Graph.hs

{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UndecidableInstances #-}

-- |
-- Module      :  Distribution.Compat.Graph
-- Copyright   :  (c) Edward Z. Yang 2016
-- License     :  BSD3
--
-- Maintainer  :  cabal-dev@haskell.org
-- Stability   :  experimental
-- Portability :  portable
--
-- A data type representing directed graphs, backed by "Data.Graph".
-- It is strict in the node type.
--
-- This is an alternative interface to "Data.Graph".  In this interface,
-- nodes (identified by the 'IsNode' type class) are associated with a
-- key and record the keys of their neighbors.  This interface is more
-- convenient than 'Data.Graph.Graph', which requires vertices to be
-- explicitly handled by integer indexes.
--
-- The current implementation has somewhat peculiar performance
-- characteristics.  The asymptotics of all map-like operations mirror
-- their counterparts in "Data.Map".  However, to perform a graph
-- operation, we first must build the "Data.Graph" representation, an
-- operation that takes /O(V + E log V)/.  However, this operation can
-- be amortized across all queries on that particular graph.
--
-- Some nodes may be broken, i.e., refer to neighbors which are not
-- stored in the graph.  In our graph algorithms, we transparently
-- ignore such edges; however, you can easily query for the broken
-- vertices of a graph using 'broken' (and should, e.g., to ensure that
-- a closure of a graph is well-formed.)  It's possible to take a closed
-- subset of a broken graph and get a well-formed graph.
module Distribution.Compat.Graph
  ( -- * Graph type
    Graph
  , IsNode (..)

    -- * Query
  , null
  , size
  , member
  , lookup

    -- * Construction
  , empty
  , insert
  , deleteKey
  , deleteLookup

    -- * Combine
  , unionLeft
  , unionRight

    -- * Graph algorithms
  , stronglyConnComp
  , SCC (..)
  , cycles
  , broken
  , neighbors
  , revNeighbors
  , closure
  , revClosure
  , topSort
  , revTopSort

    -- * Conversions

    -- ** Maps
  , toMap

    -- ** Lists
  , fromDistinctList
  , toList
  , keys

    -- ** Sets
  , keysSet

    -- ** Graphs
  , toGraph

    -- * Node type
  , Node (..)
  , nodeValue
  ) where

import Distribution.Compat.Prelude hiding (empty, lookup, null, toList)
import Prelude ()

import Data.Array ((!))
import Data.Graph (SCC (..))
import Distribution.Utils.Structured (Structure (..), Structured (..))

import qualified Data.Array as Array
import qualified Data.Foldable as Foldable
import qualified Data.Graph as G
import qualified Data.Map.Strict as Map
import qualified Data.Set as Set
import qualified Data.Tree as Tree
import qualified Distribution.Compat.Prelude as Prelude

-- | A graph of nodes @a@.  The nodes are expected to have instance
-- of class 'IsNode'.
data Graph a = Graph
  { graphMap :: !(Map (Key a) a)
  , -- Lazily cached graph representation
    graphForward :: G.Graph
  , graphAdjoint :: G.Graph
  , graphVertexToNode :: G.Vertex -> a
  , graphKeyToVertex :: Key a -> Maybe G.Vertex
  , graphBroken :: [(a, [Key a])]
  }

-- NB: Not a Functor! (or Traversable), because you need
-- to restrict Key a ~ Key b.  We provide our own mapping
-- functions.

-- General strategy is most operations are deferred to the
-- Map representation.

instance Show a => Show (Graph a) where
  show = show . toList

instance (IsNode a, Read a, Show (Key a)) => Read (Graph a) where
  readsPrec d s = map (\(a, r) -> (fromDistinctList a, r)) (readsPrec d s)

instance (IsNode a, Binary a, Show (Key a)) => Binary (Graph a) where
  put x = put (toList x)
  get = fmap fromDistinctList get

instance Structured a => Structured (Graph a) where
  structure p = Nominal (typeRep p) 0 "Graph" [structure (Proxy :: Proxy a)]

instance (Eq (Key a), Eq a) => Eq (Graph a) where
  g1 == g2 = graphMap g1 == graphMap g2

instance Foldable.Foldable Graph where
  elem x = Foldable.elem x . graphMap
  fold = Foldable.fold . graphMap
  foldl f z = Foldable.foldl f z . graphMap
  foldl' f z = Foldable.foldl' f z . graphMap
  foldr f z = Foldable.foldr f z . graphMap
  foldr' f z = Foldable.foldr' f z . graphMap
  foldMap f = Foldable.foldMap f . graphMap
  length = Foldable.length . graphMap
  maximum = Foldable.maximum . graphMap
  minimum = Foldable.minimum . graphMap
  null = Foldable.null . graphMap
  product = Foldable.product . graphMap
  sum = Foldable.sum . graphMap
  toList = Foldable.toList . graphMap

instance (NFData a, NFData (Key a)) => NFData (Graph a) where
  rnf
    Graph
      { graphMap = m
      , graphForward = gf
      , graphAdjoint = ga
      , graphVertexToNode = vtn
      , graphKeyToVertex = ktv
      , graphBroken = b
      } = gf `seq` ga `seq` vtn `seq` ktv `seq` b `seq` rnf m

-- TODO: Data instance?

