```-- (c) The University of Glasgow 2006

{-# LANGUAGE CPP, ScopedTypeVariables #-}

module Digraph(
Graph, graphFromEdgedVertices,

SCC(..), Node, flattenSCC, flattenSCCs,
stronglyConnCompG,
topologicalSortG, dfsTopSortG,
verticesG, edgesG, hasVertexG,
reachableG, reachablesG, transposeG,
outdegreeG, indegreeG,
vertexGroupsG, emptyG,
componentsG,

findCycle,

-- For backwards compatability with the simpler version of Digraph
stronglyConnCompFromEdgedVertices, stronglyConnCompFromEdgedVerticesR,
) where

#include "HsVersions.h"

------------------------------------------------------------------------------
-- A version of the graph algorithms described in:
--
-- ``Lazy Depth-First Search and Linear IntGraph Algorithms in Haskell''
--   by David King and John Launchbury
--
-- Also included is some additional code for printing tree structures ...
--
-- If you ever find yourself in need of algorithms for classifying edges,
-- or finding connected/biconnected components, consult the history; Sigbjorn
-- Finne contributed some implementations in 1997, although we've since
-- removed them since they were not used anywhere in GHC.
------------------------------------------------------------------------------

import Util        ( minWith, count )
import Outputable
import Maybes      ( expectJust )

-- Extensions
import Control.Monad    ( filterM, liftM, liftM2 )

-- std interfaces
import Data.Maybe
import Data.Array
import Data.List hiding (transpose)
import Data.Array.ST
import qualified Data.Map as Map
import qualified Data.Set as Set

import qualified Data.Graph as G
import Data.Graph hiding (Graph, Edge, transposeG, reachable)
import Data.Tree

{-
************************************************************************
*                                                                      *
*      Graphs and Graph Construction
*                                                                      *
************************************************************************

Note [Nodes, keys, vertices]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
* A 'node' is a big blob of client-stuff

* Each 'node' has a unique (client) 'key', but the latter
is in Ord and has fast comparison

* Digraph then maps each 'key' to a Vertex (Int) which is
arranged densely in 0.n
-}

data Graph node = Graph {
gr_int_graph      :: IntGraph,
gr_vertex_to_node :: Vertex -> node,
gr_node_to_vertex :: node -> Maybe Vertex
}

data Edge node = Edge node node

-- The payload is user data, just carried around in this module
-- The keys are ordered
-- The [key] are the dependencies of the node;
--    it's ok to have extra keys in the dependencies that
--    are not the key of any Node in the graph

emptyGraph :: Graph a
emptyGraph = Graph (array (1, 0) []) (error "emptyGraph") (const Nothing)

graphFromEdgedVertices
:: Ord key                      -- We only use Ord for efficiency,
-- it doesn't effect the result, so
-- it can be safely used with Unique's.
=> [Node key payload]           -- The graph; its ok for the
-- out-list to contain keys which arent
-- a vertex key, they are ignored
graphFromEdgedVertices []             = emptyGraph
graphFromEdgedVertices edged_vertices = Graph graph vertex_fn (key_vertex . key_extractor)
where key_extractor (_, k, _) = k
(bounds, vertex_fn, key_vertex, numbered_nodes) = reduceNodesIntoVertices edged_vertices key_extractor
graph = array bounds [ (v, sort \$ mapMaybe key_vertex ks)
| (v, (_, _, ks)) <- numbered_nodes]
-- We normalize outgoing edges by sorting on node order, so
-- that the result doesn't depend on the order of the edges

reduceNodesIntoVertices
:: Ord key
=> [node]
-> (node -> key)
-> (Bounds, Vertex -> node, key -> Maybe Vertex, [(Vertex, node)])
reduceNodesIntoVertices nodes key_extractor = (bounds, (!) vertex_map, key_vertex, numbered_nodes)
where
max_v           = length nodes - 1
bounds          = (0, max_v) :: (Vertex, Vertex)

-- Keep the order intact to make the result depend on input order
numbered_nodes  = zip [0..] nodes
vertex_map      = array bounds numbered_nodes

key_map = Map.fromList
[ (key_extractor node, v) | (v, node) <- numbered_nodes ]
key_vertex k = Map.lookup k key_map

{-
************************************************************************
*                                                                      *
*      SCC
*                                                                      *
************************************************************************
-}

= (Node key payload,  -- Tip of the path
[payload])         -- Rest of the path;
--  [a,b,c] means c depends on b, b depends on a

-- | Find a reasonably short cycle a->b->c->a, in a strongly
-- connected component.  The input nodes are presumed to be
-- a SCC, so you can start anywhere.
