Problem:
Score : 700 points

Problem Statement

We have a rooted binary tree with N vertices, where the vertices are numbered 1 to N.
Vertex 1 is the root, and the parent of Vertex i (i \geq 2) is Vertex \left[ \frac{i}{2} \right].

Each vertex has one item in it. The item in Vertex i has a value of V_i and a weight of W_i.
Now, process the following query Q times:

Given are a vertex v of the tree and a positive integer L.
Let us choose some (possibly none) of the items in v and the ancestors of v so that their total weight is at most L.
Find the maximum possible total value of the chosen items.

Here, Vertex u is said to be an ancestor of Vertex v when u is an indirect parent of v, that is, there exists a sequence of vertices w_1,w_2,\ldots,w_k (k\geq 2) where w_1=v, w_k=u, and w_{i+1} is the parent of w_i for each i.

Constraints

All values in input are integers.

1 \leq N < 2^{18}

1 \leq Q \leq 10^5

1 \leq V_i \leq 10^5

1 \leq W_i \leq 10^5

For the values v and L given in each query, 1 \leq v \leq N and 1 \leq L \leq 10^5.

Input

Let v_i and L_i be the values v and L given in the i-th query.
Then, Input is given from Standard Input in the following format:

N
V_1 W_1
:
V_N W_N
Q
v_1 L_1
:
v_Q L_Q

Output

For each integer i from 1 through Q,
the i-th line should contain the response to the i-th query.

Sample Input 1

3
1 2
2 3
3 4
3
1 1
2 5
3 5

Sample Output 1

0
3
3

In the first query, we are given only one choice: the item with (V, W)=(1,2). Since L = 1, we cannot actually choose it, so our response should be 0.

In the second query, we are given two choices: the items with (V, W)=(1,2) and (V, W)=(2,3). Since L = 5, we can choose both of them, so our response should be 3.

Sample Input 2

15
123 119
129 120
132 112
126 109
118 103
115 109
102 100
130 120
105 105
132 115
104 102
107 107
127 116
121 104
121 115
8
8 234
9 244
10 226
11 227
12 240
13 237
14 206
15 227

Sample Output 2

256
255
250
247
255
259
223
253

Solution in Haskell (GHC 8.8.3) (judge verdict: Accepted)

{-# OPTIONS_GHC -O2 #-}
{-# LANGUAGE BangPatterns, BinaryLiterals, CPP, DerivingStrategies  #-}
{-# LANGUAGE DerivingVia, FlexibleContexts, FlexibleInstances       #-}
{-# LANGUAGE GeneralizedNewtypeDeriving, KindSignatures, LambdaCase #-}
{-# LANGUAGE MagicHash, MultiParamTypeClasses, MultiWayIf           #-}
{-# LANGUAGE NumericUnderscores, OverloadedStrings, PatternSynonyms #-}
{-# LANGUAGE RankNTypes, RecordWildCards, ScopedTypeVariables       #-}
{-# LANGUAGE StandaloneDeriving, TupleSections, TypeApplications    #-}
{-# LANGUAGE TypeFamilies, TypeInType, UnboxedTuples, ViewPatterns  #-}

import           Control.Applicative
import           Control.Exception
import           Control.Monad
import           Control.Monad.Primitive
import           Control.Monad.Reader
import           Control.Monad.ST
import           Control.Monad.State.Strict
import           Data.Bifunctor
import           Data.Bool
import qualified Data.ByteString               as B
import qualified Data.ByteString.Builder       as B
import qualified Data.ByteString.Char8         as C
import qualified Data.ByteString.Internal      as B
import qualified Data.ByteString.Unsafe        as B
import           Data.Char
import qualified Data.Foldable                 as F
import           Data.Function
import           Data.Functor.Identity
import qualified Data.IntMap.Strict            as IM
import qualified Data.IntSet                   as IS
import qualified Data.List                     as L
import qualified Data.Map.Strict               as M
import           Data.Monoid
import           Data.Ord
import           Data.Primitive
import           Data.Proxy
import           Data.Ratio
import           Data.Semigroup
import qualified Data.Set                      as S
import           Data.Tuple
import qualified Data.Vector                   as V
import qualified Data.Vector.Generic           as G
import qualified Data.Vector.Generic.Mutable   as GM
import qualified Data.Vector.Mutable           as VM
import qualified Data.Vector.Primitive         as P
import qualified Data.Vector.Primitive.Mutable as PM
import qualified Data.Vector.Unboxed           as U
import qualified Data.Vector.Unboxed.Mutable   as UM
import           Debug.Trace
import           Foreign                       hiding (void)
import           GHC.Exts
import           GHC.TypeLits
import qualified System.IO                     as IO
import           Unsafe.Coerce