-- | The 'IsNode' class is used for datatypes which represent directed
-- graph nodes.  A node of type @a@ is associated with some unique key of
-- type @'Key' a@; given a node we can determine its key ('nodeKey')
-- and the keys of its neighbors ('nodeNeighbors').
class Ord (Key a) => IsNode a where
  type Key a
  nodeKey :: a -> Key a
  nodeNeighbors :: a -> [Key a]

instance (IsNode a, IsNode b, Key a ~ Key b) => IsNode (Either a b) where
  type Key (Either a b) = Key a
  nodeKey (Left x) = nodeKey x
  nodeKey (Right x) = nodeKey x
  nodeNeighbors (Left x) = nodeNeighbors x
  nodeNeighbors (Right x) = nodeNeighbors x

-- | A simple, trivial data type which admits an 'IsNode' instance.
data Node k a = N a k [k]
  deriving (Show, Eq)

-- | Get the value from a 'Node'.
nodeValue :: Node k a -> a
nodeValue (N a _ _) = a

instance Functor (Node k) where
  fmap f (N a k ks) = N (f a) k ks

instance Ord k => IsNode (Node k a) where
  type Key (Node k a) = k
  nodeKey (N _ k _) = k
  nodeNeighbors (N _ _ ks) = ks

-- TODO: Maybe introduce a typeclass for items which just
-- keys (so, Key associated type, and nodeKey method).  But
-- I didn't need it here, so I didn't introduce it.

-- Query

-- | /O(1)/. Is the graph empty?
null :: Graph a -> Bool
null = Map.null . toMap

-- | /O(1)/. The number of nodes in the graph.
size :: Graph a -> Int
size = Map.size . toMap

-- | /O(log V)/. Check if the key is in the graph.
member :: IsNode a => Key a -> Graph a -> Bool
member k g = Map.member k (toMap g)

-- | /O(log V)/. Lookup the node at a key in the graph.
lookup :: IsNode a => Key a -> Graph a -> Maybe a
lookup k g = Map.lookup k (toMap g)

-- Construction

-- | /O(1)/. The empty graph.
empty :: IsNode a => Graph a
empty = fromMap Map.empty

-- | /O(log V)/. Insert a node into a graph.
insert :: IsNode a => a -> Graph a -> Graph a
insert !n g = fromMap (Map.insert (nodeKey n) n (toMap g))

-- | /O(log V)/. Delete the node at a key from the graph.
deleteKey :: IsNode a => Key a -> Graph a -> Graph a
deleteKey k g = fromMap (Map.delete k (toMap g))

-- | /O(log V)/. Lookup and delete.  This function returns the deleted
-- value if it existed.
deleteLookup :: IsNode a => Key a -> Graph a -> (Maybe a, Graph a)
deleteLookup k g =
  let (r, m') = Map.updateLookupWithKey (\_ _ -> Nothing) k (toMap g)
   in (r, fromMap m')

-- Combining

-- | /O(V + V')/. Right-biased union, preferring entries
-- from the second map when conflicts occur.
-- @'nodeKey' x = 'nodeKey' (f x)@.
unionRight :: IsNode a => Graph a -> Graph a -> Graph a
unionRight g g' = fromMap (Map.union (toMap g') (toMap g))

-- | /O(V + V')/. Left-biased union, preferring entries from
-- the first map when conflicts occur.
unionLeft :: IsNode a => Graph a -> Graph a -> Graph a
unionLeft = flip unionRight

-- Graph-like operations

-- | /Ω(V + E)/. Compute the strongly connected components of a graph.
-- Requires amortized construction of graph.
stronglyConnComp :: Graph a -> [SCC a]
stronglyConnComp g = map decode forest
  where
    forest = G.scc (graphForward g)
    decode (Tree.Node v [])
      | mentions_itself v = CyclicSCC [graphVertexToNode g v]
      | otherwise = AcyclicSCC (graphVertexToNode g v)
    decode other = CyclicSCC (dec other [])
      where
        dec (Tree.Node v ts) vs =
          graphVertexToNode g v : foldr dec vs ts
    mentions_itself v = v `elem` (graphForward g ! v)

-- Implementation copied from 'stronglyConnCompR' in 'Data.Graph'.

