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 (Python):
import sys
import numpy as np

read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines

A = 256

def main(N, V, W, v, L):
    def precompute(N, V, W):
        U = 10**5 + 10
        dp = np.zeros((A, U), np.int32)
        for v in range(1, min(A, N + 1)):
            p = v // 2
            dp[v] = dp[p]
            dp[v, W[v]:] = np.maximum(dp[v, W[v]:], dp[p, :-W[v]] + V[v])
            for i in range(1, U):
                dp[v][i] = max(dp[v][i], dp[v][i - 1])
        return dp

    dp = precompute(N, V, W)
    values = np.empty((2**18)//A, np.int32)
    weights = np.empty((2**18)//A, np.int32)

    def solve(i, lim, values, weights, dp):
        if i < A:
            return dp[i][lim]
        values[0] = 0
        weights[0] = 0
        p = 0
        while i >= A:
            for j in range(1 << p):
                values[j + (1 << p)] = values[j] + V[i]
                weights[j + (1 << p)] = weights[j] + W[i]
            p += 1
            i >>= 1
        best = 0
        for n in range(1 << p):
            if weights[n] > lim:
                continue
            x = dp[i][lim - weights[n]] + values[n]
            best = max(best, x)
        return best

    Q = len(v)
    for i in range(Q):
        x = solve(v[i], L[i], values, weights, dp)
        print(x)

        
if sys.argv[-1] == 'ONLINE_JUDGE':
    from numba.pycc import CC
    cc = CC('my_module')
    cc.export('main', '(i4,i4[:],i4[:],i4[:],i4[:])')(main)
    cc.compile()
    exit()

from my_module import main

N = int(readline())
stdin = np.array(read().split(), np.int32)
V = np.zeros(N + 1, np.int32)
W = np.zeros(N + 1, np.int32)
V[1:] = stdin[:N + N:2]
W[1:] = stdin[1:N + N:2]
query = stdin[N + N + 1:]
v, L = query[::2].copy(), query[1::2].copy()

main(N, V, W, v, L)

Solution (Python):
import sys
import numpy as np

read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines

A = 1024

def main(N, V, W, v, L):
    def precompute(N, V, W):
        U = 10**5 + 10
        dp = np.zeros((A, U), np.int32)
        for v in range(1, min(A, N + 1)):
            p = v // 2
            dp[v] = dp[p]
            dp[v, W[v]:] = np.maximum(dp[v, W[v]:], dp[p, :-W[v]] + V[v])
            for i in range(1, U):
                dp[v][i] = max(dp[v][i], dp[v][i - 1])
        return dp

    dp = precompute(N, V, W)
    values = np.empty(A, np.int32)
    weights = np.empty(A, np.int32)

    def solve(i, lim, values, weights, dp):
        if i < A:
            return dp[i][lim]
        values[0] = 0
        weights[0] = 0
        p = 0
        while i >= A:
            for j in range(1 << p):
                values[j + (1 << p)] = values[j] + V[i]
                weights[j + (1 << p)] = weights[j] + W[i]
            p += 1
            i >>= 1
        best = 0
        for n in range(1 << p):
            if weights[n] > lim:
                continue
            x = dp[i][lim - weights[n]] + values[n]
            best = max(best, x)
        return best

    Q = len(v)
    for i in range(Q):
        x = solve(v[i], L[i], values, weights, dp)
        print(x)

        
if sys.argv[-1] == 'ONLINE_JUDGE':
    from numba.pycc import CC
    cc = CC('my_module')
    cc.export('main', '(i4,i4[:],i4[:],i4[:],i4[:])')(main)
    cc.compile()
    exit()

from my_module import main

N = int(readline())
stdin = np.array(read().split(), np.int32)
V = np.zeros(N + 1, np.int32)
W = np.zeros(N + 1, np.int32)
V[1:] = stdin[:N + N:2]
W[1:] = stdin[1:N + N:2]
query = stdin[N + N + 1:]
v, L = query[::2].copy(), query[1::2].copy()

main(N, V, W, v, L)