#define MOD 1000000007

main :: IO ()
main = do
    n <- readLn :: IO Int
    (vws,qs) <- U.splitAt (2*n) . U.unfoldrN (2*n+200001) (runParser int) <$> C.getContents
    let q = U.head qs
    B.hPutBuilder IO.stdout
        . U.foldr' (\x b -> B.intDec x <> B.char7 '\n' <> b) mempty
        $ solve n (tuples2N n vws) q (U.force . tuples2N q $ U.tail qs)

inf :: Int
inf = 0x3f3f3f3f

wSize, cacheSize, cacheWidth :: Int
wSize = 100001
ix :: Int -> Int -> Int
ix i j = i * wSize + j
{-# INLINE ix #-}
#ifdef DEBUG
cacheSize = 4
cacheWidth = 2
#else
cacheSize = 0x400
cacheWidth = 10
#endif

solve :: Int -> U.Vector Int -> Int -> U.Vector Int -> U.Vector Int
solve n0 (U.cons inf -> !tr) q qs = U.map (uncurry query.decode) qs
  where
    !table = U.create $ do
        dp <- UM.unsafeNew (cacheSize * wSize)
        UM.set (UM.take wSize dp) 0
        flip U.imapM_ (U.take cacheSize tr) $ \i vw -> when (i > 0) $ do
            let (!v, !w) = decode vw
            let !p = unsafeShiftRL i 1
            UM.unsafeCopy
                (UM.unsafeSlice (ix i 0) wSize dp)
                (UM.unsafeSlice (ix p 0) wSize dp)
            U.forM_ (U.generate (wSize - w) id) $ \j -> do
                v' <- (+v) <$!> UM.unsafeRead dp (ix p j)
                UM.unsafeModify dp (max v') $! (ix i (j + w))
        return dp

    query v l
        | v < cacheSize = table U.! ix v l
        | otherwise =
            let !pathSize = 64 - countLeadingZeros v - cacheWidth
                !base = ix (unsafeShiftR v pathSize) l
            in U.foldl' max 0
                . U.map (\encoded ->
                    let (score, weight) = decode encoded
                    in score + U.unsafeIndex table (base - weight))
                . U.filter ((<= l) . getWeight)
                . U.generate (shiftL 1 pathSize)
                $ \set ->
                    U.sum
                    . U.map (U.unsafeIndex tr . unsafeShiftR v)
                    . U.filter ((== 1) . (.&. 1) . unsafeShiftR set)
                    $ U.generate pathSize id

encode :: Int -> Int -> Int
encode x y = unsafeShiftL x 32 .|. y
{-# INLINE encode #-}

decode :: Int -> (Int, Int)
decode xy = (x, y)
  where
    !x = unsafeShiftRL xy 32 .&. 0xffff_ffff
    !y = xy .&. 0xffff_ffff
{-# INLINE decode #-}

getScore :: Int -> Int
getScore xy = unsafeShiftRL xy 32 .&. 0xffff_ffff
{-# INLINE getScore #-}

getWeight :: Int -> Int
getWeight xy = xy .&. 0xffff_ffff
{-# INLINE getWeight #-}


tuples2N :: Int -> U.Vector Int -> U.Vector Int
tuples2N n = U.unfoldrN n $ \v ->
    let !x = U.unsafeIndex v 0
        !y = U.unsafeIndex v 1
    in Just (encode x y, U.unsafeDrop 2 v)

-------------------------------------------------------------------------------
-- Utils
-------------------------------------------------------------------------------
rep :: Monad m => Int -> (Int -> m ()) -> m ()
rep !n = U.forM_ $ U.generate n id
{-# INLINE rep #-}
rev :: Monad m => Int -> (Int -> m ()) -> m ()
rev !n = U.forM_ $ U.iterateN n (subtract 1) (n - 1)
{-# INLINE rev #-}
infixl 8 `shiftRL`, `unsafeShiftRL`
shiftRL = unsafeShiftRL
{-# INLINE shiftRL #-}
unsafeShiftRL (I# x#) (I# i#) = I# (uncheckedIShiftRL# x# i#)
{-# INLINE unsafeShiftRL #-}
type Parser a = StateT C.ByteString Maybe a
runParser :: Parser a -> C.ByteString -> Maybe (a, C.ByteString)
runParser = runStateT
{-# INLINE runParser #-}
int :: Parser Int
int = coerce $ C.readInt . C.dropWhile isSpace
{-# INLINE int #-}
int1 :: Parser Int
int1 = fmap (subtract 1) int
{-# INLINE int1 #-}
char :: Parser Char
char = coerce C.uncons
{-# INLINE char #-}
byte :: Parser Word8
byte = coerce B.uncons
{-# INLINE byte #-}