{-# LANGUAGE TemplateHaskell #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE OverloadedStrings #-}
--------------------------------------------------------------------------------
-- |
-- Module      :  Data.PlanarGraph
-- Copyright   :  (C) Frank Staals
-- License     :  see the LICENSE file
-- Maintainer  :  Frank Staals
--
-- Data type for representing connected planar graphs
--------------------------------------------------------------------------------
module Data.PlanarGraph( -- $setup
                         -- * The Planar Graph type
                         PlanarGraph
                       , embedding, vertexData, dartData, faceData, rawDartData
                       , edgeData

                       , World(..)
                       , DualOf

                       -- * Representing edges: Arcs and Darts
                       , Arc(..)
                       , Direction(..), rev

                       , Dart(..), arc, direction
                       , twin, isPositive

                       -- * Vertices

                       , VertexId(..), VertexId'

                       -- * Building a planar graph

                       , planarGraph, planarGraph', fromAdjacencyLists
                       , toAdjacencyLists
                       , buildFromJSON

                       -- * Quering a planar graph

                       , numVertices, numDarts, numEdges, numFaces
                       , darts', darts, edges', edges, vertices', vertices, faces', faces
                       , traverseVertices, traverseDarts, traverseFaces

                       , tailOf, headOf, endPoints
                       , incidentEdges, incomingEdges, outgoingEdges, neighboursOf
                       , nextIncidentEdge, prevIncidentEdge

                       -- * Associated Data

                       , HasDataOf(..), endPointDataOf, endPointData

                       , dual

                       -- * Faces

                       , FaceId(..), FaceId'
                       , leftFace, rightFace
                       , boundaryDart, boundary, boundary', boundaryVertices
                       , nextEdge, prevEdge

                       -- * Edge Oracle

                       , EdgeOracle
                       , edgeOracle, buildEdgeOracle
                       , findEdge
                       , hasEdge, findDart

                       , allDarts
                       ) where


import           Control.Applicative (Alternative(..))
import           Control.DeepSeq
import           Control.Lens hiding ((.=))
import           Control.Monad.ST (ST)
import           Control.Monad.State.Strict
import           Data.Aeson
import           Data.Bifunctor
import           Data.Bitraversable
import           Data.Ext
import qualified Data.Foldable as F
import           Data.Maybe (catMaybes, isJust, fromJust, fromMaybe)
import           Data.Permutation
import           Data.Traversable (fmapDefault,foldMapDefault)
import           Data.Type.Equality (gcastWith, (:~:)(..))
import qualified Data.Vector as V
import qualified Data.Vector.Generic as GV
import qualified Data.Vector.Mutable as MV
import qualified Data.Vector.Unboxed as UV
import qualified Data.Vector.Unboxed.Mutable as UMV
import           GHC.Generics (Generic)
import           Unsafe.Coerce (unsafeCoerce)
-- import           Data.Yaml.Util


--------------------------------------------------------------------------------
-- $setup
-- >>> :{
-- let dart i s = Dart (Arc i) (read s)
--     (aA:aB:aC:aD:aE:aG:_) = take 6 [Arc 0..]
--     myGraph :: PlanarGraph () Primal () String ()
--     myGraph = planarGraph [ [ (Dart aA Negative, "a-")
--                             , (Dart aC Positive, "c+")
--                             , (Dart aB Positive, "b+")
--                             , (Dart aA Positive, "a+")
--                             ]
--                           , [ (Dart aE Negative, "e-")
--                             , (Dart aB Negative, "b-")
--                             , (Dart aD Negative, "d-")
--                             , (Dart aG Positive, "g+")
--                             ]
--                           , [ (Dart aE Positive, "e+")
--                             , (Dart aD Positive, "d+")
--                             , (Dart aC Negative, "c-")
--                             ]
--                           , [ (Dart aG Negative, "g-")
--                             ]
--                           ]
-- :}
--
--
-- This represents the following graph. Note that the graph is undirected, the
-- arrows are just to indicate what the Positive direction of the darts is.
--
-- ![myGraph](docs/Data/PlanarGraph/testG.png)

--------------------------------------------------------------------------------

-- | An Arc is a directed edge in a planar graph. The type s is used to tie
-- this arc to a particular graph.
newtype Arc s = Arc { _unArc :: Int } deriving (Eq,Ord,Enum,Bounded,Generic,NFData)

instance Show (Arc s) where
  show (Arc i) = "Arc " ++ show i

-- | Darts have a direction which is either Positive or Negative (shown as +1
-- or -1, respectively).
data Direction = Negative | Positive deriving (Eq,Ord,Bounded,Enum,Generic)

instance NFData Direction

instance Show Direction where
  show Positive = "+1"
  show Negative = "-1"

instance Read Direction where
  readsPrec _ "-1" = [(Negative,"")]
  readsPrec _ "+1" = [(Positive,"")]
  readsPrec _ _    = []

-- | Reverse the direcion
rev          :: Direction -> Direction
rev Negative = Positive
rev Positive = Negative

-- | A dart represents a bi-directed edge. I.e. a dart has a direction, however
-- the dart of the oposite direction is always present in the planar graph as
-- well.
data Dart s = Dart { _arc       :: !(Arc s)
                   , _direction :: !Direction
                   } deriving (Eq,Ord,Generic)
makeLenses ''Dart

instance NFData (Dart s)

instance Show (Dart s) where
  show (Dart a d) = "Dart (" ++ show a ++ ") " ++ show d

