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 Lua (LuaJIT 2.1.0) (judge verdict: Accepted)

local bls, brs = bit.lshift, bit.rshift
local mmi, mma = math.min, math.max
local bxor = bit.bxor

local function grayCode(x)
  return bxor(x, brs(x, 1))
end

-- add_func(idx), rm_func(idx), work_func()
local function grayWalk(size, add_func, rm_func, work_func)
  local prv = 0
  local total = bls(1, size) - 1
  local bpos = {}
  for i = 1, size do
    bpos[bls(1, i - 1)] = i
  end
  for i = 1, total do
    local v = grayCode(i)
    if prv < v then
      prv, v = v, v - prv
      add_func(bpos[v])
    else
      prv, v = v, prv - v
      rm_func(bpos[v])
    end
    work_func()
  end
end

local n = io.read("*n")
local v, w = {}, {}
for i = 1, n do
  v[i], w[i] = io.read("*n", "*n")
end
local head_tbl = {}
local box = {}
local ten5 = 100000

local tgt = 0
local weight, val = 0, 0
local tbl = nil
local function add_func(idx)
  weight = weight + w[brs(tgt, idx - 1)]
  val = val + v[brs(tgt, idx - 1)]
end
local function rm_func(idx)
  weight = weight - w[brs(tgt, idx - 1)]
  val = val - v[brs(tgt, idx - 1)]
end
local function work_func()
  if weight <= ten5 then
    tbl[weight] = mma(tbl[weight], val)
  end
end

for i = 512, mmi(n, 1023) do
  tgt = i
  head_tbl[i - 511] = {}
  tbl = head_tbl[i - 511]
  for j = 1, ten5 do
    tbl[j] = 0
  end
  weight, val = 0, 0
  grayWalk(10, add_func, rm_func, work_func)
  for i = 2, ten5 do
    tbl[i] = mma(tbl[i], tbl[i - 1])
  end
end

local q = io.read("*n")
local ans, wlim = 0, 0
local function work_func2()
  if weight <= wlim then
    ans = mma(ans, val)
  end
end
local function work_func3()
  if weight <= wlim then
    ans = mma(ans, val)
  end
  if weight < wlim then
    ans = mma(ans, val + tbl[wlim - weight])
  end
end
for iq = 1, q do
  local pos, wlim_tmp = io.read("*n", "*n")
  wlim = wlim_tmp
  local deg = 0
  do
    local tmp = pos
    while 0 < tmp do
      deg, tmp = deg + 1, brs(tmp, 1)
    end
  end
  if pos < 512 then
    ans = 0
    weight, val = 0, 0
    tgt = pos
    grayWalk(deg, add_func, rm_func, work_func2)
    print(ans)
  elseif pos < 1024 then
    print(head_tbl[pos - 511][wlim])
  else
    deg = deg - 10
    tbl = head_tbl[brs(pos, deg) - 511]
    ans = tbl[wlim]
    weight, val = 0, 0
    tgt = pos
    grayWalk(deg, add_func, rm_func, work_func3)
    print(ans)
  end
end