Problem:
Score : 600 points

Problem Statement

There is a cave consisting of N rooms and M one-directional passages. The rooms are numbered 1 through N.

Takahashi is now in Room 1, and Room N has the exit. The i-th passage connects Room s_i and Room t_i (s_i < t_i) and can only be traversed in the direction from Room s_i to Room t_i. It is known that, for each room except Room N, there is at least one passage going from that room.

Takahashi will escape from the cave. Each time he reaches a room (assume that he has reached Room 1 at the beginning), he will choose a passage uniformly at random from the ones going from that room and take that passage.

Aoki, a friend of Takahashi's, can block one of the passages (or do nothing) before Takahashi leaves Room 1. However, it is not allowed to block a passage so that Takahashi is potentially unable to reach Room N.

Let E be the expected number of passages Takahashi takes before he reaches Room N. Find the value of E when Aoki makes a choice that minimizes E.

Constraints

2 \leq N \leq 600

N-1 \leq M \leq \frac{N(N-1)}{2}

s_i < t_i

If i != j, (s_i, t_i) \neq (s_j, t_j). (Added 21:23 JST)

For every v = 1, 2, ..., N-1, there exists i such that v = s_i.

Input

Input is given from Standard Input in the following format:

N M
s_1 t_1
:
s_M t_M

Output

Print the value of E when Aoki makes a choice that minimizes E.
Your output will be judged as correct when the absolute or relative error from the judge's output is at most 10^{-6}.

Sample Input 1

4 6
1 4
2 3
1 3
1 2
3 4
2 4

Sample Output 1

1.5000000000

If Aoki blocks the passage from Room 1 to Room 2, Takahashi will go along the path 1 → 3 → 4 with probability \frac{1}{2} and 1 → 4 with probability \frac{1}{2}. E = 1.5 here, and this is the minimum possible value of E.

Sample Input 2

3 2
1 2
2 3

Sample Output 2

2.0000000000

Blocking any one passage makes Takahashi unable to reach Room N, so Aoki cannot block a passage.

Sample Input 3

10 33
3 7
5 10
8 9
1 10
4 6
2 5
1 7
6 10
1 4
1 3
8 10
1 5
2 6
6 9
5 6
5 8
3 6
4 8
2 7
2 9
6 7
1 2
5 9
6 8
9 10
3 9
7 8
4 5
2 10
5 7
3 5
4 7
4 9

Sample Output 3

3.0133333333

Solution in Haskell (GHC 7.10.3) (judge verdict: Runtime Error)

{-# LANGUAGE
  ScopedTypeVariables, BangPatterns, TupleSections, ExplicitForAll,
  LambdaCase, MultiWayIf, Unsafe, RecordWildCards, FlexibleContexts, CPP,
  NoMonomorphismRestriction, GADTs, PatternGuards, MagicHash, UnboxedTuples,
  RankNTypes, EmptyDataDecls, EmptyCase, ViewPatterns, PolyKinds,
  TypeFamilies, OverloadedStrings, FlexibleInstances, UndecidableInstances,
  DefaultSignatures, GeneralizedNewtypeDeriving, StandaloneDeriving,
  DeriveGeneric, DeriveFunctor, DeriveDataTypeable, DeriveFoldable,
  DeriveTraversable, DeriveDataTypeable #-}
{-# OPTIONS_GHC -O2 #-}

