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-- | Static and Dynamic Inductive Graphs
module Data.Graph.Inductive.Graph (
-- * General Type Defintions
-- ** Node and Edge Types
Node,LNode,UNode,
Edge,LEdge,UEdge,
-- ** Types Supporting Inductive Graph View
Adj,Context,MContext,Decomp,GDecomp,UDecomp,
Path,LPath(..),UPath,
-- * Graph Type Classes
-- | We define two graph classes:
--
-- Graph: static, decomposable graphs.
-- Static means that a graph itself cannot be changed
--
-- DynGraph: dynamic, extensible graphs.
-- Dynamic graphs inherit all operations from static graphs
-- but also offer operations to extend and change graphs.
--
-- Each class contains in addition to its essential operations those
-- derived operations that might be overwritten by a more efficient
-- implementation in an instance definition.
--
-- Note that labNodes is essentially needed because the default definition
-- for matchAny is based on it: we need some node from the graph to define
-- matchAny in terms of match. Alternatively, we could have made matchAny
-- essential and have labNodes defined in terms of ufold and matchAny.
-- However, in general, labNodes seems to be (at least) as easy to define
-- as matchAny. We have chosen labNodes instead of the function nodes since
-- nodes can be easily derived from labNodes, but not vice versa.
Graph(..),
DynGraph(..),
-- * Operations
-- ** Graph Folds and Maps
ufold,gmap,nmap,emap,
-- ** Graph Projection
nodes,edges,newNodes,gelem,
-- ** Graph Construction and Destruction
insNode,insEdge,delNode,delEdge,delLEdge,
insNodes,insEdges,delNodes,delEdges,
buildGr,mkUGraph,
-- ** Graph Inspection
context,lab,neighbors,
suc,pre,lsuc,lpre,
out,inn,outdeg,indeg,deg,
equal,
-- ** Context Inspection
node',lab',labNode',neighbors',
suc',pre',lpre',lsuc',
out',inn',outdeg',indeg',deg',
) where
import Data.List (sortBy)
{- Signatures:
-- basic operations
empty :: Graph gr => gr a b
isEmpty :: Graph gr => gr a b -> Bool
match :: Graph gr => Node -> gr a b -> Decomp gr a b
mkGraph :: Graph gr => [LNode a] -> [LEdge b] -> gr a b
(&) :: DynGraph gr => Context a b -> gr a b -> gr a b
-- graph folds and maps
ufold :: Graph gr => ((Context a b) -> c -> c) -> c -> gr a b -> c
gmap :: Graph gr => (Context a b -> Context c d) -> gr a b -> gr c d
nmap :: Graph gr => (a -> c) -> gr a b -> gr c b
emap :: Graph gr => (b -> c) -> gr a b -> gr a c
-- graph projection
matchAny :: Graph gr => gr a b -> GDecomp g a b
nodes :: Graph gr => gr a b -> [Node]
edges :: Graph gr => gr a b -> [Edge]
labNodes :: Graph gr => gr a b -> [LNode a]
labEdges :: Graph gr => gr a b -> [LEdge b]
newNodes :: Graph gr => Int -> gr a b -> [Node]
noNodes :: Graph gr => gr a b -> Int
nodeRange :: Graph gr => gr a b -> (Node,Node)
gelem :: Graph gr => Node -> gr a b -> Bool
-- graph construction & destruction
insNode :: DynGraph gr => LNode a -> gr a b -> gr a b
insEdge :: DynGraph gr => LEdge b -> gr a b -> gr a b
delNode :: Graph gr => Node -> gr a b -> gr a b
delEdge :: DynGraph gr => Edge -> gr a b -> gr a b
delLEdge :: (DynGraph gr, Eq b) =>
LEdge b -> gr a b -> gr a b
