf :: AdjacencyIntMap -> AdjacencyIntMap -> 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.
isSubgraphOfempty
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
isEmpty :: AdjacencyIntMap -> Bool Source #
Check if a graph is empty. Complexity: O(1) time.
isEmptyempty
== True isEmpty (overlay
empty
empty
) == True isEmpty (vertex
x) == False isEmpty (removeVertex
x $vertex
x) == True isEmpty (removeEdge
x y $edge
x y) == False
hasVertex :: Int -> AdjacencyIntMap -> Bool Source #
Check if a graph contains a given vertex. Complexity: O(log(n)) time.
hasVertex xempty
== False hasVertex x (vertex
x) == True hasVertex 1 (vertex
2) == False hasVertex x .removeVertex
x == const False
vertexCount :: AdjacencyIntMap -> Int Source #
The number of vertices in a graph. Complexity: O(1) time.
vertexCountempty
== 0 vertexCount (vertex
x) == 1 vertexCount ==length
.vertexList
edgeCount :: AdjacencyIntMap -> Int Source #
vertexList :: AdjacencyIntMap -> [Int] Source #
adjacencyList :: AdjacencyIntMap -> [(Int, [Int])] Source #
vertexIntSet :: AdjacencyIntMap -> IntSet Source #
postIntSet :: Int -> AdjacencyIntMap -> IntSet Source #
Standard families of graphs
path :: [Int] -> AdjacencyIntMap Source #
circuit :: [Int] -> AdjacencyIntMap Source #
clique :: [Int] -> AdjacencyIntMap Source #
stars :: [(Int, [Int])] -> AdjacencyIntMap Source #
The stars formed by overlaying a list of star
s. An inverse of
adjacencyList
.
Complexity: O(L * log(n)) 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 (uncurrystar
) stars .adjacencyList
== idoverlay
(stars xs) (stars ys) == stars (xs ++ ys)
tree :: Tree Int -> AdjacencyIntMap 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 []]]) ==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)]
Graph transformation
removeVertex :: Int -> AdjacencyIntMap -> AdjacencyIntMap Source #
removeEdge :: Int -> Int -> AdjacencyIntMap -> AdjacencyIntMap 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
replaceVertex :: Int -> Int -> AdjacencyIntMap -> AdjacencyIntMap Source #
The function
replaces vertex replaceVertex
x yx
with vertex y
in a
given AdjacencyIntMap
. 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 :: (Int -> Bool) -> Int -> AdjacencyIntMap -> AdjacencyIntMap 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
gmap :: (Int -> Int) -> AdjacencyIntMap -> AdjacencyIntMap 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
AdjacencyIntMap
.
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) -> AdjacencyIntMap -> AdjacencyIntMap 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
Algorithms
dfsForest :: AdjacencyIntMap -> Forest Int Source #
Compute the depth-first search forest of a graph that corresponds to
searching from each of the graph vertices in the Ord
a
order.
dfsForestempty
== []forest
(dfsForest $edge
1 1) ==vertex
1forest
(dfsForest $edge
1 2) ==edge
1 2forest
(dfsForest $edge
2 1) ==vertices
[1,2]isSubgraphOf
(forest
$ dfsForest x) x == TrueisDfsForestOf
(dfsForest x) x == True dfsForest .forest
. dfsForest == dfsForest dfsForest (vertices
vs) == map (\v -> Node v []) (nub
$sort
vs)dfsForestFrom
(vertexList
x) x == dfsForest x dfsForest $ 3 * (1 + 4) * (1 + 5) == [ Node { rootLabel = 1 , subForest = [ Node { rootLabel = 5 , subForest = [] }]} , Node { rootLabel = 3 , subForest = [ Node { rootLabel = 4 , subForest = [] }]}]
dfsForestFrom :: [Int] -> AdjacencyIntMap -> Forest Int Source #
Compute the depth-first search forest of a graph, searching from each of the given vertices in order. Note that the resulting forest does not necessarily span the whole graph, as some vertices may be unreachable.