-- | Get the twin of this dart (edge)
--
-- >>> twin (dart 0 "+1")
-- Dart (Arc 0) -1
-- >>> twin (dart 0 "-1")
-- Dart (Arc 0) +1
twin            :: Dart s -> Dart s
twin (Dart a d) = Dart a (rev d)

-- | test if a dart is Positive
isPositive   :: Dart s -> Bool
isPositive d = d^.direction == Positive


instance Enum (Dart s) where
  toEnum x
    | even x    = Dart (Arc $ x `div` 2) Positive
    | otherwise = Dart (Arc $ x `div` 2) Negative
  -- get the back edge by adding one

  fromEnum (Dart (Arc i) d) = case d of
                                Positive -> 2*i
                                Negative -> 2*i + 1


-- | Enumerates all darts such that
-- allDarts !! i = d   <=> i == fromEnum d
allDarts :: [Dart s]
allDarts = concatMap (\a -> [Dart a Positive, Dart a Negative]) [Arc 0..]

-- | The world in which the graph lives
data World = Primal | Dual deriving (Show,Eq)

-- | We can take the dual of a world. For the Primal this gives us the Dual,
-- for the Dual this gives us the Primal.
type family DualOf (sp :: World) where
  DualOf Primal = Dual
  DualOf Dual   = Primal

-- | The Dual of the Dual is the Primal.
dualDualIdentity :: forall w. DualOf (DualOf w) :~: w
dualDualIdentity = unsafeCoerce Refl
          -- manual proof:
          --    DualOf (DualOf Primal) = Primal
          --    DualOf (DualOf Dual)   = Dual


-- | A vertex in a planar graph. A vertex is tied to a particular planar graph
-- by the phantom type s, and to a particular world w.
newtype VertexId s (w :: World) = VertexId { _unVertexId :: Int }
                                deriving (Eq,Ord,Enum,ToJSON,FromJSON,Generic,NFData)
-- VertexId's are in the range 0...|orbits|-1

-- | Shorthand for vertices in the primal.
type VertexId' s = VertexId s Primal

unVertexId :: Getter (VertexId s w) Int
unVertexId = to _unVertexId

instance Show (VertexId s w) where
  show (VertexId i) = "VertexId " ++ show i

--------------------------------------------------------------------------------
-- * The graph type itself

-- | A *connected* Planar graph with bidirected edges. I.e. the edges (darts) are
-- directed, however, for every directed edge, the edge in the oposite
-- direction is also in the graph.
--
-- The types v, e, and f are the are the types of the data associated with the
-- vertices, edges, and faces, respectively.
--
-- The orbits in the embedding are assumed to be in counterclockwise
-- order. Therefore, every dart directly bounds the face to its right.
data PlanarGraph s (w :: World) v e f = PlanarGraph { _embedding   :: Permutation (Dart s)
                                                    , _vertexData  :: V.Vector v
                                                    , _rawDartData :: V.Vector e
                                                    , _faceData    :: V.Vector f
                                                    , _dual        :: PlanarGraph s (DualOf w) f e v
                                                    } deriving (Generic)

instance (Show v, Show e, Show f) => Show (PlanarGraph s w v e f) where
  show (PlanarGraph e v r f _) = unwords [ "PlanarGraph"
                                         , "embedding =", show e
                                         , ", vertexData =", show v
                                         , ", rawDartData =", show r
                                         , ", faceData =", show f
                                         ]

instance (Eq v, Eq e, Eq f) => Eq (PlanarGraph s w v e f) where
  (PlanarGraph e v r f _) == (PlanarGraph e' v' r' f' _) =  e == e' && v == v'
                                                         && r == r' && f == f'


instance (ToJSON v, ToJSON e, ToJSON f)
         => ToJSON (PlanarGraph s w v e f) where
  toJSON     = object . encodeJSON
  toEncoding = pairs . mconcat . encodeJSON



-- | Enclodes the planar graph as a JSON object.  note that every face is
-- stored together with a dart bounding the face. (and so that the face lies to
-- its right. Otherwise we cannot reconstruct the face data appropriately.
encodeJSON   :: (ToJSON f, ToJSON e, ToJSON v, KeyValue t)
             => PlanarGraph s w v e f -> [t]
encodeJSON g = [ "vertices"    .= ((\(v,x) -> v :+ x)             <$> vertices g)
               , "darts"       .= ((\(e,x) -> endPoints e g :+ x) <$> darts g)
               , "faces"       .= ((\(f,x) -> f :+ x)             <$> faces g)
               , "adjacencies" .= toAdjacencyLists g
               ]
  -- where
  --   faces'' = withBoundaryDart  <$> faces g
  --   withBoundaryDart (f,x) = let d = boundaryDart f g
  --                            in (f, endPoints d g, x)

instance (FromJSON v, FromJSON e, FromJSON f)
         => FromJSON (PlanarGraph s Primal v e f) where
  parseJSON = withObject "" $ \v -> buildFromJSON <$> v .: "vertices"
                                                  <*> v .: "darts"
                                                  <*> v .: "faces"
                                                  <*> v .: "adjacencies"