import Prelude
import Data.Bits
import Data.List
import Data.Maybe
import Data.Tuple
import Data.Ord
import Data.Int
import Data.Word
import Data.Char
import Data.Ratio
import Data.Function
import Data.STRef
import Data.IORef
import Data.Monoid
-- import Data.Semigroup -- for execution in GHC >= 8, uncomment this line
import Data.Functor
import Data.Data
import Data.Typeable
import GHC.Generics
import System.IO
import Control.Arrow
import Control.Applicative
import Control.Monad
import Control.Monad.Primitive
import Control.Monad.State.Strict
import Control.Monad.ST
import Control.Monad.ST.Lazy (strictToLazyST, lazyToStrictST)
import qualified Control.Monad.ST.Lazy as STL
-- import Control.Monad.ST.Safe
import Control.DeepSeq
import Data.Coerce
import qualified Data.ByteString.Char8 as BS
import qualified Data.ByteString.Lazy.Char8 as BSL
import qualified Data.ByteString.Builder as BSB
import Data.IntMap (IntMap)
import qualified Data.IntMap.Strict as IMS
import qualified Data.IntMap.Lazy as IML
import Data.IntSet (IntSet)
import qualified Data.IntSet as IS
import qualified Data.Sequence as Seq
import Data.Sequence (Seq)
import qualified Data.Array.IArray as A
import qualified Data.Array.MArray.Safe as A
import qualified Data.Array.MArray as A
import Data.Array (Array)
import Data.Array.Unboxed (UArray)
import Data.Array.IArray (IArray)
import Data.Array.MArray.Safe (MArray)
import Data.Array.IO.Safe (IOArray, IOUArray)
import Data.Array.ST.Safe (STArray, STUArray, runSTArray, runSTUArray)
import qualified Data.Vector as V
import qualified Data.Vector.Mutable as VM
import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VUM
import qualified Data.Vector.Storable as VS
import qualified Data.Vector.Storable.Mutable as VSM
import qualified Data.Vector.Primitive as VP
import qualified Data.Vector.Primitive.Mutable as VPM
import qualified Data.Vector.Generic as VG
import qualified Data.Vector.Generic.Mutable as VGM
import qualified Data.Vector.Fusion.Bundle.Monadic as VFBM
import qualified Data.Vector.Fusion.Bundle as VFB
import qualified Data.Vector.Fusion.Stream.Monadic as VFSM
import qualified Data.Vector.Fusion.Bundle.Size as VFBS
import qualified Data.Vector.Fusion.Util as VFU
import qualified Data.Attoparsec.ByteString.Char8 as Atto
import Debug.Trace
import GHC.Exts (build, Int(..), Int#,
                 (+#), (*#), (-#), (<#), (>=#), (==#), quotRemInt#,
                 remInt#, uncheckedIShiftRL#, andI#, orI#,
                isTrue#)
rIntS1 = subtract 1 <$> rIntS

main :: IO ()
main = do
  [n,m] <- map readInt . words <$> getLine
  gd <- getDirectedGraphForward n m
        $ fromJust . evalStateT (liftA2 (,,()) rIntS1 rIntS1) <$> BS.getLine
  print $ query n gd
  return ()

data Folder = F {-# UNPACK #-} !Double {-# UNPACK #-} !Double

{- -- For execution in GHC >= 8, uncomment this region
instance Semigroup Folder where
  F m0 t0 <> F m1 t1 = F (max m0 m1) (t0 + t1)
-}

instance Monoid Folder where
  mappend = (<>)
  mempty = F (-1/0) 0
  mconcat = foldl' (<>) mempty

query :: Int -> GraphData () -> Double
query n gd
  = VU.minimum
    $ (\f -> VU.izipWith f means probs)
    $ \ !i !meani !probi ->
        let outi = edgesOut gd i
            !lenOut = VU.length outi
            meanRmv = (\(F m t) -> 1 + (t-m) / toDouble (lenOut-1))
                        $ VU.foldl' (\x y -> x `mappend` F y y) mempty
                        $ VU.backpermute means $ VU.map fst outi
        in if lenOut <= 1
           then plainRes
           else plainRes - probi * (meani - meanRmv)
  where
    !plainRes = VU.head means
    means = meanJumps n gd
    probs = arrvProbs n gd

meanJumps :: Int -> GraphData () -> VU.Vector Double
meanJumps !n !gd = VU.create $ do
  dp <- VUM.replicate n 0
  forM_ [n-2,n-3..0] $ \ !i -> do
    let outi = edgesOut gd i
    !sumDP <- VU.foldM' (\ !v (!j,()) -> (v+) <$> VUM.read dp j) 0 outi
    VUM.unsafeWrite dp i $! 1 + sumDP / toDouble (VU.length outi)
  return dp