Solution (Python):
import sys
import numpy as np

read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines

A = 512

def main(N, V, W, v, L):
    def precompute(N, V, W):
        U = 10**5 + 10
        dp = np.zeros((A, U), np.int32)
        for v in range(1, min(A, N + 1)):
            p = v // 2
            dp[v] = dp[p]
            dp[v, W[v]:] = np.maximum(dp[v, W[v]:], dp[p, :-W[v]] + V[v])
            for i in range(1, U):
                dp[v][i] = max(dp[v][i], dp[v][i - 1])
        return dp

    dp = precompute(N, V, W)
    values = np.empty(A, np.int32)
    weights = np.empty(A, np.int32)

    def solve(i, lim, values, weights, dp):
        if i < A:
            return dp[i][lim]
        values[0] = 0
        weights[0] = 0
        p = 0
        while i >= A:
            for j in range(1 << p):
                values[j + (1 << p)] = values[j] + V[i]
                weights[j + (1 << p)] = weights[j] + W[i]
            p += 1
            i >>= 1
        best = 0
        for n in range(1 << p):
            if weights[n] > lim:
                continue
            x = dp[i][lim - weights[n]] + values[n]
            best = max(best, x)
        return best

    Q = len(v)
    for i in range(Q):
        x = solve(v[i], L[i], values, weights, dp)
        print(x)

        
if sys.argv[-1] == 'ONLINE_JUDGE':
    from numba.pycc import CC
    cc = CC('my_module')
    cc.export('main', '(i4,i4[:],i4[:],i4[:],i4[:])')(main)
    cc.compile()
    exit()

from my_module import main

N = int(readline())
stdin = np.array(read().split(), np.int32)
V = np.zeros(N + 1, np.int32)
W = np.zeros(N + 1, np.int32)
V[1:] = stdin[:N + N:2]
W[1:] = stdin[1:N + N:2]
query = stdin[N + N + 1:]
v, L = query[::2].copy(), query[1::2].copy()

main(N, V, W, v, L)

Solution (Python):
import sys
import numpy as np

read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines


def main(N, V, W, v, L):
    def precompute(N, V, W):
        U = 10**5 + 10
        dp = np.zeros((1024, U), np.int32)
        for v in range(1, min(1024, N + 1)):
            p = v // 2
            dp[v] = dp[p]
            dp[v, W[v]:] = np.maximum(dp[v, W[v]:], dp[p, :-W[v]] + V[v])
            for i in range(1, U):
                dp[v][i] = max(dp[v][i], dp[v][i - 1])
        return dp

    dp = precompute(N, V, W)
    values = np.empty(1024, np.int32)
    weights = np.empty(1024, np.int32)

    def solve(i, lim, values, weights, dp):
        if i < 1024:
            return dp[i][lim]
        values[0] = 0
        weights[0] = 0
        p = 0
        while i >= 1024:
            for j in range(1 << p):
                values[j + (1 << p)] = values[j] + V[i]
                weights[j + (1 << p)] = weights[j] + W[i]
            p += 1
            i >>= 1
        best = 0
        for n in range(1 << p):
            if weights[n] > lim:
                continue
            x = dp[i][lim - weights[n]] + values[n]
            best = max(best, x)
        return best

    Q = len(v)
    for i in range(Q):
        x = solve(v[i], L[i], values, weights, dp)
        print(x)

        
if sys.argv[-1] == 'ONLINE_JUDGE':
    from numba.pycc import CC
    cc = CC('my_module')
    cc.export('main', '(i4,i4[::1],i4[::1],i4[::1],i4[::1])')(main)
    cc.compile()
    exit()

from my_module import main

N = int(readline())
stdin = np.array(read().split(), np.int32)
V = np.zeros(N + 1, np.int32)
W = np.zeros(N + 1, np.int32)
V[1:] = stdin[:N + N:2]
W[1:] = stdin[1:N + N:2]
query = stdin[N + N + 1:]
v, L = query[::2].copy(), query[1::2].copy()

main(N, V, W, v, L)