insNodes :: DynGraph gr => [LNode a] -> gr a b -> gr a b
insEdges :: DynGraph gr => [LEdge b] -> gr a b -> gr a b
delNodes :: Graph gr => [Node] -> gr a b -> gr a b
delEdges :: DynGraph gr => [Edge] -> gr a b -> gr a b
buildGr :: DynGraph gr => [Context a b] -> gr a b
mkUGraph :: DynGraph gr => [Node] -> [Edge] -> gr () ()
-- graph inspection
context :: Graph gr => gr a b -> Node -> Context a b
lab :: Graph gr => gr a b -> Node -> Maybe a
neighbors :: Graph gr => gr a b -> Node -> [Node]
suc :: Graph gr => gr a b -> Node -> [Node]
pre :: Graph gr => gr a b -> Node -> [Node]
lsuc :: Graph gr => gr a b -> Node -> [(Node,b)]
lpre :: Graph gr => gr a b -> Node -> [(Node,b)]
out :: Graph gr => gr a b -> Node -> [LEdge b]
inn :: Graph gr => gr a b -> Node -> [LEdge b]
outdeg :: Graph gr => gr a b -> Node -> Int
indeg :: Graph gr => gr a b -> Node -> Int
deg :: Graph gr => gr a b -> Node -> Int
-- context inspection
node' :: Context a b -> Node
lab' :: Context a b -> a
labNode' :: Context a b -> LNode a
neighbors' :: Context a b -> [Node]
suc' :: Context a b -> [Node]
pre' :: Context a b -> [Node]
lpre' :: Context a b -> [(Node,b)]
lsuc' :: Context a b -> [(Node,b)]
out' :: Context a b -> [LEdge b]
inn' :: Context a b -> [LEdge b]
outdeg' :: Context a b -> Int
indeg' :: Context a b -> Int
deg' :: Context a b -> Int
-}
-- | Unlabeled node
type Node = Int
-- | Labeled node
type LNode a = (Node,a)
-- | Quasi-unlabeled node
type UNode = LNode ()
-- | Unlabeled edge
type Edge = (Node,Node)
-- | Labeled edge
type LEdge b = (Node,Node,b)
-- | Quasi-unlabeled edge
type UEdge = LEdge ()
-- | Unlabeled path
type Path = [Node]
-- | Labeled path
newtype LPath a = LP [LNode a]
instance Show a => Show (LPath a) where
show (LP xs) = show xs
-- | Quasi-unlabeled path
type UPath = [UNode]
-- | Labeled links to or from a 'Node'.
type Adj b = [(b,Node)]
-- | Links to the 'Node', the 'Node' itself, a label, links from the 'Node'.
type Context a b = (Adj b,Node,a,Adj b) -- Context a b "=" Context' a b "+" Node
type MContext a b = Maybe (Context a b)
-- | 'Graph' decomposition - the context removed from a 'Graph', and the rest
-- of the 'Graph'.
type Decomp g a b = (MContext a b,g a b)
-- | The same as 'Decomp', only more sure of itself.
type GDecomp g a b = (Context a b,g a b)
-- | Unlabeled context.
type UContext = ([Node],Node,[Node])
-- | Unlabeled decomposition.
type UDecomp g = (Maybe UContext,g)
-- | Minimum implementation: 'empty', 'isEmpty', 'match', 'mkGraph', 'labNodes'
class Graph gr where
-- essential operations
-- | An empty 'Graph'.
empty :: gr a b
-- | True if the given 'Graph' is empty.
isEmpty :: gr a b -> Bool
-- | Decompose a 'Graph' into the 'MContext' found for the given node and the
-- remaining 'Graph'.
match :: Node -> gr a b -> Decomp gr a b
-- | Create a 'Graph' from the list of 'LNode's and 'LEdge's.
mkGraph :: [LNode a] -> [LEdge b] -> gr a b
-- | A list of all 'LNode's in the 'Graph'.
labNodes :: gr a b -> [LNode a]
-- derived operations
-- | Decompose a graph into the 'Context' for an arbitrarily-chosen 'Node'
-- and the remaining 'Graph'.