-- | Helper function to build the graph from JSON data
--
-- running time: \(O(n)\)
buildFromJSON             :: V.Vector (VertexId' s :+ v)
                          -> V.Vector ((VertexId' s, VertexId' s) :+ e)
                          -> V.Vector (FaceId' s :+ f)
                          -> [(VertexId' s, V.Vector (VertexId' s))]
                          -> PlanarGraph s Primal v e f
buildFromJSON vs es fs as = g&vertexData .~ reorder vs _unVertexId
                             &dartData   .~ ds
                             &faceData   .~ reorder fs (_unVertexId._unFaceId)
  where
    g = fromAdjacencyLists as
    oracle = edgeOracle g
    findEdge' (u,v) = fromJust $ findDart u v oracle

    ds = es&traverse %~ \(e:+x) -> (findEdge' e,x)
    -- TODO: do we have enough infor to reconstruct the face info?

    -- make sure we order the data values appropriately
    reorder v f = V.create $ do
                               v' <- MV.new (V.length v)
                               forM_ v $ \(i :+ x) ->
                                 MV.write v' (f i) x
                               pure v'






-- ** lenses and getters

-- | Get the embedding, reprsented as a permutation of the darts, of this
-- graph.
embedding :: Getter (PlanarGraph s w v e f) (Permutation (Dart s))
embedding = to _embedding

vertexData :: Lens (PlanarGraph s w v e f) (PlanarGraph s w v' e f)
                   (V.Vector v) (V.Vector v')
vertexData = lens _vertexData (\g vD -> updateData (const vD) id id g)

rawDartData :: Lens (PlanarGraph s w v e f) (PlanarGraph s w v e' f)
                    (V.Vector e) (V.Vector e')
rawDartData = lens _rawDartData (\g dD -> updateData id (const dD) id g)

faceData :: Lens (PlanarGraph s w v e f) (PlanarGraph s w v e f')
                 (V.Vector f) (V.Vector f')
faceData = lens _faceData (\g fD -> updateData id id (const fD) g)

-- | Get the dual graph of this graph.
dual :: Getter (PlanarGraph s w v e f) (PlanarGraph s (DualOf w) f e v)
dual = to _dual


-- FIXME: So I guess the two darts associated with an edge can store different
-- data. This is useful. Make sure that works as expected.

-- | lens to access the Dart Data
--
--
dartData :: Lens (PlanarGraph s w v e f) (PlanarGraph s w v e' f)
                (V.Vector (Dart s, e)) (V.Vector (Dart s, e'))
dartData = lens darts (\g dD -> updateData id (const $ reorderEdgeData dD) id g)

-- | edgeData is just an alias for 'dartData'
edgeData :: Lens (PlanarGraph s w v e f) (PlanarGraph s w v e' f)
                 (V.Vector (Dart s, e)) (V.Vector (Dart s, e'))
edgeData = dartData

-- | Helper function to update the data in a planar graph. Takes care to update
-- both the data in the original graph as well as in the dual.
updateData :: forall s w v e f v' e' f'
           .  (V.Vector v -> V.Vector v')
           -> (V.Vector e -> V.Vector e')
           -> (V.Vector f -> V.Vector f')
           -> PlanarGraph s w v  e  f
           -> PlanarGraph s w v' e' f'
updateData = gcastWith proof updateData'
  where
    proof :: DualOf (DualOf w) :~: w
    proof = dualDualIdentity

-- | The function that does the actual work for 'updateData'
updateData'  :: (DualOf (DualOf w) ~ w)
             => (V.Vector v -> V.Vector v')
             -> (V.Vector e -> V.Vector e')
             -> (V.Vector f -> V.Vector f')
             -> PlanarGraph s w v  e  f
             -> PlanarGraph s w v' e' f'
updateData' fv fe ff (PlanarGraph em vtxData dData fData dg) = g'
  where
    vtxData' = fv vtxData
    dData'   = fe dData
    fData'   = ff fData

    g'       = PlanarGraph em              vtxData' dData' fData'   dg'
    dg'      = PlanarGraph (dg^.embedding) fData'   dData' vtxData' g'


-- | Reorders the edge data to be in the right order to set edgeData
reorderEdgeData    :: Foldable f => f (Dart s, e) -> V.Vector e
reorderEdgeData ds = V.create $ do
                                  v <- MV.new (F.length ds)
                                  forM_ (F.toList ds) $ \(d,x) ->
                                    MV.write v (fromEnum d) x
                                  pure v

-- | Traverse the vertices
--
-- (^.vertexData) <$> traverseVertices (\i x -> Just (i,x)) myGraph
-- Just [(VertexId 0,()),(VertexId 1,()),(VertexId 2,()),(VertexId 3,())]
-- >>> traverseVertices (\i x -> print (i,x)) myGraph >> pure ()
-- (VertexId 0,())
-- (VertexId 1,())
-- (VertexId 2,())
-- (VertexId 3,())
traverseVertices   :: Applicative m
                   => (VertexId s w -> v -> m v')
                   -> PlanarGraph s w v e f
                   -> m (PlanarGraph s w v' e f)
traverseVertices f = itraverseOf (vertexData.itraversed) (\i -> f (VertexId i))