findCycle :: forall payload key. Ord key
=> [Node key payload]     -- The nodes.  The dependencies can
-- contain extra keys, which are ignored
-> Maybe [payload]        -- A cycle, starting with node
-- so each depends on the next
findCycle graph
= go Set.empty (new_work root_deps []) []
where
env :: Map.Map key (Node key payload)
env = Map.fromList [ (key, node) | node@(_, key, _) <- graph ]

-- Find the node with fewest dependencies among the SCC modules
-- This is just a heuristic to find some plausible root module
root = fst (minWith snd [ (node, count (`Map.member` env) deps)
| node@(_,_,deps) <- graph ])

-- 'go' implements Dijkstra's algorithm, more or less
go :: Set.Set key   -- Visited
-> [WorkItem key payload]        -- Work list, items length n
-> [WorkItem key payload]        -- Work list, items length n+1
-> Maybe [payload]               -- Returned cycle
-- Invariant: in a call (go visited ps qs),
--            visited = union (map tail (ps ++ qs))

go _       [] [] = Nothing  -- No cycles
go visited [] qs = go visited qs []
go visited (((payload,key,deps), path) : ps) qs
| key == root_key           = Just (root_payload : reverse path)
| key `Set.member` visited  = go visited ps qs
| key `Map.notMember` env   = go visited ps qs
| otherwise                 = go (Set.insert key visited)
ps (new_qs ++ qs)
where
new_qs = new_work deps (payload : path)

new_work deps path = [ (n, path) | Just n <- map (`Map.lookup` env) deps ]

{-
************************************************************************
*                                                                      *
*      Strongly Connected Component wrappers for Graph
*                                                                      *
************************************************************************

Note: the components are returned topologically sorted: later components
depend on earlier ones, but not vice versa i.e. later components only have
edges going from them to earlier ones.
-}

stronglyConnCompG :: Graph node -> [SCC node]
stronglyConnCompG graph = decodeSccs graph forest
where forest = {-# SCC "Digraph.scc" #-} scc (gr_int_graph graph)

decodeSccs :: Graph node -> Forest Vertex -> [SCC node]
decodeSccs Graph { gr_int_graph = graph, gr_vertex_to_node = vertex_fn } forest
= map decode forest
where
decode (Node v []) | mentions_itself v = CyclicSCC [vertex_fn v]
| otherwise         = AcyclicSCC (vertex_fn v)
decode other = CyclicSCC (dec other [])
where dec (Node v ts) vs = vertex_fn v : foldr dec vs ts
mentions_itself v = v `elem` (graph ! v)

-- The following two versions are provided for backwards compatability:
stronglyConnCompFromEdgedVertices
:: Ord key
stronglyConnCompFromEdgedVertices
= map (fmap get_node) . stronglyConnCompFromEdgedVerticesR
where get_node (n, _, _) = n

-- The "R" interface is used when you expect to apply SCC to
-- (some of) the result of SCC, so you dont want to lose the dependency info
stronglyConnCompFromEdgedVerticesR
:: Ord key
stronglyConnCompFromEdgedVerticesR = stronglyConnCompG . graphFromEdgedVertices

{-
************************************************************************
*                                                                      *
*      Misc wrappers for Graph
*                                                                      *
************************************************************************
-}

topologicalSortG :: Graph node -> [node]
topologicalSortG graph = map (gr_vertex_to_node graph) result
where result = {-# SCC "Digraph.topSort" #-} topSort (gr_int_graph graph)

dfsTopSortG :: Graph node -> [[node]]
dfsTopSortG graph =
map (map (gr_vertex_to_node graph) . flatten) \$ dfs g (topSort g)
where
g = gr_int_graph graph

reachableG :: Graph node -> node -> [node]
reachableG graph from = map (gr_vertex_to_node graph) result
where from_vertex = expectJust "reachableG" (gr_node_to_vertex graph from)
result = {-# SCC "Digraph.reachable" #-} reachable (gr_int_graph graph) [from_vertex]

reachablesG :: Graph node -> [node] -> [node]
reachablesG graph froms = map (gr_vertex_to_node graph) result
where result = {-# SCC "Digraph.reachable" #-}
reachable (gr_int_graph graph) vs
vs = [ v | Just v <- map (gr_node_to_vertex graph) froms ]

hasVertexG :: Graph node -> node -> Bool
hasVertexG graph node = isJust \$ gr_node_to_vertex graph node

verticesG :: Graph node -> [node]
verticesG graph = map (gr_vertex_to_node graph) \$ vertices (gr_int_graph graph)