arrvProbs :: Int -> GraphData () -> VU.Vector Double
arrvProbs !n !gd = VU.create $ do
  dp <- VUM.replicate n 0
  VUM.write dp 0 1
  forM_ [0..n-2] $ \ !i -> do
    p_i <- VUM.unsafeRead dp i
    let outi = edgesOut gd i
        !distProb = p_i / toDouble (VU.length outi)
    VU.forM_ outi $ VUM.modify dp (+distProb) . fst
  return dp
    

type Vertex = Int
data GraphData d = GraphData {-# UNPACK #-} !(VU.Vector (Vertex,d))
                             {-# UNPACK #-} !(VU.Vector Int)
data MGraphData s d = MGraphData {-# UNPACK #-} !(VUM.MVector s (Vertex,d))
                                 {-# UNPACK #-} !(VU.Vector Int)
data TreeData d = TreeData {-# UNPACK #-} !Vertex
                           {-# UNPACK #-} !(GraphData d)

edgesOut :: (VU.Unbox d)
  => GraphData d -> Vertex -> VU.Vector (Vertex,d)
edgesOut (GraphData connTable degCount) vert
  = VU.slice beg (end-beg) connTable
  where
    !end = degCount VU.! (vert+1)
    !beg = degCount VU.! vert

type Depth = Int
{-# INLINE getTreeWithDepths #-}
getTreeWithDepths :: (PrimMonad m, VU.Unbox d)
  => Int -> Vertex -> (m (Int,Int,d)) -> m (TreeData d,VU.Vector Depth)
getTreeWithDepths !numVert !root getEdge = do
  !gd <- getUndirectedGraphMut numVert (numVert-1) getEdge
  !depths <- dfsFindTreeAndDepthInMGraph gd root
  !gdFrozen <- unsafeFreezeGraphData gd
  return (TreeData root gdFrozen, depths)

dfsFindTreeAndDepthInMGraph :: (PrimMonad m, VU.Unbox d)
  => MGraphData (PrimState m) d -> Vertex -> m (VU.Vector Depth)
dfsFindTreeAndDepthInMGraph (MGraphData connTable degCount) !root = do
  depths <- VUM.replicate (VU.length degCount - 1) (-1)
  let dfs !parent !this !depth = do
        !prevDepth <- VUM.read depths this
        when (prevDepth < 0) $ do
          VUM.unsafeWrite depths this depth
          let !beg = VU.unsafeIndex degCount this
              !end = VU.unsafeIndex degCount (this+1) - 1
          forM_ [beg..end] $ \ !i -> do
            (!vertex,!d) <- VUM.read connTable i
            if vertex == parent
              then do !vbeg <- VUM.unsafeRead connTable beg
                      VUM.unsafeWrite connTable beg (vertex,d)
                      VUM.unsafeWrite connTable i vbeg
              else dfs this vertex (depth+1)
  dfs (-1) root 0
  VU.unsafeFreeze depths

{-# INLINE unsafeFreezeGraphData #-}
unsafeFreezeGraphData :: (PrimMonad m, VU.Unbox d) =>
  MGraphData (PrimState m) d -> m (GraphData d)
unsafeFreezeGraphData (MGraphData connTable degCount)
  = (`GraphData` degCount) <$!> VU.unsafeFreeze connTable 


{-# INLINE getDirectedGraphForward #-}
getDirectedGraphForward :: (PrimMonad m, VU.Unbox d) =>
  Int -> Int -> m (Int,Int,d) -> m (GraphData d)
getDirectedGraphForward !numVert !numEdges getEdge = do
  getDirectedGraphMutForward numVert numEdges getEdge >>= unsafeFreezeGraphData


{-# INLINE getUndirectedGraph #-}
getUndirectedGraph
  :: (PrimMonad m, VU.Unbox d) =>
  Int -> Int -> (m (Int,Int,d)) -> m (GraphData d)
getUndirectedGraph !numVert !numEdges getEdge
  = getUndirectedGraphMut numVert numEdges getEdge >>= unsafeFreezeGraphData