Solution (Python):
import sys
import numpy as np

read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines


def main(N, V, W, v, L):
    def precompute(N, V, W):
        U = 10**5 + 10
        dp = np.zeros((1024, U), np.int32)
        for v in range(1, min(1024, N + 1)):
            p = v // 2
            dp[v] = dp[p]
            dp[v, W[v]:] = np.maximum(dp[v, W[v]:], dp[p, :-W[v]] + V[v])
            for i in range(1, U):
                dp[v][i] = max(dp[v][i], dp[v][i - 1])
        return dp

    dp = precompute(N, V, W)
    values = np.empty(1024, np.int32)
    weights = np.empty(1024, np.int32)

    def solve(i, lim, values, weights, dp):
        if i < 1024:
            return dp[i][lim]
        values[0] = 0
        weights[0] = 0
        p = 0
        while i >= 1024:
            for j in range(1 << p):
                values[j + (1 << p)] = values[j] + V[i]
                weights[j + (1 << p)] = weights[j] + W[i]
            p += 1
            i >>= 1
        best = 0
        for n in range(1 << p):
            if weights[n] > lim:
                continue
            x = dp[i][lim - weights[n]] + values[n]
            best = max(best, x)
        return best

    Q = len(v)
    for i in range(Q):
        x = solve(v[i], L[i], values, weights, dp)
        print(x)

        
if sys.argv[-1] == 'ONLINE_JUDGE':
    from numba.pycc import CC
    cc = CC('my_module')
    cc.export('main', '(i4,i4[:],i4[:],i4[:],i4[:])')(main)
    cc.compile()
    exit()

from my_module import main

N = int(readline())
stdin = np.array(read().split(), np.int32)
V = np.zeros(N + 1, np.int32)
W = np.zeros(N + 1, np.int32)
V[1:] = stdin[:N + N:2]
W[1:] = stdin[1:N + N:2]
query = stdin[N + N + 1:]
v, L = query[::2], query[1::2]

main(N, V, W, v, L)

Solution (Python):
import sys
input = lambda: sys.stdin.readline().rstrip()
N = int(input())
M = 10 ** 5 + 1
K = 10
A = [(0, 0)] * max(N + 1, 1 << K)
X = [[], [-1] * M]
X[1][0] = 0
v, w = map(int, input().split())
X[1][w] = v
A[1] = (v, w)
for i in range(2, min(N + 1, 1 << K)):
    v, w = map(int, input().split())
    if i % 2 == 0:
        X.append(X[i // 2][:])
    else:
        X.append([])
        X[i] = X[i // 2]
    
    for j in range(w, M)[::-1]:
        if X[i][j-w] >= 0:
            X[i][j] = max(X[i][j], X[i][j-w] + v)
    
    A[i] = (v, w)

for i in range(1 << K, N + 1):
    v, w = map(int, input().split())
    A[i] = (v, w)

if len(X) < 1 << K:
    for i in range(len(X), 1 << K):
        X.append([-1] * M)

for i in range(1 << K - 1, 1 << K):
    for j in range(1, M):
        X[i][j] = max(X[i][j], X[i][j-1])