-- | /Ω(V + E)/. Compute the cycles of a graph.
-- Requires amortized construction of graph.
cycles :: Graph a -> [[a]]
cycles g = [vs | CyclicSCC vs <- stronglyConnComp g]

-- | /O(1)/.  Return a list of nodes paired with their broken
-- neighbors (i.e., neighbor keys which are not in the graph).
-- Requires amortized construction of graph.
broken :: Graph a -> [(a, [Key a])]
broken g = graphBroken g

-- | Lookup the immediate neighbors from a key in the graph.
-- Requires amortized construction of graph.
neighbors :: Graph a -> Key a -> Maybe [a]
neighbors g k = do
  v <- graphKeyToVertex g k
  return (map (graphVertexToNode g) (graphForward g ! v))

-- | Lookup the immediate reverse neighbors from a key in the graph.
-- Requires amortized construction of graph.
revNeighbors :: Graph a -> Key a -> Maybe [a]
revNeighbors g k = do
  v <- graphKeyToVertex g k
  return (map (graphVertexToNode g) (graphAdjoint g ! v))

-- | Compute the subgraph which is the closure of some set of keys.
-- Returns @Nothing@ if one (or more) keys are not present in
-- the graph.
-- Requires amortized construction of graph.
closure :: Graph a -> [Key a] -> Maybe [a]
closure g ks = do
  vs <- traverse (graphKeyToVertex g) ks
  return (decodeVertexForest g (G.dfs (graphForward g) vs))

-- | Compute the reverse closure of a graph from some set
-- of keys.  Returns @Nothing@ if one (or more) keys are not present in
-- the graph.
-- Requires amortized construction of graph.
revClosure :: Graph a -> [Key a] -> Maybe [a]
revClosure g ks = do
  vs <- traverse (graphKeyToVertex g) ks
  return (decodeVertexForest g (G.dfs (graphAdjoint g) vs))

flattenForest :: Tree.Forest a -> [a]
flattenForest = concatMap Tree.flatten

decodeVertexForest :: Graph a -> Tree.Forest G.Vertex -> [a]
decodeVertexForest g = map (graphVertexToNode g) . flattenForest

-- | Topologically sort the nodes of a graph.
-- Requires amortized construction of graph.
topSort :: Graph a -> [a]
topSort g = map (graphVertexToNode g) $ G.topSort (graphForward g)

-- | Reverse topologically sort the nodes of a graph.
-- Requires amortized construction of graph.
revTopSort :: Graph a -> [a]
revTopSort g = map (graphVertexToNode g) $ G.topSort (graphAdjoint g)

-- Conversions

-- | /O(1)/. Convert a map from keys to nodes into a graph.
-- The map must satisfy the invariant that
-- @'fromMap' m == 'fromList' ('Data.Map.elems' m)@;
-- if you can't fulfill this invariant use @'fromList' ('Data.Map.elems' m)@
-- instead.  The values of the map are assumed to already
-- be in WHNF.
fromMap :: IsNode a => Map (Key a) a -> Graph a
fromMap m =
  Graph
    { graphMap = m
    , -- These are lazily computed!
      graphForward = g
    , graphAdjoint = G.transposeG g
    , graphVertexToNode = vertex_to_node
    , graphKeyToVertex = key_to_vertex
    , graphBroken = broke
    }
  where
    try_key_to_vertex k = maybe (Left k) Right (key_to_vertex k)

    (brokenEdges, edges) =
      unzip $
        [ partitionEithers (map try_key_to_vertex (nodeNeighbors n))
        | n <- ns
        ]
    broke = filter (not . Prelude.null . snd) (zip ns brokenEdges)

    g = Array.listArray bounds edges

    ns = Map.elems m -- sorted ascending
    vertices = zip (map nodeKey ns) [0 ..]
    vertex_map = Map.fromAscList vertices
    key_to_vertex k = Map.lookup k vertex_map

    vertex_to_node vertex = nodeTable ! vertex

    nodeTable = Array.listArray bounds ns
    bounds = (0, Map.size m - 1)

-- | /O(V log V)/. Convert a list of nodes (with distinct keys) into a graph.
fromDistinctList :: (IsNode a, Show (Key a)) => [a] -> Graph a
fromDistinctList =
  fromMap
    . Map.fromListWith (\_ -> duplicateError)
    . map (\n -> n `seq` (nodeKey n, n))
  where
    duplicateError n =
      error $
        "Graph.fromDistinctList: duplicate key: "
          ++ show (nodeKey n)

-- Map-like operations

-- | /O(V)/. Convert a graph into a list of nodes.
toList :: Graph a -> [a]
toList g = Map.elems (toMap g)

-- | /O(V)/. Convert a graph into a list of keys.
keys :: Graph a -> [Key a]
keys g = Map.keys (toMap g)

-- | /O(V)/. Convert a graph into a set of keys.
keysSet :: Graph a -> Set.Set (Key a)
keysSet g = Map.keysSet (toMap g)

-- | /O(1)/. Convert a graph into a map from keys to nodes.
-- The resulting map @m@ is guaranteed to have the property that
-- @'Prelude.all' (\(k,n) -> k == 'nodeKey' n) ('Data.Map.toList' m)@.
toMap :: Graph a -> Map (Key a) a
toMap = graphMap

-- Graph-like operations

-- | /O(1)/. Convert a graph into a 'Data.Graph.Graph'.
-- Requires amortized construction of graph.
toGraph :: Graph a -> (G.Graph, G.Vertex -> a, Key a -> Maybe G.Vertex)
toGraph g = (graphForward g, graphVertexToNode g, graphKeyToVertex g)