dfsForestFrom vsempty
== []forest
(dfsForestFrom [1] $edge
1 1) ==vertex
1forest
(dfsForestFrom [1] $edge
1 2) ==edge
1 2forest
(dfsForestFrom [2] $edge
1 2) ==vertex
2forest
(dfsForestFrom [3] $edge
1 2) ==empty
forest
(dfsForestFrom [2,1] $edge
1 2) ==vertices
[1,2]isSubgraphOf
(forest
$ dfsForestFrom vs x) x == TrueisDfsForestOf
(dfsForestFrom (vertexList
x) x) x == True dfsForestFrom (vertexList
x) x ==dfsForest
x dfsForestFrom vs (vertices
vs) == map (\v -> Node v []) (nub
vs) dfsForestFrom [] x == [] dfsForestFrom [1,4] $ 3 * (1 + 4) * (1 + 5) == [ Node { rootLabel = 1 , subForest = [ Node { rootLabel = 5 , subForest = [] } , Node { rootLabel = 4 , subForest = [] }]
dfs :: [Int] -> AdjacencyIntMap -> [Int] Source #
Compute the list of vertices visited by the depth-first search in a graph, when searching from each of the given vertices in order.
dfs vs $empty
== [] dfs [1] $edge
1 1 == [1] dfs [1] $edge
1 2 == [1,2] dfs [2] $edge
1 2 == [2] dfs [3] $edge
1 2 == [] dfs [1,2] $edge
1 2 == [1,2] dfs [2,1] $edge
1 2 == [2,1] dfs [] $ x == [] dfs [1,4] $ 3 * (1 + 4) * (1 + 5) == [1,5,4]isSubgraphOf
(vertices
$ dfs vs x) x == True
reachable :: Int -> AdjacencyIntMap -> [Int] Source #
Compute the list of vertices that are reachable from a given source vertex in a graph. The vertices in the resulting list appear in the depth-first order.
reachable x $empty
== [] reachable 1 $vertex
1 == [1] reachable 1 $vertex
2 == [] reachable 1 $edge
1 1 == [1] reachable 1 $edge
1 2 == [1,2] reachable 4 $path
[1..8] == [4..8] reachable 4 $circuit
[1..8] == [4..8] ++ [1..3] reachable 8 $clique
[8,7..1] == [8] ++ [1..7]isSubgraphOf
(vertices
$ reachable x y) y == True
topSort :: AdjacencyIntMap -> Maybe [Int] Source #
Compute the topological sort of a graph or return Nothing
if the graph
is cyclic.
topSort (1 * 2 + 3 * 1) == Just [3,1,2] topSort (1 * 2 + 2 * 1) == Nothing fmap (flipisTopSortOf
x) (topSort x) /= Just FalseisJust
. topSort ==isAcyclic
isAcyclic :: AdjacencyIntMap -> Bool Source #
Correctness properties
isDfsForestOf :: Forest Int -> AdjacencyIntMap -> Bool Source #
Check if a given forest is a correct depth-first search forest of a graph. The implementation is based on the paper "Depth-First Search and Strong Connectivity in Coq" by François Pottier.
isDfsForestOf []empty
== True isDfsForestOf [] (vertex
1) == False isDfsForestOf [Node 1 []] (vertex
1) == True isDfsForestOf [Node 1 []] (vertex
2) == False isDfsForestOf [Node 1 [], Node 1 []] (vertex
1) == False isDfsForestOf [Node 1 []] (edge
1 1) == True isDfsForestOf [Node 1 []] (edge
1 2) == False isDfsForestOf [Node 1 [], Node 2 []] (edge
1 2) == False isDfsForestOf [Node 2 [], Node 1 []] (edge
1 2) == True isDfsForestOf [Node 1 [Node 2 []]] (edge
1 2) == True isDfsForestOf [Node 1 [], Node 2 []] (vertices
[1,2]) == True isDfsForestOf [Node 2 [], Node 1 []] (vertices
[1,2]) == True isDfsForestOf [Node 1 [Node 2 []]] (vertices
[1,2]) == False isDfsForestOf [Node 1 [Node 2 [Node 3 []]]] (path
[1,2,3]) == True isDfsForestOf [Node 1 [Node 3 [Node 2 []]]] (path
[1,2,3]) == False isDfsForestOf [Node 3 [], Node 1 [Node 2 []]] (path
[1,2,3]) == True isDfsForestOf [Node 2 [Node 3 []], Node 1 []] (path
[1,2,3]) == True isDfsForestOf [Node 1 [], Node 2 [Node 3 []]] (path
[1,2,3]) == False
isTopSortOf :: [Int] -> AdjacencyIntMap -> Bool Source #
Check if a given list of vertices is a correct topological sort of a graph.
isTopSortOf [3,1,2] (1 * 2 + 3 * 1) == True isTopSortOf [1,2,3] (1 * 2 + 3 * 1) == False isTopSortOf [] (1 * 2 + 3 * 1) == False isTopSortOf []empty
== True isTopSortOf [x] (vertex
x) == True isTopSortOf [x] (edge
x x) == False