{-# INLINE getUndirectedGraphMut #-}
getUndirectedGraphMut
  :: (PrimMonad m, VU.Unbox d)
  => Int -> Int -> (m (Int,Int,d)) -> m (MGraphData (PrimState m) d)
getUndirectedGraphMut !numVert !numEdges getEdge = do
  !degCount <- VUM.replicate (numVert+1) 0
  !connTable <- VUM.new (shiftL numEdges 1)
  write degCount connTable
  MGraphData connTable <$!> VU.unsafeFreeze degCount
  where
    write !degCount !connTable = do
      !temp <- do
        !temp <- VUM.new numEdges
        VU.forM_ (VU.generate numEdges id) $ \ !i -> do
          (!a,!b,!d) <- getEdge
          VUM.unsafeWrite temp i (a,b,d)
          VUM.modify degCount (+1) (a+1)
          VUM.modify degCount (+1) (b+1)
        VU.unsafeFreeze temp
      VU.foldM'
        (\ !total !i -> do
            !di <- VUM.unsafeRead degCount i
            VUM.unsafeWrite degCount i total
            return $! total+di)
        0 (VU.generate (numVert+1) id)
      VU.forM_ temp $ \(!a,!b,!d) -> do
        !ia <- VUM.read degCount (a+1)
        VUM.unsafeWrite degCount (a+1) $! ia+1
        VUM.unsafeWrite connTable ia $! (b,d)
        !ib <- VUM.read degCount (b+1)
        VUM.unsafeWrite degCount (b+1) $! ib+1
        VUM.unsafeWrite connTable ib $! (a,d)
      
getDirectedGraphMutForward :: (PrimMonad m, VU.Unbox d)
  => Int -> Int -> (m (Int,Int,d))
  -> m (MGraphData (PrimState m) d)
getDirectedGraphMutForward !numVert !numEdges getEdge = do
  !degCount <- VUM.replicate (numVert+1) 0
  !connTable <- VUM.new numEdges
  write degCount connTable
  MGraphData connTable <$!> VU.unsafeFreeze degCount
  where
    write !degCount !connTable = do
      !temp <- do
        !temp <- VUM.new numEdges
        VU.forM_ (VU.generate numEdges id) $ \ !i -> do
          (!a,!b,!d) <- getEdge
          VUM.unsafeWrite temp i (a,b,d)
          VUM.modify degCount (+1) (a+1)
        VU.unsafeFreeze temp
      VU.foldM'
        (\ !total !i -> do
            !di <- VUM.unsafeRead degCount i
            VUM.unsafeWrite degCount i total
            return $! total+di)
        0 (VU.generate (numVert+1) id)
      VU.forM_ temp $ \(!a,!b,!d) -> do
        !ia <- VUM.read degCount (a+1)
        VUM.unsafeWrite degCount (a+1) $! ia+1
        VUM.unsafeWrite connTable ia $! (b,d)


#define IL(f) {-# INLINE f #-}; f

class ShowAsBuilder a where
  showAsBuilder :: a -> BSB.Builder
  default showAsBuilder :: (Show a) => a -> BSB.Builder
  showAsBuilder = BSB.string8 . show

#define INS(t,f) instance ShowAsBuilder t where \
  { {-# INLINE showAsBuilder #-}; showAsBuilder=f }
INS(Int,BSB.intDec)
INS(Int8,BSB.int8Dec)
INS(Int16,BSB.int16Dec)
INS(Int32,BSB.int32Dec)
INS(Int64,BSB.int64Dec)
INS(Word,BSB.wordDec)
INS(Word8,BSB.word8Dec)
INS(Word16,BSB.word16Dec)
INS(Word32,BSB.word32Dec)
INS(Word64,BSB.word64Dec)
INS(Integer,BSB.integerDec)
INS(Float,BSB.floatDec)
INS(Double,BSB.doubleDec)
-- INS(String,BSB.string8) -- Inconsistent with Show
-- INS(BS.ByteString,BSB.byteString) -- Inconsistent with Show
-- INS(BSL.ByteString,BSB.lazyByteString) -- Inconsisitent with Show
#undef INS

-- Inconsistent with show
instance (ShowAsBuilder a, VG.Vector v a) => ShowAsBuilder (v a) where
  showAsBuilder = v2BSpcSep