matchAny :: gr a b -> GDecomp gr a b
-- | The number of 'Node's in a 'Graph'.
noNodes :: gr a b -> Int
-- | The minimum and maximum 'Node' in a 'Graph'.
nodeRange :: gr a b -> (Node,Node)
-- | A list of all 'LEdge's in the 'Graph'.
labEdges :: gr a b -> [LEdge b]
-- default implementation of derived operations
matchAny g = case labNodes g of
[] -> error "Match Exception, Empty Graph"
(v,_):_ -> (c,g') where (Just c,g') = match v g
noNodes = length . labNodes
nodeRange g = (minimum vs,maximum vs) where vs = map fst (labNodes g)
labEdges = ufold (\(_,v,_,s)->((map (\(l,w)->(v,w,l)) s)++)) []
class Graph gr => DynGraph gr where
-- | Merge the 'Context' into the 'DynGraph'.
(&) :: Context a b -> gr a b -> gr a b
-- | Fold a function over the graph.
ufold :: Graph gr => ((Context a b) -> c -> c) -> c -> gr a b -> c
ufold f u g | isEmpty g = u
| otherwise = f c (ufold f u g')
where (c,g') = matchAny g
-- | Map a function over the graph.
gmap :: DynGraph gr => (Context a b -> Context c d) -> gr a b -> gr c d
gmap f = ufold (\c->(f c&)) empty
-- | Map a function over the 'Node' labels in a graph.
nmap :: DynGraph gr => (a -> c) -> gr a b -> gr c b
nmap f = gmap (\(p,v,l,s)->(p,v,f l,s))
-- | Map a function over the 'Edge' labels in a graph.
emap :: DynGraph gr => (b -> c) -> gr a b -> gr a c
emap f = gmap (\(p,v,l,s)->(map1 f p,v,l,map1 f s))
where map1 g = map (\(l,v)->(g l,v))
-- | List all 'Node's in the 'Graph'.
nodes :: Graph gr => gr a b -> [Node]
nodes = map fst . labNodes
-- | List all 'Edge's in the 'Graph'.
edges :: Graph gr => gr a b -> [Edge]
edges = map (\(v,w,_)->(v,w)) . labEdges
-- | List N available 'Node's, i.e. 'Node's that are not used in the 'Graph'.
newNodes :: Graph gr => Int -> gr a b -> [Node]
newNodes i g = [n+1..n+i] where (_,n) = nodeRange g
-- | 'True' if the 'Node' is present in the 'Graph'.
gelem :: Graph gr => Node -> gr a b -> Bool
gelem v g = case match v g of {(Just _,_) -> True; _ -> False}
-- | Insert a 'LNode' into the 'Graph'.
insNode :: DynGraph gr => LNode a -> gr a b -> gr a b
insNode (v,l) = (([],v,l,[])&)
-- | Insert a 'LEdge' into the 'Graph'.
insEdge :: DynGraph gr => LEdge b -> gr a b -> gr a b
insEdge (v,w,l) g = (pr,v,la,(l,w):su) & g'
where (Just (pr,_,la,su),g') = match v g
-- | Remove a 'Node' from the 'Graph'.
delNode :: Graph gr => Node -> gr a b -> gr a b
delNode v = delNodes [v]
-- | Remove an 'Edge' from the 'Graph'.
delEdge :: DynGraph gr => Edge -> gr a b -> gr a b
delEdge (v,w) g = case match v g of
(Nothing,_) -> g
(Just (p,v',l,s),g') -> (p,v',l,filter ((/=w).snd) s) & g'
-- | Remove an 'LEdge' from the 'Graph'.
delLEdge :: (DynGraph gr, Eq b) => LEdge b -> gr a b -> gr a b
delLEdge (v,w,b) g = case match v g of
(Nothing,_) -> g
(Just (p,v',l,s),g') -> (p,v',l,filter (\(x,n) -> x /= b || n /= w) s) & g'
-- | Insert multiple 'LNode's into the 'Graph'.
