-- -- Copyright (c) 2018 Andreas Klebinger -- {-# LANGUAGE TypeFamilies, ScopedTypeVariables #-} {-# LANGUAGE TupleSections #-} {-# LANGUAGE GeneralizedNewtypeDeriving #-} {-# LANGUAGE CPP #-} {-# LANGUAGE Rank2Types #-} {-# LANGUAGE BangPatterns #-} {-# LANGUAGE DataKinds #-} module CFG ( CFG, CfgEdge(..), EdgeInfo(..), EdgeWeight(..) , TransitionSource(..) --Modify the CFG , addWeightEdge, addEdge , delEdge, delNode , addNodesBetween, shortcutWeightMap , reverseEdges, filterEdges , addImmediateSuccessor , mkWeightInfo, adjustEdgeWeight, setEdgeWeight --Query the CFG , infoEdgeList, edgeList , getSuccessorEdges, getSuccessors , getSuccEdgesSorted , getEdgeInfo , getCfgNodes, hasNode -- Loop Information , loopMembers, loopLevels, loopInfo --Construction/Misc , getCfg, getCfgProc, pprEdgeWeights, sanityCheckCfg --Find backedges and update their weight , optimizeCFG , mkGlobalWeights ) where #include "HsVersions.h" import GhcPrelude import BlockId import Cmm import CmmUtils import CmmSwitch import Hoopl.Collections import Hoopl.Label import Hoopl.Block import qualified Hoopl.Graph as G import Util import Digraph import Maybes import Unique import qualified Dominators as Dom import Data.IntMap.Strict (IntMap) import Data.IntSet (IntSet) import qualified Data.IntMap.Strict as IM import qualified Data.Map as M import qualified Data.IntSet as IS import qualified Data.Set as S import Data.Tree import Data.Bifunctor import Outputable -- DEBUGGING ONLY --import Debug -- import Debug.Trace --import OrdList --import Debug.Trace import PprCmm () -- For Outputable instances import qualified DynFlags as D import Data.List (sort, nub, partition) import Data.STRef.Strict import Control.Monad.ST import Data.Array.MArray import Data.Array.ST import Data.Array.IArray import Data.Array.Unsafe (unsafeFreeze) import Data.Array.Base (unsafeRead, unsafeWrite) import Control.Monad type Prob = Double type Edge = (BlockId, BlockId) type Edges = [Edge] newtype EdgeWeight = EdgeWeight { weightToDouble :: Double } deriving (Eq,Ord,Enum,Num,Real,Fractional) instance Outputable EdgeWeight where ppr (EdgeWeight w) = doublePrec 5 w type EdgeInfoMap edgeInfo = LabelMap (LabelMap edgeInfo) -- | A control flow graph where edges have been annotated with a weight. -- Implemented as IntMap (IntMap <edgeData>) -- We must uphold the invariant that for each edge A -> B we must have: -- A entry B in the outer map. -- A entry B in the map we get when looking up A. -- Maintaining this invariant is useful as any failed lookup now indicates -- an actual error in code which might go unnoticed for a while -- otherwise. type CFG = EdgeInfoMap EdgeInfo data CfgEdge = CfgEdge { edgeFrom :: !BlockId , edgeTo :: !BlockId , edgeInfo :: !EdgeInfo } -- | Careful! Since we assume there is at most one edge from A to B -- the Eq instance does not consider weight. instance Eq CfgEdge where (==) (CfgEdge from1 to1 _) (CfgEdge from2 to2 _) = from1 == from2 && to1 == to2 -- | Edges are sorted ascending pointwise by weight, source and destination instance Ord CfgEdge where compare (CfgEdge from1 to1 (EdgeInfo {edgeWeight = weight1})) (CfgEdge from2 to2 (EdgeInfo {edgeWeight = weight2})) | weight1 < weight2 || weight1 == weight2 && from1 < from2 || weight1 == weight2 && from1 == from2 && to1 < to2 = LT | from1 == from2 && to1 == to2 && weight1 == weight2 = EQ | otherwise = GT instance Outputable CfgEdge where ppr (CfgEdge from1 to1 edgeInfo) = parens (ppr from1 <+> text "-(" <> ppr edgeInfo <> text ")->" <+> ppr to1) -- | Can we trace back a edge to a specific Cmm Node -- or has it been introduced during assembly codegen. We use this to maintain -- some information which would otherwise be lost during the -- Cmm <-> asm transition. -- See also Note [Inverting Conditional Branches] data TransitionSource = CmmSource { trans_cmmNode :: (CmmNode O C) , trans_info :: BranchInfo } | AsmCodeGen deriving (Eq) data BranchInfo = NoInfo -- ^ Unknown, but not heap or stack check. | HeapStackCheck -- ^ Heap or stack check deriving Eq instance Outputable BranchInfo where ppr NoInfo = text "regular" ppr HeapStackCheck = text "heap/stack" isHeapOrStackCheck :: TransitionSource -> Bool isHeapOrStackCheck (CmmSource { trans_info = HeapStackCheck}) = True isHeapOrStackCheck _ = False -- | Information about edges data EdgeInfo = EdgeInfo { transitionSource :: !TransitionSource , edgeWeight :: !EdgeWeight } deriving (Eq) instance Outputable EdgeInfo where ppr edgeInfo = text "weight:" <+> ppr (edgeWeight edgeInfo) -- | Convenience function, generate edge info based -- on weight not originating from cmm. mkWeightInfo :: EdgeWeight -> EdgeInfo mkWeightInfo = EdgeInfo AsmCodeGen -- | Adjust the weight between the blocks using the given function. -- If there is no such edge returns the original map. adjustEdgeWeight :: CFG -> (EdgeWeight -> EdgeWeight) -> BlockId -> BlockId -> CFG adjustEdgeWeight cfg f from to | Just info <- getEdgeInfo from to cfg , !weight <- edgeWeight info , !newWeight <- f weight = addEdge from to (info { edgeWeight = newWeight}) cfg | otherwise = cfg -- | Set the weight between the blocks to the given weight. -- If there is no such edge returns the original map. setEdgeWeight :: CFG -> EdgeWeight -> BlockId -> BlockId -> CFG setEdgeWeight cfg !weight from to | Just info <- getEdgeInfo from to cfg = addEdge from to (info { edgeWeight = weight}) cfg | otherwise = cfg getCfgNodes :: CFG -> [BlockId] getCfgNodes m = mapKeys m -- | Is this block part of this graph? hasNode :: CFG -> BlockId -> Bool hasNode m node = -- Check the invariant that each node must exist in the first map or not at all. ASSERT( found || not (any (mapMember node) m)) found where found = mapMember node m -- | Check if the nodes in the cfg and the set of blocks are the same. -- In a case of a missmatch we panic and show the difference. sanityCheckCfg :: CFG -> LabelSet -> SDoc -> Bool sanityCheckCfg m blockSet msg | blockSet == cfgNodes = True | otherwise = pprPanic "Block list and cfg nodes don't match" ( text "difference:" <+> ppr diff $$ text "blocks:" <+> ppr blockSet $$ text "cfg:" <+> pprEdgeWeights m $$ msg ) False where cfgNodes = setFromList $ getCfgNodes m :: LabelSet diff = (setUnion cfgNodes blockSet) `setDifference` (setIntersection cfgNodes blockSet) :: LabelSet -- | Filter the CFG with a custom function f. -- Paramaeters are `f from to edgeInfo` filterEdges :: (BlockId -> BlockId -> EdgeInfo -> Bool) -> CFG -> CFG filterEdges f cfg = mapMapWithKey filterSources cfg where filterSources from m = mapFilterWithKey (\to w -> f from to w) m {- Note [Updating the CFG during shortcutting] See Note [What is shortcutting] in the control flow optimization code (CmmContFlowOpt.hs) for a slightly more in depth explanation on shortcutting. In the native backend we shortcut jumps at the assembly level. (AsmCodeGen.hs) This means we remove blocks containing only one jump from the code and instead redirecting all jumps targeting this block to the deleted blocks jump target. However we want to have an accurate representation of control flow in the CFG. So we add/remove edges accordingly to account for the eliminated blocks and new edges. If we shortcut A -> B -> C to A -> C: * We delete edges A -> B and B -> C * Replacing them with the edge A -> C We also try to preserve jump weights while doing so. Note that: * The edge B -> C can't have interesting weights since the block B consists of a single unconditional jump without branching. * We delete the edge A -> B and add the edge A -> C. * The edge A -> B can be one of many edges originating from A so likely has edge weights we want to preserve. For this reason we simply store the edge info from the original A -> B edge and apply this information to the new edge A -> C. Sometimes we have a scenario where jump target C is not represented by an BlockId but an immediate value. I'm only aware of this happening without tables next to code currently. Then we go from A ---> B - -> IMM to A - -> IMM where the dashed arrows are not stored in the CFG. In that case we simply delete the edge A -> B. In terms of implementation the native backend first builds a mapping from blocks suitable for shortcutting to their jump targets. Then it redirects all jump instructions to these blocks using the built up mapping. This function (shortcutWeightMap) takes the same mapping and applies the mapping to the CFG in the way layed out above. -} shortcutWeightMap :: LabelMap (Maybe BlockId) -> CFG -> CFG shortcutWeightMap cuts cfg = foldl' applyMapping cfg $ mapToList cuts where -- takes the tuple (B,C) from the notation in [Updating the CFG during shortcutting] applyMapping :: CFG -> (BlockId,Maybe BlockId) -> CFG --Shortcut immediate applyMapping m (from, Nothing) = mapDelete from . fmap (mapDelete from) $ m --Regular shortcut applyMapping m (from, Just to) = let updatedMap :: CFG updatedMap = fmap (shortcutEdge (from,to)) $ (mapDelete from m :: CFG ) --Sometimes we can shortcut multiple blocks like so: -- A -> B -> C -> D -> E => A -> E -- so we check for such chains. in case mapLookup to cuts of Nothing -> updatedMap Just dest -> applyMapping updatedMap (to, dest) --Redirect edge from B to C shortcutEdge :: (BlockId, BlockId) -> LabelMap EdgeInfo -> LabelMap EdgeInfo shortcutEdge (from, to) m = case mapLookup from m of Just info -> mapInsert to info $ mapDelete from m Nothing -> m -- | Sometimes we insert a block which should unconditionally be executed -- after a given block. This function updates the CFG for these cases. -- So we get A -> B => A -> A' -> B -- \ \ -- -> C => -> C -- addImmediateSuccessor :: BlockId -> BlockId -> CFG -> CFG addImmediateSuccessor node follower cfg = updateEdges . addWeightEdge node follower uncondWeight $ cfg where uncondWeight = fromIntegral . D.uncondWeight . D.cfgWeightInfo $ D.unsafeGlobalDynFlags targets = getSuccessorEdges cfg node successors = map fst targets :: [BlockId] updateEdges = addNewSuccs . remOldSuccs remOldSuccs m = foldl' (flip (delEdge node)) m successors addNewSuccs m = foldl' (\m' (t,info) -> addEdge follower t info m') m targets -- | Adds a new edge, overwrites existing edges if present addEdge :: BlockId -> BlockId -> EdgeInfo -> CFG -> CFG addEdge from to info cfg = mapAlter addFromToEdge from $ mapAlter addDestNode to cfg where -- Simply insert the edge into the edge list. addFromToEdge Nothing = Just $ mapSingleton to info addFromToEdge (Just wm) = Just $ mapInsert to info wm -- We must add the destination node explicitly addDestNode Nothing = Just $ mapEmpty addDestNode n@(Just _) = n -- | Adds a edge with the given weight to the cfg -- If there already existed an edge it is overwritten. -- `addWeightEdge from to weight cfg` addWeightEdge :: BlockId -> BlockId -> EdgeWeight -> CFG -> CFG addWeightEdge from to weight cfg = addEdge from to (mkWeightInfo weight) cfg delEdge :: BlockId -> BlockId -> CFG -> CFG delEdge from to m = mapAlter remDest from m where remDest Nothing = Nothing remDest (Just wm) = Just $ mapDelete to wm delNode :: BlockId -> CFG -> CFG delNode node cfg = fmap (mapDelete node) -- < Edges to the node (mapDelete node cfg) -- < Edges from the node -- | Destinations from bid ordered by weight (descending) getSuccEdgesSorted :: CFG -> BlockId -> [(BlockId,EdgeInfo)] getSuccEdgesSorted