-- | Traverses the darts
--
-- >>> traverseDarts (\d x -> print (d,x)) myGraph >> pure ()
-- (Dart (Arc 0) +1,"a+")
-- (Dart (Arc 0) -1,"a-")
-- (Dart (Arc 1) +1,"b+")
-- (Dart (Arc 1) -1,"b-")
-- (Dart (Arc 2) +1,"c+")
-- (Dart (Arc 2) -1,"c-")
-- (Dart (Arc 3) +1,"d+")
-- (Dart (Arc 3) -1,"d-")
-- (Dart (Arc 4) +1,"e+")
-- (Dart (Arc 4) -1,"e-")
-- (Dart (Arc 5) +1,"g+")
-- (Dart (Arc 5) -1,"g-")
traverseDarts   :: Applicative m
                => (Dart s -> e -> m e')
                -> PlanarGraph s w v e f
                -> m (PlanarGraph s w v e' f)
traverseDarts f = itraverseOf (rawDartData.itraversed) (\i -> f (toEnum i))

-- | Traverses the faces
--
-- >>> traverseFaces (\i x -> print (i,x)) myGraph >> pure ()
-- (FaceId 0,())
-- (FaceId 1,())
-- (FaceId 2,())
-- (FaceId 3,())
traverseFaces   :: Applicative m
                => (FaceId s w -> f -> m f')
                -> PlanarGraph s w v e f
                -> m (PlanarGraph s w v e f')
traverseFaces f = itraverseOf (faceData.itraversed) (\i -> f (FaceId $ VertexId i))

--------------------------------------------------------------------------------
-- ** Constructing a Planar graph

-- | Construct a planar graph
--
-- running time: \(O(n)\).
planarGraph'      :: Permutation (Dart s) -> PlanarGraph s w () () ()
planarGraph' perm = pg
  where
    pg = PlanarGraph perm vData eData fData (computeDual pg)
        -- note the lazy calculation of computeDual that refers to pg itself
    d  = size perm
    e  = d `div` 2
    v  = V.length (perm^.orbits)
    f  = e - v + 2

    vData  = V.replicate v ()
    eData  = V.replicate d ()
    fData  = V.replicate f ()

-- | Construct a planar graph, given the darts in cyclic order around each
-- vertex.
--
-- running time: \(O(n)\).
planarGraph    :: [[(Dart s,e)]] -> PlanarGraph s Primal () e ()
planarGraph ds = (planarGraph' perm)&dartData .~ (V.fromList . concat $ ds)
  where
    n     = sum . map length $ ds
    perm  = toCycleRep n $ map (map fst) ds



-- | Construct a planar graph from a adjacency matrix. For every vertex, all
-- vertices should be given in counter clockwise order.
--
-- pre: No self-loops, and no multi-edges
--
-- running time: \(O(n)\).
fromAdjacencyLists      :: forall s w h. (Foldable h, Functor h)
                        => [(VertexId s w, h (VertexId s w))]
                        -> PlanarGraph s w () () ()
fromAdjacencyLists adjM = planarGraph' . toCycleRep n $ perm
  where
    n    = sum . fmap length $ perm
    perm = map toOrbit  $ adjM'

    adjM' = fmap (second F.toList) adjM

    -- -- | Assign Arcs
    -- adjM' = (^._1) . foldr assignArcs (SP [] 0) $ adjM

    -- Build an edgeOracle, so that we can query the arcId assigned to
    -- an edge in O(1) time.
    oracle :: EdgeOracle s w Int
    oracle = fmap (^.core) . assignArcs . buildEdgeOracle
           . map (second $ map ext)  $ adjM'

    toOrbit (u,adjU) = concatMap (toDart u) adjU

    -- if u = v we have a self-loop, so we add both a positive and a negative dart
    toDart u v = let Just a = findEdge u v oracle
                 in case u `compare` v of
                      LT -> [Dart (Arc a) Positive]
                      EQ -> [Dart (Arc a) Positive, Dart (Arc a) Negative]
                      GT -> [Dart (Arc a) Negative]


assignArcs   :: EdgeOracle s w e -> EdgeOracle s w (Int :+ e)
assignArcs o = evalState (traverse f o) 0
  where
    f   :: e -> State Int (Int :+ e)
    f e = do i <- get ; put (i+1) ; pure (i :+ e)

-- | Produces the adjacencylists for all vertices in the graph. For every vertex, the
-- adjacent vertices are given in counter clockwise order.
--
-- Note that in case a vertex u as a self loop, we have that this vertexId occurs
-- twice in the list of neighbours, i.e.: u : [...,u,..,u,...]. Similarly, if there are
-- multiple darts between a pair of edges they occur multiple times.
--
-- running time: \(O(n)\)
toAdjacencyLists    :: PlanarGraph s w v e f -> [(VertexId s w, V.Vector (VertexId s w))]
toAdjacencyLists pg = map (\u -> (u,neighboursOf u pg)) . V.toList . vertices' $ pg
-- TODO: something weird happens when we have self-loops here.