insNodes :: DynGraph gr => [LNode a] -> gr a b -> gr a b
insNodes vs g = foldr insNode g vs
-- | Insert multiple 'LEdge's into the 'Graph'.
insEdges :: DynGraph gr => [LEdge b] -> gr a b -> gr a b
insEdges es g = foldr insEdge g es
-- | Remove multiple 'Node's from the 'Graph'.
delNodes :: Graph gr => [Node] -> gr a b -> gr a b
delNodes [] g = g
delNodes (v:vs) g = delNodes vs (snd (match v g))
-- | Remove multiple 'Edge's from the 'Graph'.
delEdges :: DynGraph gr => [Edge] -> gr a b -> gr a b
delEdges es g = foldr delEdge g es
-- | Build a 'Graph' from a list of 'Context's.
buildGr :: DynGraph gr => [Context a b] -> gr a b
buildGr = foldr (&) empty
-- mkGraph :: DynGraph gr => [LNode a] -> [LEdge b] -> gr a b
-- mkGraph vs es = (insEdges es . insNodes vs) empty
-- | Build a quasi-unlabeled 'Graph'.
mkUGraph :: Graph gr => [Node] -> [Edge] -> gr () ()
mkUGraph vs es = mkGraph (labUNodes vs) (labUEdges es)
labUEdges = map (\(v,w)->(v,w,()))
labUNodes = map (\v->(v,()))
-- | Find the context for the given 'Node'. Causes an error if the 'Node' is
-- not present in the 'Graph'.
context :: Graph gr => gr a b -> Node -> Context a b
context g v = case match v g of
(Nothing,_) -> error ("Match Exception, Node: "++show v)
(Just c,_) -> c
-- | Find the label for a 'Node'.
lab :: Graph gr => gr a b -> Node -> Maybe a
lab g v = fst (match v g) >>= return.lab'
-- | Find the neighbors for a 'Node'.
neighbors :: Graph gr => gr a b -> Node -> [Node]
neighbors = (\(p,_,_,s) -> map snd (p++s)) .: context
-- | Find all 'Node's that have a link from the given 'Node'.
suc :: Graph gr => gr a b -> Node -> [Node]
suc = map snd .: context4
-- | Find all 'Node's that link to to the given 'Node'.
pre :: Graph gr => gr a b -> Node -> [Node]
pre = map snd .: context1
-- | Find all 'Node's that are linked from the given 'Node' and the label of
-- each link.
lsuc :: Graph gr => gr a b -> Node -> [(Node,b)]
lsuc = map flip2 .: context4
-- | Find all 'Node's that link to the given 'Node' and the label of each link.
lpre :: Graph gr => gr a b -> Node -> [(Node,b)]
lpre = map flip2 .: context1
-- | Find all outward-bound 'LEdge's for the given 'Node'.
out :: Graph gr => gr a b -> Node -> [LEdge b]
out g v = map (\(l,w)->(v,w,l)) (context4 g v)
-- | Find all inward-bound 'LEdge's for the given 'Node'.
inn :: Graph gr => gr a b -> Node -> [LEdge b]
inn g v = map (\(l,w)->(w,v,l)) (context1 g v)
-- | The outward-bound degree of the 'Node'.
outdeg :: Graph gr => gr a b -> Node -> Int
outdeg = length .: context4
-- | The inward-bound degree of the 'Node'.
indeg :: Graph gr => gr a b -> Node -> Int
indeg = length .: context1
-- | The degree of the 'Node'.
deg :: Graph gr => gr a b -> Node -> Int
deg = (\(p,_,_,s) -> length p+length s) .: context
-- | The 'Node' in a 'Context'.
node' :: Context a b -> Node
node' (_,v,_,_) = v
-- | The label in a 'Context'.
lab' :: Context a b -> a
lab' (_,_,l,_) = l
-- | The 'LNode' from a 'Context'.