m bid = let destMap = mapFindWithDefault mapEmpty bid m cfgEdges = mapToList destMap sortedEdges = sortWith (negate . edgeWeight . snd) cfgEdges in --pprTrace "getSuccEdgesSorted" (ppr bid <+> text "map:" <+> ppr m) sortedEdges -- | Get successors of a given node with edge weights. getSuccessorEdges :: HasDebugCallStack => CFG -> BlockId -> [(BlockId,EdgeInfo)] getSuccessorEdges m bid = maybe lookupError mapToList (mapLookup bid m) where lookupError = pprPanic "getSuccessorEdges: Block does not exist" $ ppr bid <+> pprEdgeWeights m getEdgeInfo :: BlockId -> BlockId -> CFG -> Maybe EdgeInfo getEdgeInfo from to m | Just wm <- mapLookup from m , Just info <- mapLookup to wm = Just $! info | otherwise = Nothing getEdgeWeight :: CFG -> BlockId -> BlockId -> EdgeWeight getEdgeWeight cfg from to = edgeWeight $ expectJust "Edgeweight for noexisting block" $ getEdgeInfo from to cfg getTransitionSource :: BlockId -> BlockId -> CFG -> TransitionSource getTransitionSource from to cfg = transitionSource $ expectJust "Source info for noexisting block" $ getEdgeInfo from to cfg reverseEdges :: CFG -> CFG reverseEdges cfg = mapFoldlWithKey (\cfg from toMap -> go (addNode cfg from) from toMap) mapEmpty cfg where -- We must preserve nodes without outgoing edges! addNode :: CFG -> BlockId -> CFG addNode cfg b = mapInsertWith mapUnion b mapEmpty cfg go :: CFG -> BlockId -> (LabelMap EdgeInfo) -> CFG go cfg from toMap = mapFoldlWithKey (\cfg to info -> addEdge to from info cfg) cfg toMap :: CFG -- | Returns a unordered list of all edges with info infoEdgeList :: CFG -> [CfgEdge] infoEdgeList m = go (mapToList m) [] where -- We avoid foldMap to avoid thunk buildup go :: [(BlockId,LabelMap EdgeInfo)] -> [CfgEdge] -> [CfgEdge] go [] acc = acc go ((from,toMap):xs) acc = go' xs from (mapToList toMap) acc go' :: [(BlockId,LabelMap EdgeInfo)] -> BlockId -> [(BlockId,EdgeInfo)] -> [CfgEdge] -> [CfgEdge] go' froms _ [] acc = go froms acc go' froms from ((to,info):tos) acc = go' froms from tos (CfgEdge from to info : acc) -- | Returns a unordered list of all edges without weights edgeList :: CFG -> [Edge] edgeList m = go (mapToList m) [] where -- We avoid foldMap to avoid thunk buildup go :: [(BlockId,LabelMap EdgeInfo)] -> [Edge] -> [Edge] go [] acc = acc go ((from,toMap):xs) acc = go' xs from (mapKeys toMap) acc go' :: [(BlockId,LabelMap EdgeInfo)] -> BlockId -> [BlockId] -> [Edge] -> [Edge] go' froms _ [] acc = go froms acc go' froms from (to:tos) acc = go' froms from tos ((from,to) : acc) -- | Get successors of a given node without edge weights. getSuccessors :: HasDebugCallStack => CFG -> BlockId -> [BlockId] getSuccessors m bid | Just wm <- mapLookup bid m = mapKeys wm | otherwise = lookupError where lookupError = pprPanic "getSuccessors: Block does not exist" $ ppr bid <+> pprEdgeWeights m pprEdgeWeights :: CFG -> SDoc pprEdgeWeights m = let edges = sort $ infoEdgeList m :: [CfgEdge] printEdge (CfgEdge from to (EdgeInfo { edgeWeight = weight })) = text "\t" <> ppr from <+> text "->" <+> ppr to <> text "[label=\"" <> ppr weight <> text "\",weight=\"" <> ppr weight <> text "\"];\n" --for the case that there are no edges from/to this node. --This should rarely happen but it can save a lot of time --to immediately see it when it does. printNode node = text "\t" <> ppr node <> text ";\n" getEdgeNodes (CfgEdge from to _) = [from,to] edgeNodes = setFromList $ concatMap getEdgeNodes edges :: LabelSet nodes = filter (\n -> (not . setMember n) edgeNodes) . mapKeys $ mapFilter null m in text "digraph {\n" <> (foldl' (<>) empty (map printEdge edges)) <> (foldl' (<>) empty (map printNode nodes)) <> text "}\n" {-# INLINE updateEdgeWeight #-} --Allows eliminating the tuple when possible -- | Invariant: The edge **must** exist already in the graph. updateEdgeWeight :: (EdgeWeight -> EdgeWeight) -> Edge -> CFG -> CFG updateEdgeWeight f (from, to) cfg | Just oldInfo <- getEdgeInfo from to cfg = let !oldWeight = edgeWeight oldInfo !newWeight = f oldWeight in addEdge from to (oldInfo {edgeWeight = newWeight}) cfg | otherwise = panic "Trying to update invalid edge" -- from to oldWeight => newWeight mapWeights :: (BlockId -> BlockId -> EdgeWeight -> EdgeWeight) -> CFG -> CFG mapWeights f cfg = foldl' (\cfg (CfgEdge from to info) -> let oldWeight = edgeWeight info newWeight = f from to oldWeight in addEdge from to (info {edgeWeight = newWeight}) cfg) cfg (infoEdgeList cfg) -- | Insert a block in the control flow between two other blocks. -- We pass a list of tuples (A,B,C) where -- * A -> C: Old edge -- * A -> B -> C : New Arc, where B is the new block. -- It's possible that a block has two jumps to the same block -- in the assembly code. However we still only store a single edge for -- these cases. -- We assign the old edge info to the edge A -> B and assign B -> C the -- weight of an unconditional jump. addNodesBetween :: CFG -> [(BlockId,BlockId,BlockId)] -> CFG addNodesBetween m updates = foldl' updateWeight m . weightUpdates $ updates where weight = fromIntegral . D.uncondWeight . D.cfgWeightInfo $ D.unsafeGlobalDynFlags -- We might add two blocks for different jumps along a single -- edge. So we end up with edges: A -> B -> C , A -> D -> C -- in this case after applying the first update the weight for A -> C -- is no longer available. So we calculate future weights before updates. weightUpdates = map getWeight getWeight :: (BlockId,BlockId,BlockId) -> (BlockId,BlockId,BlockId,EdgeInfo) getWeight (from,between,old) | Just edgeInfo <- getEdgeInfo from old m = (from,between,old,edgeInfo) | otherwise = pprPanic "Can't find weight for edge that should have one" ( text "triple" <+> ppr (from,between,old) $$ text "updates" <+> ppr updates $$ text "cfg:" <+> pprEdgeWeights m ) updateWeight :: CFG -> (BlockId,BlockId,BlockId,EdgeInfo) -> CFG updateWeight m (from,between,old,edgeInfo) = addEdge from between edgeInfo . addWeightEdge between old weight . delEdge from old $ m {- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ Note [CFG Edge Weights] ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Edge weights assigned do not currently represent a specific cost model and rather just a ranking of which blocks should be placed next to each other given their connection type in the CFG. This is especially relevant if we whenever two blocks will jump to the same target. A B \ / C Should A or B be placed in front of C? The block layout algorithm decides this based on which edge (A,C)/(B,C) is heavier. So we make a educated guess on which branch should be preferred. We rank edges in this order: * Unconditional Control Transfer - They will always transfer control to their target. Unless there is a info table we can turn the jump into a fallthrough as well. We use 20k as default, so it's easy to spot if values have been modified but unlikely that we run into issues with overflow. * If branches (likely) - We assume branches marked as likely are taken more than 80% of the time. By ranking them below unconditional jumps we make sure we prefer the unconditional if there is a conditional and unconditional edge towards a block. * If branches (regular) - The false branch can potentially be turned into a fallthrough so we prefer it slightly over the true branch. * Unlikely branches - These can be assumed to be taken less than 20% of the time. So we given them one of the lowest priorities. * Switches - Switches at this level are implemented as jump tables so have a larger number of successors. So without more information we can only say that each individual successor is unlikely to be jumped to and we rank them accordingly. * Calls - We currently ignore calls completly: * By the time we return from a call there is a good chance that the address we return to has already been evicted from cache eliminating a main advantage sequential placement brings. * Calls always require a info table in front of their return address. This reduces the chance that we return to the same cache line further. -} -- | Generate weights for a Cmm proc based on some simple heuristics. getCfgProc :: D.CfgWeights -> RawCmmDecl -> CFG getCfgProc _ (CmmData {}) = mapEmpty getCfgProc weights (CmmProc _info _lab _live graph) = getCfg weights graph getCfg :: D.CfgWeights -> CmmGraph -> CFG getCfg weights graph = foldl' insertEdge edgelessCfg $ concatMap getBlockEdges blocks where D.CFGWeights { D.uncondWeight = uncondWeight , D.condBranchWeight = condBranchWeight , D.switchWeight = switchWeight , D.callWeight = callWeight , D.likelyCondWeight = likelyCondWeight , D.unlikelyCondWeight = unlikelyCondWeight -- Last two are used in other places --, D.infoTablePenalty = infoTablePenalty --, D.backEdgeBonus = backEdgeBonus } = weights -- Explicitly add all nodes to the cfg to ensure they are part of the -- CFG. edgelessCfg = mapFromList $ zip (map G.entryLabel blocks) (repeat mapEmpty) insertEdge :: CFG -> ((BlockId,BlockId),EdgeInfo) -> CFG insertEdge m ((from,to),weight) = mapAlter f from m where f :: Maybe (LabelMap EdgeInfo) -> Maybe (LabelMap EdgeInfo) f Nothing = Just $ mapSingleton to weight f (Just destMap) = Just $ mapInsert to weight destMap getBlockEdges :: CmmBlock -> [((BlockId,BlockId),EdgeInfo)] getBlockEdges block = case branch of CmmBranch dest -> [mkEdge dest uncondWeight] CmmCondBranch cond t f l | l == Nothing -> [mkEdge f condBranchWeight, mkEdge t condBranchWeight] | l == Just True -> [mkEdge f unlikelyCondWeight, mkEdge t likelyCondWeight] | l == Just False -> [mkEdge f likelyCondWeight, mkEdge t unlikelyCondWeight] where mkEdgeInfo = -- pprTrace "Info" (ppr branchInfo <+> ppr cond) EdgeInfo (CmmSource branch branchInfo) . fromIntegral mkEdge target weight = ((bid,target), mkEdgeInfo weight) branchInfo = foldRegsUsed (panic "foldRegsDynFlags") (\info r -> if r == SpLim || r == HpLim || r == BaseReg then HeapStackCheck else info) NoInfo cond (CmmSwitch _e ids) -> let switchTargets = switchTargetsToList ids --Compiler performance hack - for very wide switches don't --consider targets for layout. adjustedWeight = if (length switchTargets > 10) then -1 else switchWeight in map (\x -> mkEdge x adjustedWeight) switchTargets (CmmCall { cml_cont = Just cont}) -> [mkEdge cont callWeight] (CmmForeignCall {Cmm.succ = cont}) -> [mkEdge cont callWeight] (CmmCall { cml_cont = Nothing }) -> [] other -> panic "Foo" $ ASSERT2(False, ppr "Unkown successor cause:" <> (ppr branch <+> text "=>" <> ppr (G.successors other))) map (\x -> ((bid,x),mkEdgeInfo 0)) $ G.successors other where bid = G.entryLabel block mkEdgeInfo = EdgeInfo (CmmSource branch NoInfo) . fromIntegral mkEdge target weight = ((bid,target), mkEdgeInfo weight) branch = lastNode block :: CmmNode O C blocks = revPostorder graph :: [CmmBlock] --Find back edges by BFS findBackEdges :: HasDebugCallStack => BlockId -> CFG -> Edges findBackEdges root cfg = --pprTraceIt "Backedges:" $ map fst . filter (\x -> snd x == Backward) $ typedEdges where edges = edgeList cfg :: [(BlockId,BlockId)] getSuccs = getSuccessors cfg :: BlockId -> [BlockId] typedEdges = classifyEdges root getSuccs edges :: [((BlockId,BlockId),EdgeType)] optimizeCFG :: D.CfgWeights -> RawCmmDecl -> CFG -> CFG optimizeCFG _ (CmmData {}) cfg = cfg optimizeCFG weights (CmmProc info _lab _live graph) cfg = {-# SCC optimizeCFG #-} -- pprTrace "Initial:" (pprEdgeWeights cfg) $ -- pprTrace "Initial:" (ppr $ mkGlobalWeights (g_entry graph) cfg) $ -- pprTrace "LoopInfo:" (ppr $ loopInfo cfg (g_entry graph)) $ favourFewerPreds . penalizeInfoTables info . increaseBackEdgeWeight (g_entry graph) $ cfg where -- | Increase the weight of all backedges in the CFG -- this helps to make loop jumpbacks the heaviest edges increaseBackEdgeWeight :: BlockId -> CFG -> CFG increaseBackEdgeWeight root cfg = let backedges = findBackEdges root cfg update weight --Keep irrelevant edges irrelevant | weight <= 0 = 0 | otherwise = weight + fromIntegral (D.backEdgeBonus weights) in foldl' (\cfg edge -> updateEdgeWeight update edge cfg) cfg backedges -- | Since we cant fall through info tables we penalize these. penalizeInfoTables :: LabelMap a -> CFG -> CFG penalizeInfoTables info cfg = mapWeights fupdate cfg where fupdate :: BlockId -> BlockId -> EdgeWeight -> EdgeWeight fupdate _ to weight | mapMember to info = weight - (fromIntegral $ D.infoTablePenalty weights) | otherwise = weight -- | If a block has two successors, favour the one with fewer -- predecessors and/or the one allowing fall through. favourFewerPreds :: CFG -> CFG favourFewerPreds cfg = let revCfg = reverseEdges $ filterEdges (\_from -> fallthroughTarget) cfg predCount n = length $ getSuccessorEdges revCfg n nodes = getCfgNodes cfg modifiers :: Int -> Int -> (EdgeWeight, EdgeWeight) modifiers preds1 preds2 | preds1 < preds2 = ( 1,-1) | preds1 == preds2 = ( 0, 0) | otherwise = (-1, 1) update :: CFG -> BlockId -> CFG update cfg node | [(s1,e1),(s2,e2)] <- getSuccessorEdges cfg node , !w1 <- edgeWeight e1 , !w2 <- edgeWeight e2 --Only change the weights if there isn't already a ordering. , w1 == w2 , (mod1,mod2) <- modifiers (predCount s1) (predCount s2) = (\cfg' -> (adjustEdgeWeight cfg' (+mod2) node s2)) (adjustEdgeWeight cfg (+mod1) node s1) | otherwise = cfg in foldl' update cfg nodes where fallthroughTarget :: BlockId -> EdgeInfo -> Bool fallthroughTarget to (EdgeInfo source _weight) | mapMember to info = False | AsmCodeGen <- source = True | CmmSource { trans_cmmNode = CmmBranch {} } <- source = True | CmmSource { trans_cmmNode = CmmCondBranch {} } <- source = True | otherwise = False -- | Determine loop membership of blocks based on SCC analysis -- This is faster but only gives yes/no answers. loopMembers :: HasDebugCallStack => CFG -> LabelMap Bool loopMembers cfg = foldl' (flip setLevel) mapEmpty sccs where mkNode :: BlockId -> Node BlockId BlockId mkNode bid = DigraphNode bid bid (getSuccessors cfg bid) nodes = map mkNode (getCfgNodes cfg) sccs = stronglyConnCompFromEdgedVerticesOrd nodes setLevel :: SCC BlockId -> LabelMap Bool -> LabelMap Bool setLevel (AcyclicSCC bid) m = mapInsert bid False m setLevel (CyclicSCC bids) m = foldl' (\m k -> mapInsert k True m) m bids loopLevels :: CFG -> BlockId -> LabelMap Int loopLevels cfg root = liLevels loopInfos where loopInfos = loopInfo cfg root data LoopInfo = LoopInfo { liBackEdges :: [(Edge)] -- ^ List of back edges , liLevels :: LabelMap Int -- ^ BlockId -> LoopLevel mapping , liLoops :: [(Edge, LabelSet)] -- ^ (backEdge, loopBody), body includes header } instance Outputable LoopInfo where ppr (LoopInfo _ _lvls loops) = text "Loops:(backEdge, bodyNodes)" $$ (vcat $ map ppr loops) {- Note [Determining the loop body] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Starting with the knowledge that: * head dominates the loop * `tail` -> `head` is a backedge We can determine all nodes by: * Deleting the loop head from the graph. * Collect all blocks which are reachable from the `tail`. We do so by performing bfs from the tail node towards the head. -} -- | Determine loop membership of blocks based on Dominator analysis. -- This is slower but gives loop levels instead of just loop membership. -- However it only detects natural loops. Irreducible control flow is not -- recognized even if it loops. But that is rare enough that we don't have -- to care about that special case. loopInfo :: HasDebugCallStack => CFG -> BlockId -> LoopInfo loopInfo cfg root = LoopInfo { liBackEdges = backEdges , liLevels = mapFromList loopCounts , liLoops = loopBodies } where revCfg = reverseEdges cfg graph = -- pprTrace "CFG - loopInfo" (pprEdgeWeights cfg) $ fmap (setFromList . mapKeys ) cfg :: LabelMap LabelSet --TODO - This should be a no op: Export constructors? Use unsafeCoerce? ... rooted = ( fromBlockId root , toIntMap $ fmap toIntSet graph) :: (Int, IntMap IntSet) tree = fmap toBlockId $ Dom.domTree rooted :: Tree BlockId -- Map from Nodes to their dominators domMap :: LabelMap LabelSet domMap = mkDomMap tree edges = edgeList cfg :: [(BlockId, BlockId)] -- We can't recompute nodes from edges, there might be blocks not connected via edges. nodes = getCfgNodes cfg :: [BlockId] -- identify back edges isBackEdge (from,to) | Just doms <- mapLookup from domMap , setMember to doms = True | otherwise = False -- See Note [Determining the loop body] -- Get the loop body associated with a back edge. findBody edge@(tail, head) = ( edge, setInsert head $ go (setSingleton tail) (setSingleton tail) ) where -- See Note [Determining the loop body] cfg' = delNode head revCfg go :: LabelSet -> LabelSet -> LabelSet go found current | setNull current = found | otherwise = go (setUnion newSuccessors found) newSuccessors where -- Really predecessors, since we use the reversed cfg. newSuccessors = setFilter (\n -> not $ setMember n found) successors :: LabelSet successors = setFromList $ concatMap (getSuccessors cfg') -- we filter head as it's no longer part of the cfg. (filter (/= head) $ setElems current) :: LabelSet