--------------------------------------------------------------------------------
-- ** Convenience functions

-- | Get the number of vertices
--
-- >>> numVertices myGraph
-- 4
numVertices :: PlanarGraph s w v e f -> Int
numVertices g = V.length (g^.embedding.orbits)

-- | Get the number of Darts
--
-- >>> numDarts myGraph
-- 12
numDarts :: PlanarGraph s w v e f -> Int
numDarts g = size (g^.embedding)

-- | Get the number of Edges
--
-- >>> numEdges myGraph
-- 6
numEdges :: PlanarGraph s w v e f -> Int
numEdges g = numDarts g `div` 2

-- | Get the number of faces
--
-- >>> numFaces myGraph
-- 4
numFaces   :: PlanarGraph s w v e f -> Int
numFaces g = numEdges g - numVertices g + 2


-- | Enumerate all vertices
--
-- >>> vertices' myGraph
-- [VertexId 0,VertexId 1,VertexId 2,VertexId 3]
vertices'   :: PlanarGraph s w v e f -> V.Vector (VertexId s w)
vertices' g = VertexId <$> V.enumFromN 0 (V.length (g^.embedding.orbits))

-- | Enumerate all vertices, together with their vertex data

-- >>> vertices myGraph
-- [(VertexId 0,()),(VertexId 1,()),(VertexId 2,()),(VertexId 3,())]
vertices   :: PlanarGraph s w v e f -> V.Vector (VertexId s w, v)
vertices g = V.zip (vertices' g) (g^.vertexData)


-- | Enumerate all darts
darts' :: PlanarGraph s w v e f -> V.Vector (Dart s)
darts' = elems . _embedding

-- | Get all darts together with their data
--
-- >>> mapM_ print $ darts myGraph
-- (Dart (Arc 0) -1,"a-")
-- (Dart (Arc 2) +1,"c+")
-- (Dart (Arc 1) +1,"b+")
-- (Dart (Arc 0) +1,"a+")
-- (Dart (Arc 4) -1,"e-")
-- (Dart (Arc 1) -1,"b-")
-- (Dart (Arc 3) -1,"d-")
-- (Dart (Arc 5) +1,"g+")
-- (Dart (Arc 4) +1,"e+")
-- (Dart (Arc 3) +1,"d+")
-- (Dart (Arc 2) -1,"c-")
-- (Dart (Arc 5) -1,"g-")
darts   :: PlanarGraph s w v e f -> V.Vector (Dart s, e)
darts g = (\d -> (d,g^.dataOf d)) <$> darts' g

-- | Enumerate all edges. We report only the Positive darts
edges' :: PlanarGraph s w v e f -> V.Vector (Dart s)
edges' = V.filter isPositive . darts'

-- | Enumerate all edges with their edge data. We report only the Positive
-- darts.
--
-- >>> mapM_ print $ edges myGraph
-- (Dart (Arc 2) +1,"c+")
-- (Dart (Arc 1) +1,"b+")
-- (Dart (Arc 0) +1,"a+")
-- (Dart (Arc 5) +1,"g+")
-- (Dart (Arc 4) +1,"e+")
-- (Dart (Arc 3) +1,"d+")
edges :: PlanarGraph s w v e f -> V.Vector (Dart s, e)
edges = V.filter (isPositive . fst) . darts


-- | The tail of a dart, i.e. the vertex this dart is leaving from
--
-- running time: \(O(1)\)
tailOf     :: Dart s -> PlanarGraph s w v e f -> VertexId s w
tailOf d g = VertexId . fst $ lookupIdx (g^.embedding) d

-- | The vertex this dart is heading in to
--
-- running time: \(O(1)\)
headOf   :: Dart s -> PlanarGraph s w v e f -> VertexId s w
headOf d = tailOf (twin d)

-- | endPoints d g = (tailOf d g, headOf d g)
--
-- running time: \(O(1)\)
endPoints :: Dart s -> PlanarGraph s w v e f -> (VertexId s w, VertexId s w)
endPoints d g = (tailOf d g, headOf d g)


-- | All edges incident to vertex v, in counterclockwise order around v.
--
-- running time: \(O(k)\), where \(k\) is the output size
incidentEdges                :: VertexId s w -> PlanarGraph s w v e f
                             -> V.Vector (Dart s)
incidentEdges (VertexId v) g = g^?!embedding.orbits.ix v
  -- TODO: The Delaunay triang. stuff seems to produce these in clockwise order instead

-- | All incoming edges incident to vertex v, in counterclockwise order around v.
incomingEdges     :: VertexId s w -> PlanarGraph s w v e f -> V.Vector (Dart s)
incomingEdges v g = V.filter (not . isPositive) $ incidentEdges v g

-- | All outgoing edges incident to vertex v, in counterclockwise order around v.
outgoingEdges     :: VertexId s w -> PlanarGraph s w v e f -> V.Vector (Dart s)
outgoingEdges v g = V.filter isPositive $ incidentEdges v g


-- | Gets the neighbours of a particular vertex, in counterclockwise order
-- around the vertex.
--
-- running time: \(O(k)\), where \(k\) is the output size
neighboursOf     :: VertexId s w -> PlanarGraph s w v e f -> V.Vector (VertexId s w)
neighboursOf v g = otherVtx <$> incidentEdges v g
  where
    otherVtx d = let u = tailOf d g in if u == v then headOf d g else u

-- | Given a dart d that points into some vertex v, report the next dart in the
-- cyclic order around v.
--
-- running time: \(O(1)\)
nextIncidentEdge     :: Dart s -> PlanarGraph s w v e f -> Dart s
nextIncidentEdge d g = let perm  = g^.embedding
                           (i,j) = lookupIdx perm d
                       in next (perm^?!orbits.ix i) j