labNode' :: Context a b -> LNode a
labNode' (_,v,l,_) = (v,l)
-- | All 'Node's linked to or from in a 'Context'.
neighbors' :: Context a b -> [Node]
neighbors' (p,_,_,s) = map snd p++map snd s
-- | All 'Node's linked to in a 'Context'.
suc' :: Context a b -> [Node]
suc' (_,_,_,s) = map snd s
-- | All 'Node's linked from in a 'Context'.
pre' :: Context a b -> [Node]
pre' (p,_,_,_) = map snd p
-- | All 'Node's linked from in a 'Context', and the label of the links.
lpre' :: Context a b -> [(Node,b)]
lpre' (p,_,_,_) = map flip2 p
-- | All 'Node's linked from in a 'Context', and the label of the links.
lsuc' :: Context a b -> [(Node,b)]
lsuc' (_,_,_,s) = map flip2 s
-- | All outward-directed 'LEdge's in a 'Context'.
out' :: Context a b -> [LEdge b]
out' (_,v,_,s) = map (\(l,w)->(v,w,l)) s
-- | All inward-directed 'LEdge's in a 'Context'.
inn' :: Context a b -> [LEdge b]
inn' (p,v,_,_) = map (\(l,w)->(w,v,l)) p
-- | The outward degree of a 'Context'.
outdeg' :: Context a b -> Int
outdeg' (_,_,_,s) = length s
-- | The inward degree of a 'Context'.
indeg' :: Context a b -> Int
indeg' (p,_,_,_) = length p
-- | The degree of a 'Context'.
deg' :: Context a b -> Int
deg' (p,_,_,s) = length p+length s
-- graph equality
--
nodeComp :: Eq b => LNode b -> LNode b -> Ordering
nodeComp n@(v,_) n'@(w,_) | n == n' = EQ
| v<w = LT
| otherwise = GT
slabNodes :: (Eq a,Graph gr) => gr a b -> [LNode a]
slabNodes = sortBy nodeComp . labNodes
edgeComp :: Eq b => LEdge b -> LEdge b -> Ordering
edgeComp e@(v,w,_) e'@(x,y,_) | e == e' = EQ
| v<x || (v==x && w<y) = LT
| otherwise = GT
slabEdges :: (Eq b,Graph gr) => gr a b -> [LEdge b]
slabEdges = sortBy edgeComp . labEdges
-- instance (Eq a,Eq b,Graph gr) => Eq (gr a b) where
-- g == g' = slabNodes g == slabNodes g' && slabEdges g == slabEdges g'
equal :: (Eq a,Eq b,Graph gr) => gr a b -> gr a b -> Bool
equal g g' = slabNodes g == slabNodes g' && slabEdges g == slabEdges g'
----------------------------------------------------------------------
-- UTILITIES
----------------------------------------------------------------------
-- auxiliary functions used in the implementation of the
-- derived class members
--
(.:) :: (c -> d) -> (a -> b -> c) -> (a -> b -> d)
-- f .: g = \x y->f (g x y)
-- f .: g = (f .) . g
-- (.:) f = ((f .) .)
-- (.:) = (.) (.) (.)
(.:) = (.) . (.)
fst4 (x,_,_,_) = x
{- not used
snd4 (_,x,_,_) = x
thd4 (_,_,x,_) = x
-}
fth4 (_,_,_,x) = x
{- not used
fst3 (x,_,_) = x
snd3 (_,x,_) = x
thd3 (_,_,x) = x
-}
flip2 (x,y) = (y,x)
-- projecting on context elements
--
-- context1 g v = fst4 (contextP g v)
context1 :: Graph gr => gr a b -> Node -> Adj b
{- not used
context2 :: Graph gr => gr a b -> Node -> Node
context3 :: Graph gr => gr a b -> Node -> a
-}
context4 :: Graph gr => gr a b -> Node -> Adj b
context1 = fst4 .: context
{- not used
context2 = snd4 .: context
context3 = thd4 .: context
-}
context4 = fth4 .: context
|