backEdges = filter isBackEdge edges loopBodies = map findBody backEdges :: [(Edge, LabelSet)] -- Block b is part of n loop bodies => loop nest level of n loopCounts = let bodies = map (first snd) loopBodies -- [(Header, Body)] loopCount n = length $ nub . map fst . filter (setMember n . snd) $ bodies in map (\n -> (n, loopCount n)) $ nodes :: [(BlockId, Int)] toIntSet :: LabelSet -> IntSet toIntSet s = IS.fromList . map fromBlockId . setElems $ s toIntMap :: LabelMap a -> IntMap a toIntMap m = IM.fromList $ map (\(x,y) -> (fromBlockId x,y)) $ mapToList m mkDomMap :: Tree BlockId -> LabelMap LabelSet mkDomMap root = mapFromList $ go setEmpty root where go :: LabelSet -> Tree BlockId -> [(Label,LabelSet)] go parents (Node lbl []) = [(lbl, parents)] go parents (Node _ leaves) = let nodes = map rootLabel leaves entries = map (\x -> (x,parents)) nodes in entries ++ concatMap (\n -> go (setInsert (rootLabel n) parents) n) leaves fromBlockId :: BlockId -> Int fromBlockId = getKey . getUnique toBlockId :: Int -> BlockId toBlockId = mkBlockId . mkUniqueGrimily -- We make the CFG a Hoopl Graph, so we can reuse revPostOrder. newtype BlockNode (e :: Extensibility) (x :: Extensibility) = BN (BlockId,[BlockId]) instance G.NonLocal (BlockNode) where entryLabel (BN (lbl,_)) = lbl successors (BN (_,succs)) = succs revPostorderFrom :: HasDebugCallStack => CFG -> BlockId -> [BlockId] revPostorderFrom cfg root = map fromNode $ G.revPostorderFrom hooplGraph root where nodes = getCfgNodes cfg hooplGraph = foldl' (\m n -> mapInsert n (toNode n) m) mapEmpty nodes fromNode :: BlockNode C C -> BlockId fromNode (BN x) = fst x toNode :: BlockId -> BlockNode C C toNode bid = BN (bid,getSuccessors cfg $ bid) -- | We take in a CFG which has on its edges weights which are -- relative only to other edges originating from the same node. -- -- We return a CFG for which each edge represents a GLOBAL weight. -- This means edge weights are comparable across the whole graph. -- -- For irreducible control flow results might be imprecise, otherwise they -- are reliable. -- -- The algorithm is based on the Paper -- "Static Branch Prediction and Program Profile Analysis" by Y Wu, JR Larus -- The only big change is that we go over the nodes in the body of loops in -- reverse post order. Which is required for diamond control flow to work probably. -- -- We also apply a few prediction heuristics (based on the same paper) {-# NOINLINE mkGlobalWeights #-} {-# SCC mkGlobalWeights #-} mkGlobalWeights :: HasDebugCallStack => BlockId -> CFG -> (LabelMap Double, LabelMap (LabelMap Double)) mkGlobalWeights root localCfg | null localCfg = panic "Error - Empty CFG" | otherwise = (blockFreqs', edgeFreqs') where -- Calculate fixpoints (blockFreqs, edgeFreqs) = calcFreqs nodeProbs backEdges' bodies' revOrder' blockFreqs' = mapFromList $ map (first fromVertex) (assocs blockFreqs) :: LabelMap Double edgeFreqs' = fmap fromVertexMap $ fromVertexMap edgeFreqs fromVertexMap :: IM.IntMap x -> LabelMap x fromVertexMap m = mapFromList . map (first fromVertex) $ IM.toList m revOrder = revPostorderFrom localCfg root :: [BlockId] loopResults@(LoopInfo backedges _levels bodies) = loopInfo localCfg root revOrder' = map toVertex revOrder backEdges' = map (bimap toVertex toVertex) backedges bodies' = map calcBody bodies estimatedCfg = staticBranchPrediction root loopResults localCfg -- Normalize the weights to probabilities and apply heuristics nodeProbs = cfgEdgeProbabilities estimatedCfg toVertex -- By mapping vertices to numbers in reverse post order we can bring any subset into reverse post -- order simply by sorting. -- TODO: The sort is redundant if we can guarantee that setElems returns elements ascending calcBody (backedge, blocks) = (toVertex $ snd backedge, sort . map toVertex $ (setElems blocks)) vertexMapping = mapFromList $ zip revOrder [0..] :: LabelMap Int blockMapping = listArray (0,mapSize vertexMapping - 1) revOrder :: Array Int BlockId -- Map from blockId to indicies starting at zero toVertex :: BlockId -> Int toVertex blockId = expectJust "mkGlobalWeights" $ mapLookup blockId vertexMapping -- Map from indicies starting at zero to blockIds fromVertex :: Int -> BlockId fromVertex vertex = blockMapping ! vertex {- Note [Static Branch Prediction] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The work here has been based on the paper "Static Branch Prediction and Program Profile Analysis" by Y Wu, JR Larus. The primary differences are that if we branch on the result of a heap check we do not apply any of the heuristics. The reason is simple: They look like loops in the control flow graph but are usually never entered, and if at most once. Currently implemented is a heuristic to predict that we do not exit loops (lehPredicts) and one to predict that backedges are more likely than any other edge. The back edge case is special as it superceeds any other heuristic if it applies. Do NOT rely solely on nofib results for benchmarking this. I recommend at least comparing megaparsec and container benchmarks. Nofib does not seeem to have many instances of "loopy" Cmm where these make a difference. TODO: * The paper containers more benchmarks which should be implemented. * If we turn the likelyhood on if/else branches into a probability instead of true/false we could implement this as a Cmm pass. + The complete Cmm code still exists and can be accessed by the heuristics + There is no chance of register allocation/codegen inserting branches/blocks + making the TransitionSource info wrong. + potential to use this information in CmmPasses. - Requires refactoring of all the code relying on the binary