-- | Given a dart d that points into some vertex v, report the next dart in the
-- cyclic order around v.
--
-- running time: \(O(1)\)
prevIncidentEdge     :: Dart s -> PlanarGraph s w v e f -> Dart s
prevIncidentEdge d g = let perm  = g^.embedding
                           (i,j) = lookupIdx perm d
                       in previous (perm^?!orbits.ix i) j


--------------------------------------------------------------------------------
-- * Access data


class HasDataOf g i where
  type DataOf g i
  -- | get the data associated with the value i.
  --
  -- running time: \(O(1)\) to read the data, \(O(n)\) to write it.
  dataOf :: i -> Lens' g (DataOf g i)

instance HasDataOf (PlanarGraph s w v e f) (VertexId s w) where
  type DataOf (PlanarGraph s w v e f) (VertexId s w) = v
  dataOf (VertexId i) = vertexData.singular (ix i)

instance HasDataOf (PlanarGraph s w v e f) (Dart s) where
  type DataOf (PlanarGraph s w v e f) (Dart s) = e
  dataOf d = rawDartData.singular (ix $ fromEnum d)

instance HasDataOf (PlanarGraph s w v e f) (FaceId s w) where
  type DataOf (PlanarGraph s w v e f) (FaceId s w) = f
  dataOf (FaceId (VertexId i)) = faceData.singular (ix i)


-- | Data corresponding to the endpoints of the dart
endPointDataOf   :: Dart s -> Getter (PlanarGraph s w v e f) (v,v)
endPointDataOf d = to $ endPointData d


-- | Data corresponding to the endpoints of the dart
--
-- running time: \(O(1)\)
endPointData     :: Dart s -> PlanarGraph s w v e f -> (v,v)
endPointData d g = let (u,v) = endPoints d g in (g^.dataOf u, g^.dataOf v)

--------------------------------------------------------------------------------
-- * The Dual graph

-- | The dual of this graph
--
-- >>> :{
--  let fromList = V.fromList
--      answer = fromList [ fromList [dart 0 "-1"]
--                        , fromList [dart 2 "+1",dart 4 "+1",dart 1 "-1",dart 0 "+1"]
--                        , fromList [dart 1 "+1",dart 3 "-1",dart 2 "-1"]
--                        , fromList [dart 4 "-1",dart 3 "+1",dart 5 "+1",dart 5 "-1"]
--                        ]
--  in (computeDual myGraph)^.embedding.orbits == answer
-- :}
-- True
--
-- running time: \(O(n)\).
computeDual :: forall s w v e f. PlanarGraph s w v e f -> PlanarGraph s (DualOf w) f e v
computeDual = gcastWith proof computeDual'
  where
    proof :: DualOf (DualOf w) :~: w
    proof = dualDualIdentity

-- | Does the actual work for dualGraph
computeDual'   :: (DualOf (DualOf w) ~ w)
               => PlanarGraph s w v e f -> PlanarGraph s (DualOf w) f e v
computeDual' g = dualG
  where
    perm  = g^.embedding
    dualG = PlanarGraph (cycleRep (elems perm) (apply perm . twin))
                        (g^.faceData)
                        (g^.rawDartData)
                        (g^.vertexData)
                        g


-- | The type to reprsent FaceId's
newtype FaceId s w = FaceId { _unFaceId :: VertexId s (DualOf w) }
                   deriving (Eq,Ord,Enum,ToJSON,FromJSON)

-- | Shorthand for FaceId's in the primal.
type FaceId' s = FaceId s Primal

instance Show (FaceId s w) where
  show (FaceId (VertexId i)) = "FaceId " ++ show i

-- | Enumerate all faces in the planar graph
faces' :: PlanarGraph s w v e f -> V.Vector (FaceId s w)
faces' = fmap FaceId . vertices' . _dual

-- | All faces with their face data.
faces   :: PlanarGraph s w v e f -> V.Vector (FaceId s w, f)
faces g = V.zip (faces' g) (g^.faceData)

-- | The face to the left of the dart
--
-- >>> leftFace (dart 1 "+1") myGraph
-- FaceId 1
-- >>> leftFace (dart 1 "-1") myGraph
-- FaceId 2
-- >>> leftFace (dart 2 "+1") myGraph
-- FaceId 2
-- >>> leftFace (dart 0 "+1") myGraph
-- FaceId 0
--
-- running time: \(O(1)\).
leftFace     :: Dart s -> PlanarGraph s w v e f -> FaceId s w
leftFace d g = FaceId . headOf d $ _dual g


-- | The face to the right of the dart
--
-- >>> rightFace (dart 1 "+1") myGraph
-- FaceId 2
-- >>> rightFace (dart 1 "-1") myGraph
-- FaceId 1
-- >>> rightFace (dart 2 "+1") myGraph
-- FaceId 1
-- >>> rightFace (dart 0 "+1") myGraph
-- FaceId 1
--
-- running time: \(O(1)\).
rightFace     :: Dart s -> PlanarGraph s w v e f -> FaceId s w
rightFace d g = FaceId . tailOf d $ _dual g


-- | Get the next edge along the face
--
-- running time: \(O(1)\).
nextEdge   :: Dart s -> PlanarGraph s w v e f -> Dart s
nextEdge d = nextIncidentEdge d . _dual

-- | Get the previous edge along the face
--
-- running time: \(O(1)\).
prevEdge :: Dart s -> PlanarGraph s w v e f -> Dart s
prevEdge d = prevIncidentEdge d . _dual


-- | Gets a dart bounding this face. I.e. a dart d such that the face lies to
-- the right of the dart.
boundaryDart   :: FaceId s w -> PlanarGraph s w v e f -> Dart s
boundaryDart f = V.head . boundary f
-- TODO: make sure that this is indeed to the right.