{-# INLINE putBuilder #-}
putBuilder = BSB.hPutBuilder stdout

printVecInLines, printVecInSpcSepLn ::
  (VG.Vector v a, ShowAsBuilder a) => v a -> IO ()
{-# INLINE printVecInLines #-}
printVecInLines = putBuilder . v2BLines
{-# INLINE printVecInSpcSepLn #-}
printVecInSpcSepLn = putBuilder . v2BSpcSepLn

v2BSpcSepLn, v2BSpcSep, v2BConcat, v2BLines ::
  (VG.Vector v a, ShowAsBuilder a)
  => v a -> BSB.Builder
{-# INLINE v2BSpcSepLn #-}
v2BSpcSepLn = v2BSpcSepLnWith showAsBuilder
{-# INLINE v2BSpcSep #-}
v2BSpcSep = v2BSpcSepWith showAsBuilder
{-# INLINE v2BConcat #-}
v2BConcat = v2BConcatWith showAsBuilder
{-# INLINE v2BLines #-}
v2BLines = v2BLinesWith showAsBuilder


v2BSpcSepLnWith, v2BSpcSepWith, v2BConcatWith, v2BLinesWith ::
  (VG.Vector v a)
  => (a -> BSB.Builder) -- ^ show function
  -> v a -> BSB.Builder
{-# INLINE v2BSpcSepLnWith #-}
v2BSpcSepLnWith = v2BSpcSepPostfWith "\n"
{-# INLINE v2BSpcSepWith #-}
v2BSpcSepWith = v2BSpcSepPostfWith ""
{-# INLINE v2BConcatWith #-}
v2BConcatWith showFct = VG.foldr ((<>) . showFct) mempty
{-# INLINE v2BLinesWith #-}
v2BLinesWith showFct
  = VG.foldr (\ a -> (showFct a <>) . (BSB.char7 '\n' <>)) mempty


v2BSpcSepPostf :: (VG.Vector v a, ShowAsBuilder a)
  => BS.ByteString -- ^ postfix
  -> v a -> BSB.Builder
{-# INLINE v2BSpcSepPostf #-}
v2BSpcSepPostf = (`v2BSpcSepPostfWith` showAsBuilder)

v2BSpcSepPostfWith :: (VG.Vector v a)
  => BS.ByteString -- ^ postfix
  -> (a -> BSB.Builder) -- ^ show function
  -> v a -> BSB.Builder
{-# INLINE v2BSpcSepPostfWith #-}
v2BSpcSepPostfWith = vecToBuilder "" " "

{-# INLINE vecToBuilder #-}
vecToBuilder :: (VG.Vector v a)
  => BS.ByteString -- ^ prefix
  -> BS.ByteString -- ^ separator
  -> BS.ByteString -- ^ postfix
  -> (a -> BSB.Builder) -- ^ show function
  -> v a -> BSB.Builder
vecToBuilder !prefix !separator !postfix
  = vecToBuilder_ (BSB.byteString prefix)
                  (BSB.byteString separator)
                  (BSB.byteString postfix)


{-# INLINE vecToBuilder_ #-}
vecToBuilder_ :: (VG.Vector v a)
  => BSB.Builder -- ^ prefix
  -> BSB.Builder -- ^ separator
  -> BSB.Builder -- ^ postfix
  -> (a -> BSB.Builder) -- ^ show function
  -> v a -> BSB.Builder
vecToBuilder_ !prefix !separator !postfix showFct vec
  = prefix <> VG.foldr
    (\ a rest !prefx -> prefx <> (showFct a <> rest separator))
    (const postfix) vec mempty