edgesG :: Graph node -> [Edge node]
edgesG graph = map (\(v1, v2) -> Edge (v2n v1) (v2n v2)) \$ edges (gr_int_graph graph)
where v2n = gr_vertex_to_node graph

transposeG :: Graph node -> Graph node
transposeG graph = Graph (G.transposeG (gr_int_graph graph))
(gr_vertex_to_node graph)
(gr_node_to_vertex graph)

outdegreeG :: Graph node -> node -> Maybe Int
outdegreeG = degreeG outdegree

indegreeG :: Graph node -> node -> Maybe Int
indegreeG = degreeG indegree

degreeG :: (G.Graph -> Table Int) -> Graph node -> node -> Maybe Int
degreeG degree graph node = let table = degree (gr_int_graph graph)
in fmap ((!) table) \$ gr_node_to_vertex graph node

vertexGroupsG :: Graph node -> [[node]]
vertexGroupsG graph = map (map (gr_vertex_to_node graph)) result
where result = vertexGroups (gr_int_graph graph)

emptyG :: Graph node -> Bool
emptyG g = graphEmpty (gr_int_graph g)

componentsG :: Graph node -> [[node]]
componentsG graph = map (map (gr_vertex_to_node graph) . flatten)
\$ components (gr_int_graph graph)

{-
************************************************************************
*                                                                      *
*      Showing Graphs
*                                                                      *
************************************************************************
-}

instance Outputable node => Outputable (Graph node) where
ppr graph = vcat [
hang (text "Vertices:") 2 (vcat (map ppr \$ verticesG graph)),
hang (text "Edges:") 2 (vcat (map ppr \$ edgesG graph))
]

instance Outputable node => Outputable (Edge node) where
ppr (Edge from to) = ppr from <+> text "->" <+> ppr to

graphEmpty :: G.Graph -> Bool
graphEmpty g = lo > hi
where (lo, hi) = bounds g

{-
************************************************************************
*                                                                      *
*      IntGraphs
*                                                                      *
************************************************************************
-}

type IntGraph = G.Graph

{-
------------------------------------------------------------
-- Depth first search numbering
------------------------------------------------------------
-}

-- Data.Tree has flatten for Tree, but nothing for Forest
preorderF           :: Forest a -> [a]
preorderF ts         = concat (map flatten ts)

{-
------------------------------------------------------------
-- Finding reachable vertices
------------------------------------------------------------
-}

-- This generalizes reachable which was found in Data.Graph
reachable    :: IntGraph -> [Vertex] -> [Vertex]
reachable g vs = preorderF (dfs g vs)

{-
------------------------------------------------------------
-- Total ordering on groups of vertices
------------------------------------------------------------

The plan here is to extract a list of groups of elements of the graph
such that each group has no dependence except on nodes in previous
groups (i.e. in particular they may not depend on nodes in their own
group) and is maximal such group.

Clearly we cannot provide a solution for cyclic graphs.

We proceed by iteratively removing elements with no outgoing edges
and their associated edges from the graph.

This probably isn't very efficient and certainly isn't very clever.
-}

type Set s    = STArray s Vertex Bool

mkEmpty      :: Bounds -> ST s (Set s)
mkEmpty bnds  = newArray bnds False

contains     :: Set s -> Vertex -> ST s Bool
contains m v  = readArray m v

include      :: Set s -> Vertex -> ST s ()
include m v   = writeArray m v True

vertexGroups :: IntGraph -> [[Vertex]]
vertexGroups g = runST (mkEmpty (bounds g) >>= \provided -> vertexGroupsS provided g next_vertices)
where next_vertices = noOutEdges g

noOutEdges :: IntGraph -> [Vertex]
noOutEdges g = [ v | v <- vertices g, null (g!v)]

vertexGroupsS :: Set s -> IntGraph -> [Vertex] -> ST s [[Vertex]]
vertexGroupsS provided g to_provide
= if null to_provide
then do {
all_provided <- allM (provided `contains`) (vertices g)
; if all_provided
then return []
else error "vertexGroup: cyclic graph"
}
else do {
mapM_ (include provided) to_provide
; to_provide' <- filterM (vertexReady provided g) (vertices g)
; rest <- vertexGroupsS provided g to_provide'
; return \$ to_provide : rest
}

vertexReady :: Set s -> IntGraph -> Vertex -> ST s Bool
vertexReady provided g v = liftM2 (&&) (liftM not \$ provided `contains` v) (allM (provided `contains`) (g!v))
```