-- | The darts bounding this face, for internal faces in clockwise order, for
-- the outer face in counter clockwise order.
--
--
-- running time: \(O(k)\), where \(k\) is the output size.
boundary              :: FaceId s w -> PlanarGraph s w v e f -> V.Vector (Dart s)
boundary (FaceId v) g = incidentEdges v $ _dual g


-- | Generates the darts incident to a face, starting with the given dart.
--
--
-- \(O(k)\), where \(k\) is the number of darts reported
boundary'     :: Dart s -> PlanarGraph s w v e f -> V.Vector (Dart s)
boundary' d g = fromMaybe (error "boundary'")  . rotateTo d $ boundary (rightFace d g) g
  where
    rotateTo     :: Eq a => a -> V.Vector a -> Maybe (V.Vector a)
    rotateTo x v = f <$> V.elemIndex x v
      where
        f i = let (a,b) = V.splitAt i v  in b <> a


-- | The vertices bounding this face, for internal faces in clockwise order, for
-- the outer face in counter clockwise order.
--
--
-- running time: \(O(k)\), where \(k\) is the output size.
boundaryVertices     :: FaceId s w -> PlanarGraph s w v e f -> V.Vector (VertexId s w)
boundaryVertices f g = fmap (flip tailOf g) $ boundary f g

-- -- | Gets the next dart along the face
-- nextDart     :: Dart s -> PlanarGraph s w v e f -> Dart s
-- nextDart d g = f rightFace e

--------------------------------------------------------------------------------
-- Testing stuff

-- testPerm :: Permutation (Dart s)
-- testPerm = let (a:b:c:d:e:g:_) = take 6 [Arc 0..]
--            in toCycleRep 12 [ [ Dart a Negative
--                               , Dart c Positive
--                               , Dart b Positive
--                               , Dart a Positive
--                               ]
--                             , [ Dart e Negative
--                               , Dart b Negative
--                               , Dart d Negative
--                               , Dart g Positive
--                               ]
--                             , [ Dart e Positive
--                               , Dart d Positive
--                               , Dart c Negative
--                               ]
--                             , [ Dart g Negative
--                               ]
--                             ]

-- data Test

-- testG :: PlanarGraph Test Primal () String ()
-- testG = planarGraph [ [ (Dart aA Negative, "a-")
--                       , (Dart aC Positive, "c+")
--                       , (Dart aB Positive, "b+")
--                       , (Dart aA Positive, "a+")
--                       ]
--                     , [ (Dart aE Negative, "e-")
--                       , (Dart aB Negative, "b-")
--                       , (Dart aD Negative, "d-")
--                       , (Dart aG Positive, "g+")
--                       ]
--                     , [ (Dart aE Positive, "e+")
--                       , (Dart aD Positive, "d+")
--                       , (Dart aC Negative, "c-")
--                       ]
--                     , [ (Dart aG Negative, "g-")
--                       ]
--                     ]
--   where
--     (aA:aB:aC:aD:aE:aG:_) = take 6 [Arc 0..]






--------------------------------------------------------------------------------


--------------------------------------------------------------------------------
-- * Testing Connectivity


-- | Edge Oracle:
--
-- main idea: store adjacency lists in such a way that we store an edge (u,v)
-- either in u's adjacency list or in v's. This can be done s.t. all adjacency
-- lists have length at most 6.
--
-- note: Every edge is stored exactly once (i.e. either at u or at v, but not both)
newtype EdgeOracle s w a =
  EdgeOracle { _unEdgeOracle :: V.Vector (V.Vector (VertexId s w :+ a)) }
                         deriving (Show,Eq)

instance Functor (EdgeOracle s w) where
  fmap = fmapDefault

instance Foldable (EdgeOracle s w) where
  foldMap = foldMapDefault

instance Traversable (EdgeOracle s w) where
  traverse f (EdgeOracle v) = EdgeOracle <$> traverse g v
    where
      -- g   :: V.Vector (VertexId :+ a) -> f (V.Vector (VertexId :+ b))
      g = traverse (bitraverse pure f)


-- | Given a planar graph, construct an edge oracle. Given a pair of vertices
-- this allows us to efficiently find the dart representing this edge in the
-- graph.
--
-- pre: No self-loops and no multi-edges!!!
--
-- running time: \(O(n)\)
edgeOracle   :: PlanarGraph s w v e f -> EdgeOracle s w (Dart s)
edgeOracle g = buildEdgeOracle [ (v, mkAdjacency v <$> incidentEdges v g)
                               | v <- F.toList $ vertices' g
                               ]
  where
    mkAdjacency v d = otherVtx v d :+ d
    otherVtx v d = let u = tailOf d g in if u == v then headOf d g else u