nature of likelyhood. - Requires refactoring `loopInfo` to work on both, Cmm Graphs and the backend CFG. -} -- | Combination of target node id and information about the branch -- we are looking at. type TargetNodeInfo = (BlockId, EdgeInfo) -- | Update branch weights based on certain heuristics. -- See Note [Static Branch Prediction] -- TODO: This should be combined with optimizeCFG {-# SCC staticBranchPrediction #-} staticBranchPrediction :: BlockId -> LoopInfo -> CFG -> CFG staticBranchPrediction _root (LoopInfo l_backEdges loopLevels l_loops) cfg = -- pprTrace "staticEstimatesOn" (ppr (cfg)) $ foldl' update cfg nodes where nodes = getCfgNodes cfg backedges = S.fromList $ l_backEdges -- Loops keyed by their back edge loops = M.fromList $ l_loops :: M.Map Edge LabelSet loopHeads = S.fromList $ map snd $ M.keys loops update :: CFG -> BlockId -> CFG update cfg node -- No successors, nothing to do. | null successors = cfg -- Mix of backedges and others: -- Always predict the backedges. | not (null m) && length m < length successors -- Heap/Stack checks "loop", but only once. -- So we simply exclude any case involving them. , not $ any (isHeapOrStackCheck . transitionSource . snd) successors = let loopChance = repeat $! pred_LBH / (fromIntegral $ length m) exitChance = repeat $! (1 - pred_LBH) / fromIntegral (length not_m) updates = zip (map fst m) loopChance ++ zip (map fst not_m) exitChance in -- pprTrace "mix" (ppr (node,successors)) $ foldl' (\cfg (to,weight) -> setEdgeWeight cfg weight node to) cfg updates -- For (regular) non-binary branches we keep the weights from the STG -> Cmm translation. | length successors /= 2 = cfg -- Only backedges - no need to adjust | length m > 0 = cfg -- A regular binary branch, we can plug addition predictors in here. | [(s1,s1_info),(s2,s2_info)] <- successors , not $ any (isHeapOrStackCheck . transitionSource . snd) successors = -- Normalize weights to total of 1 let !w1 = max (edgeWeight s1_info) (0) !w2 = max (edgeWeight s2_info) (0) -- Of both weights are <= 0 we set both to 0.5 normalizeWeight w = if w1 + w2 == 0 then 0.5 else w/(w1+w2) !cfg' = setEdgeWeight cfg (normalizeWeight w1) node s1 !cfg'' = setEdgeWeight cfg' (normalizeWeight w2) node s2 -- Figure out which heuristics apply to these successors heuristics = map ($ ((s1,s1_info),(s2,s2_info))) [lehPredicts, phPredicts, ohPredicts, ghPredicts, lhhPredicts, chPredicts , shPredicts, rhPredicts] -- Apply result of a heuristic. Argument is the likelyhood -- predicted for s1. applyHeuristic :: CFG -> Maybe Prob -> CFG applyHeuristic cfg Nothing = cfg applyHeuristic cfg (Just (s1_pred :: Double)) | s1_old == 0 || s2_old == 0 || isHeapOrStackCheck (transitionSource s1_info) || isHeapOrStackCheck (transitionSource s2_info) = cfg | otherwise = let -- Predictions from heuristic s1_prob = EdgeWeight s1_pred :: EdgeWeight s2_prob = 1.0 - s1_prob -- Update d = (s1_old * s1_prob) + (s2_old * s2_prob) :: EdgeWeight s1_prob' = s1_old * s1_prob / d !s2_prob' = s2_old * s2_prob / d !cfg_s1 = setEdgeWeight cfg s1_prob' node s1 in -- pprTrace "Applying heuristic!" (ppr (node,s1,s2) $$ ppr (s1_prob', s2_prob')) $ setEdgeWeight cfg_s1 s2_prob' node s2 where -- Old weights s1_old = getEdgeWeight cfg node s1 s2_old = getEdgeWeight cfg node s2 in -- pprTraceIt "RegularCfgResult" $ foldl' applyHeuristic cfg'' heuristics -- Branch on heap/stack check | otherwise = cfg where -- Chance that loops are taken. pred_LBH = 0.875 -- successors successors = getSuccessorEdges cfg node -- backedges (m,not_m) = partition (\succ -> S.member (node, fst succ) backedges) successors -- Heuristics return nothing if they don't say anything about this branch -- or Just (prob_s1) where prob_s1 is the likelyhood for s1 to be the -- taken branch. s1 is the branch in the true case. -- Loop exit heuristic. -- We are unlikely to leave a loop unless it's to enter another one. pred_LEH = 0.75 -- If and only if no successor is a loopheader, -- then we will likely not exit the current loop body. lehPredicts :: (TargetNodeInfo,TargetNodeInfo) -> Maybe Prob lehPredicts ((s1,_s1_info),(s2,_s2_info)) | S.member s1 loopHeads || S.member s2 loopHeads = Nothing | otherwise = --pprTrace "lehPredict:" (ppr $ compare s1Level s2Level) $ case compare s1Level s2Level of EQ -> Nothing LT -> Just (1-pred_LEH) --s1 exits to a shallower loop level (exits loop) GT -> Just (pred_LEH) --s1 exits to a deeper loop level where s1Level = mapLookup s1 loopLevels s2Level = mapLookup s2 loopLevels -- Comparing to a constant is unlikely to be equal. ohPredicts (s1,_s2) | CmmSource { trans_cmmNode = src1 } <- getTransitionSource node (fst s1) cfg , CmmCondBranch cond ltrue _lfalse likely <- src1 , likely == Nothing , CmmMachOp mop args <- cond , MO_Eq {} <- mop , not (null [x | x@CmmLit{} <- args]) = if fst s1 == ltrue then Just 0.3 else Just 0.7 | otherwise = Nothing -- TODO: These are all the other heuristics from the paper. -- Not all will apply, for now we just stub them out as Nothing. phPredicts = const Nothing ghPredicts = const Nothing lhhPredicts = const Nothing chPredicts = const Nothing shPredicts = const Nothing rhPredicts = const Nothing -- We normalize all edge weights as probabilities between 0 and 1. -- Ignoring rounding errors all outgoing edges sum up to 1. cfgEdgeProbabilities :: CFG -> (BlockId -> Int) -> IM.IntMap (IM.IntMap Prob) cfgEdgeProbabilities cfg toVertex = mapFoldlWithKey foldEdges IM.empty cfg where foldEdges = (\m from toMap -> IM.insert (toVertex from) (normalize toMap) m) normalize :: (LabelMap EdgeInfo) -> (IM.IntMap Prob) normalize weightMap | edgeCount <= 1 = mapFoldlWithKey (\m k _ -> IM.insert (toVertex k) 1.0 m) IM.empty weightMap | otherwise = mapFoldlWithKey (\m k _ -> IM.insert (toVertex k) (normalWeight k) m) IM.empty weightMap where edgeCount = mapSize weightMap -- Negative weights are generally allowed but are mapped to zero. -- We then check if there is at least one non-zero edge and if not -- assign uniform weights to all branches. minWeight = 0 :: Prob weightMap' = fmap (\w -> max (weightToDouble . edgeWeight $ w) minWeight) weightMap totalWeight = sum weightMap' normalWeight :: BlockId -> Prob normalWeight bid | totalWeight == 0 = 1.0 / fromIntegral edgeCount | Just w <- mapLookup bid weightMap' = w/totalWeight | otherwise = panic "impossible" -- This is the fixpoint algorithm from -- "Static Branch Prediction and Program Profile Analysis" by Y Wu, JR Larus -- The adaption to Haskell is my own. calcFreqs :: IM.IntMap (IM.IntMap Prob) -> [(Int,Int)] -> [(Int, [Int])] -> [Int] -> (Array Int Double, IM.IntMap (IM.IntMap Prob)) calcFreqs graph backEdges loops revPostOrder = runST $ do visitedNodes <- newArray (0,nodeCount-1) False :: ST s (STUArray s Int Bool) blockFreqs <- newArray (0,nodeCount-1) 0.0 :: ST s (STUArray s Int Double) edgeProbs <- newSTRef graph edgeBackProbs <- newSTRef graph -- let traceArray a = do -- vs <- forM [0..nodeCount-1] $ \i -> readArray a i >>= (\v -> return (i,v)) -- trace ("array: " ++ show vs) $ return () let -- See #1600, we need to inline or unboxing makes perf worse. -- {-# INLINE getFreq #-} {-# INLINE visited #-} visited b = unsafeRead visitedNodes b getFreq b = unsafeRead blockFreqs b -- setFreq :: forall s. Int -> Double -> ST s () setFreq b f = unsafeWrite blockFreqs b f -- setVisited :: forall s. Node -> ST s () setVisited b = unsafeWrite visitedNodes b True -- Frequency/probability that edge is taken. getProb' arr b1 b2 = readSTRef arr >>= (\graph -> return . fromMaybe (error "getFreq 1") . IM.lookup b2 . fromMaybe (error "getFreq 2") $ (IM.lookup b1 graph) ) setProb' arr b1 b2 prob = do g <- readSTRef arr let !m = fromMaybe (error "Foo") $ IM.lookup b1 g !m' = IM.insert b2 prob m writeSTRef arr $! (IM.insert b1 m' g) getEdgeFreq b1 b2 = getProb' edgeProbs b1 b2 setEdgeFreq b1 b2 = setProb' edgeProbs b1 b2 getProb b1 b2 = fromMaybe (error "getProb") $ do m' <- IM.lookup b1 graph IM.lookup b2 m' getBackProb b1 b2 = getProb' edgeBackProbs b1 b2 setBackProb b1 b2 = setProb' edgeBackProbs b1 b2 let -- calcOutFreqs :: Node -> ST s () calcOutFreqs bhead block = do !f <- getFreq block forM (successors block) $ \bi -> do let !prob = getProb block bi let !succFreq = f * prob setEdgeFreq block bi succFreq -- traceM $ "SetOut: " ++ show (block, bi, f, prob, succFreq) when (bi == bhead) $ setBackProb block bi succFreq let propFreq block head = do -- traceM ("prop:" ++ show (block,head)) -- traceShowM block !v <- visited block if v then return () --Dont look at nodes twice else if block == head then setFreq block 1.0 -- Loop header frequency is always 1 else do let preds = IS.elems $ predecessors block irreducible <- (fmap or) $ forM preds $ \bp -> do !bp_visited <- visited bp let bp_backedge = isBackEdge bp block return (not bp_visited && not bp_backedge) if irreducible then return () -- Rare we don't care else do setFreq block 0 !cycleProb <- sum <$> (forM preds $ \pred -> do if isBackEdge pred block then getBackProb pred block else do !f <- getFreq block !prob <- getEdgeFreq pred block setFreq block $! f + prob return 0) -- traceM $ "cycleProb:" ++ show cycleProb let limit = 1 - 1/512 -- Paper uses 1 - epsilon, but this works. -- determines how large likelyhoods in loops can grow. !cycleProb <- return $ min cycleProb limit -- <- return $ if cycleProb > limit then limit else cycleProb -- traceM $ "cycleProb:" ++ show cycleProb !f <- getFreq block setFreq block (f / (1.0 - cycleProb)) setVisited block calcOutFreqs head block -- Loops, by nesting, inner to outer forM_ loops $ \(head, body) -> do forM_ [0 .. nodeCount - 1] (\i -> unsafeWrite visitedNodes i True) -- Mark all nodes as visited. forM_ body (\i -> unsafeWrite visitedNodes i False) -- Mark all blocks reachable from head as not visited forM_ body $ \block -> propFreq block head -- After dealing with all loops, deal with non-looping parts of the CFG forM_ [0 .. nodeCount - 1] (\i -> unsafeWrite visitedNodes i False) -- Everything in revPostOrder is reachable forM_ revPostOrder $ \block -> propFreq block (head revPostOrder) -- trace ("Final freqs:") $ return () -- let freqString = pprFreqs freqs -- trace (unlines freqString) $ return () -- trace (pprFre) $ return () graph' <- readSTRef edgeProbs freqs' <- unsafeFreeze blockFreqs return (freqs', graph') where -- How can these lookups fail? Consider the CFG [A -> B] predecessors :: Int -> IS.IntSet predecessors b = fromMaybe IS.empty $ IM.lookup b revGraph successors :: Int -> [Int] successors b = fromMaybe (lookupError "succ" b graph)$ IM.keys <$> IM.lookup b graph lookupError s b g = pprPanic ("Lookup error " ++ s) $ ( text "node" <+> ppr b $$ text "graph" <+> vcat (map (\(k,m) -> ppr (k,m :: IM.IntMap Double)) $ IM.toList g) ) nodeCount = IM.foldl' (\count toMap -> IM.foldlWithKey' countTargets count toMap) (IM.size graph) graph where countTargets = (\count k _ -> countNode k + count ) countNode n = if IM.member n graph then 0 else 1 isBackEdge from to = S.member (from,to) backEdgeSet backEdgeSet = S.fromList backEdges revGraph :: IntMap IntSet revGraph = IM.foldlWithKey' (\m from toMap -> addEdges m from toMap) IM.empty graph where addEdges m0 from toMap = IM.foldlWithKey' (\m k _ -> addEdge m from k) m0 toMap addEdge m0 from to = IM.insertWith IS.union to (IS.singleton from) m0