IL(evalVals) :: [a] -> [a]
evalVals xs = build $ \c n -> foldr (c $!) n xs
IL(forceVals) :: (NFData a) => [a] -> [a]
forceVals xs = build $ \c n -> foldr (c $!!) n xs

getVecGLn :: (VG.Vector v a) =>
           Int -> StateT BS.ByteString Maybe a -> IO (v a)
getVecGLn n s = VG.unfoldrN n (runStateT s) <$> BS.getLine
{-# INLINE getVecGLn #-}
getVecGRest :: (VG.Vector v a) =>
            Int -> StateT BSL.ByteString Maybe a -> IO (v a)
getVecGRest n s = VG.unfoldrN n (runStateT s) <$> BSL.getContents
{-# INLINE getVecGRest #-}
getVecLn :: Int -> StateT BS.ByteString Maybe a -> IO (V.Vector a)
getVecLn = getVecGLn
{-# INLINE getVecLn #-}
getVecRest :: Int -> StateT BSL.ByteString Maybe a -> IO (V.Vector a)
getVecRest = getVecGRest
{-# INLINE getVecRest #-}
getVecULn :: (VU.Unbox a) =>
  Int -> StateT BS.ByteString Maybe a -> IO (VU.Vector a)
getVecULn = getVecGLn
{-# INLINE getVecULn #-}
getVecURest :: (VU.Unbox a) =>
  Int -> StateT BSL.ByteString Maybe a -> IO (VU.Vector a)
getVecURest = getVecGRest
{-# INLINE getVecURest #-}

rInt :: StateT BSL.ByteString Maybe Int
IL(rInt) = StateT $ BSL.readInt . BSL.dropWhile (<'!')
rIntS :: StateT BS.ByteString Maybe Int
IL(rIntS) = StateT $ BS.readInt . BS.dropWhile (<'!')

unlessM, whenM :: (Monad m) => m Bool -> m () -> m ()
IL(whenM) = (. flip when) . (>>=)
IL(unlessM) = (. flip unless) . (>>=)

IL(wrA) = A.writeArray
IL(rdA) = A.readArray
IL(mdA) = \arr f !i -> do
  ai <- rdA arr i
  let fai = f ai 
  wrA arr i fai
  return (ai,fai)
{-# INLINE mdA' #-}
mdA' = \arr f !i -> do
  !ai <- rdA arr i
  let !fai = f ai
  wrA arr i fai
  return (ai,fai)
IL(swapA) = \arr !i !j -> do
  ai <- rdA arr i
  wrA arr i =<< rdA arr j
  wrA arr j ai

#define D(f,r,d)\
  IL(f) :: Integral a=>a->d; f=fromIntegral;\
  IL(r) :: String->d; r=read
#define C(f,r,g,h,d) D(f,r,d);\
  g,h :: RealFrac a=>a->d; IL(g)=floor; IL(h)=ceiling
C(_toInteger_,readInteger,floorInteger,ceilInteger,Integer)
C(toInt,readInt,floorInt,ceilInt,Int)
C(toI8,readI8,floorI8,ceilI8,Int8)
C(toI16,readI16,floorI16,ceilI16,Int16)
C(toI32,readI32,floorI32,ceilI32,Int32)
C(toI64,readI64,floorI64,ceilI64,Int64)
C(toWord,readWord,floorWord,ceilWord,Word)
C(toW8,readW8,floorW8,ceilW8,Word8)
C(toW16,readW16,floorW16,ceilW16,Word16)
C(toW32,readW32,floorW32,ceilW32,Word32)
C(toW64,readW64,floorW64,ceilW64,Word64)
D(toDouble,readDouble,Double)
D(toFloat,readFloat,Float)
#undef D
#undef C

#define N(f,g,h,a,m)\
  IL(f) :: forall e i s. (C(a,m)A.Ix i) => (i,i) -> e -> m (a i e);\
  f=A.newArray;\
  IL(g) :: forall e i s. (C(a,m)A.Ix i) => (i,i) -> m (a i e);\
  g=A.newArray_;\
  IL(h) :: forall e i s. (C(a,m)A.Ix i) => (i,i) -> [e] -> m (a i e);\
  h=A.newListArray
#define C(a,m)
N(newIOA,newIOA_,newIOAL,IOArray,IO)
N(newSTA,newSTA_,newSTAL,STArray s,ST s)
#undef C
#define C(a,m) MArray (a) e (m), 
N(newIOUA,newIOUA_,newIOUAL,IOUArray,IO)
N(newSTUA,newSTUA_,newSTUAL,STUArray s,ST s)
#undef C
#undef N

#undef IL