-- | Builds an edge oracle that can be used to efficiently test if two vertices
-- are connected by an edge.
--
-- running time: \(O(n)\)
buildEdgeOracle        :: forall f s w e. (Foldable f)
                       => [(VertexId s w, f (VertexId s w :+ e))] -> EdgeOracle s w e
buildEdgeOracle inAdj' = EdgeOracle $ V.create $ do
                          counts <- UV.thaw initCounts
                          marks  <- UMV.replicate (UMV.length counts) False
                          outV   <- MV.new (UMV.length counts)
                          build counts marks outV initQ
                          pure outV
    -- main idea: maintain a vector with counts; i.e. how many unprocessed
    -- vertices are adjacent to u, and a bit vector with marks to keep track if
    -- a vertex has been processed yet. When we process a vertex, we keep only
    -- the adjacencies of unprocessed verticese.
  where
    -- Convert to a vector representation
    inAdj = V.create $ do
              mv <- MV.new (length inAdj')
              forM_ inAdj' $ \(VertexId i,adjI) ->
                MV.write mv i (V.fromList . F.toList $ adjI)
              pure mv

    initCounts = V.convert . fmap GV.length $ inAdj
    -- initial vertices available for processing
    initQ = GV.ifoldr (\i k q -> if k <= 6 then i : q else q) [] initCounts

    -- | Construct the adjacencylist for vertex i. I.e. by retaining only adjacent
    -- vertices that have not been processed yet.
    extractAdj         :: UMV.MVector s' Bool -> Int
                       -> ST s' (V.Vector (VertexId s w :+ e))
    extractAdj marks i = let p = fmap not . UMV.read marks . (^.core.unVertexId)
                         in GV.filterM  p $ inAdj V.! i

    -- | Decreases the number of adjacencies that vertex j has
    -- if it has <= 6 adjacencies left it has become available for processing
    decrease                          :: UMV.MVector s' Int -> (VertexId s w :+ e')
                                      -> ST s' (Maybe Int)
    decrease counts (VertexId j :+ _) = do k <- UMV.read counts j
                                           let k'  = k - 1
                                           UMV.write counts j k'
                                           pure $ if k' <= 6 then Just j else Nothing

    -- The actual algorithm that builds the items
    build :: UMV.MVector s' Int -> UMV.MVector s' Bool
          -> MV.MVector s' (V.Vector (VertexId s w :+ e)) -> [Int] -> ST s' ()
    build _      _     _    []    = pure ()
    build counts marks outV (i:q) = do
             b <- UMV.read marks i
             nq <- if b then pure []
                        else do
                          adjI <- extractAdj marks i
                          MV.write outV i adjI
                          UMV.write marks i True
                          V.toList <$> mapM (decrease counts) adjI
             build counts marks outV (catMaybes nq <> q)



-- | Test if u and v are connected by an edge.
--
-- running time: \(O(1)\)
hasEdge     :: VertexId s w -> VertexId s w -> EdgeOracle s w a -> Bool
hasEdge u v = isJust . findEdge u v


-- | Find the edge data corresponding to edge (u,v) if such an edge exists
--
-- running time: \(O(1)\)
findEdge :: VertexId s w -> VertexId s w -> EdgeOracle s w a -> Maybe a
findEdge  (VertexId u) (VertexId v) (EdgeOracle os) = find' u v <|> find' v u
  where
    find' j i = fmap (^.extra) . F.find (\(VertexId k :+ _) -> j == k) $ os V.! i

-- | Given a pair of vertices (u,v) returns the dart, oriented from u to v,
-- corresponding to these vertices.
--
-- running time: \(O(1)\)
findDart :: VertexId s w -> VertexId s w -> EdgeOracle s w (Dart s) -> Maybe (Dart s)
findDart (VertexId u) (VertexId v) (EdgeOracle os) = find' twin u v <|> find' id v u
    -- looks up j in the adjacencylist of i and applies f to the result
  where
    find' f j i = fmap (f . (^.extra)) . F.find (\(VertexId k :+ _) -> j == k) $ os V.! i

--------------------------------------------------------------------------------

-- data Test

-- type Vertex = VertexId Test Primal

-- testEdges :: [(Vertex,[Vertex])]
-- testEdges = map (\(i,vs) -> (VertexId i, map VertexId vs))
--             [ (0, [1])
--             , (1, [0,1,2,4])
--             , (2, [1,3,4])
--             , (3, [2,5])
--             , (4, [1,2,5])
--             , (5, [3,4])
--             ]


-- myGraph :: PlanarGraph Test Primal () String ()
-- myGraph = planarGraph [ [ (Dart aA Negative, "a-")
--                             , (Dart aC Positive, "c+")
--                             , (Dart aB Positive, "b+")
--                             , (Dart aA Positive, "a+")
--                             ]
--                           , [ (Dart aE Negative, "e-")
--                             , (Dart aB Negative, "b-")
--                             , (Dart aD Negative, "d-")
--                             , (Dart aG Positive, "g+")
--                             ]
--                           , [ (Dart aE Positive, "e+")
--                             , (Dart aD Positive, "d+")
--                             , (Dart aC Negative, "c-")
--                             ]
--                           , [ (Dart aG Negative, "g-")
--                             ]
--                           ]
--   where
--     -- dart i s = Dart (Arc i) (read s)
--     (aA:aB:aC:aD:aE:aG:_) = take 6 [Arc 0..]