Q = int(input())
for _ in range(Q):
    i, maw = map(int, input().split())
    if i < 1 << K - 1:
        V = []
        W = []
        while i:
            v, w = A[i]
            V.append(v)
            W.append(w)
            i //= 2
        k = len(V)
        ma = 0
        WW = [0] * (1 << k)
        VV = [0] * (1 << k)
        for i in range(1, 1 << k):
            a = i & (-i)
            b = a.bit_length() - 1
            ww = WW[i-a] + W[b]
            WW[i] = ww
            if ww > maw: continue
            vv = VV[i-a] + V[b]
            VV[i] = vv
            ma = max(ma, vv)
        print(ma)
    else:
        V = []
        W = []
        while i >= 1 << K:
            v, w = A[i]
            V.append(v)
            W.append(w)
            i //= 2
        ii = i
        k = len(V)
        WW = [0] * (1 << k)
        VV = [0] * (1 << k)
        ma = X[ii][maw]
        for i in range(1, 1 << k):
            a = i & (-i)
            b = a.bit_length() - 1
            ww = WW[i-a] + W[b]
            WW[i] = ww
            if ww > maw: continue
            vv = VV[i-a] + V[b]
            VV[i] = vv
            ma = max(ma, X[ii][maw - ww] + vv)
        print(ma)

Solution (Python):
import sys
input = sys.stdin.readline

N = int(input())
LIMIT = min(N,4195)

V = [0] * N
W = [0] * N
for i in range(N):
    nV, nW = map(int, input().split())
    V[i] = nV
    W[i] = nW

pre = [[] for i in range(LIMIT)]

for i in range(LIMIT):
    if i == 0:
        pre[i] = [(0,0), (W[i], V[i])]
        continue

    curr = pre[i]
    prev = pre[(i - 1)//2]
    nV, nW = V[i], W[i]
    
    size = len(prev)

    left = 0
    right = 0
    while left < size or right < size:
        if left == size or prev[left][0] > prev[right][0] + nW or (prev[left][0] == prev[right][0] + nW and prev[left][1] < prev[right][1] + nV):
            if len(curr) == 0 or prev[right][1] + nV > curr[-1][1]:
                curr.append((prev[right][0] + nW, prev[right][1] + nV))
            right += 1
        else:
            if len(curr) == 0 or prev[left][1] > curr[-1][1]:
                curr.append(prev[left])
            left += 1
            
Q = int(input())
out = [0] * Q
for t in range(Q):
    vert, L = map(int, input().split())
    vert -= 1

    stack = []
    while vert >= LIMIT:
        stack.append((V[vert], W[vert]))
        vert = (vert - 1) // 2
        
    prev = pre[vert]
    best = 0
    
    for i in range(1 << len(stack)):
        sV = 0
        sW = 0
        red = i
        for nV, nW in stack:
            if red & 1:
                sV += nV
                sW += nW
            red >>= 1

        limit = L - sW
        if limit < 0:
            continue
        
        lo = -1
        hi = len(prev)
        while hi - lo > 1:
            test = (hi + lo) // 2
            if prev[test][0] <= limit:
                lo = test
            else:
                hi = test
                
        if lo >= 0:
            new = prev[lo][1] + sV
            best = max(best, new)
        
    out[t] = best

print('\n'.join(map(str,out)))

Solution (Python):
import sys
import numpy as np
from numba import njit

read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines


@njit('(i4,i4[:],i4[:])', cache=True)
def precompute(N, V, W):
    U = 10**5 + 10
    dp = np.zeros((1024, U), np.int32)
    for v in range(1, min(1024, N + 1)):
        p = v // 2
        dp[v] = dp[p]
        dp[v, W[v]:] = np.maximum(dp[v, W[v]:], dp[p, :-W[v]] + V[v])
        for i in range(1, U):
            dp[v][i] = max(dp[v][i], dp[v][i - 1])
    return dp

@njit('(i4,i4[:],i4[:],i4[:],i4[:])', cache=True)
def main(N, V, W, v, L):
    dp = precompute(N, V, W)
    values = np.empty(1024, np.int32)
    weights = np.empty(1024, np.int32)

    def solve(i, lim, values, weights, dp):
        if i < 1024:
            return dp[i][lim]
        values[0] = 0
        weights[0] = 0
        p = 0
        while i >= 1024:
            for j in range(1 << p):
                values[j + (1 << p)] = values[j] + V[i]
                weights[j + (1 << p)] = weights[j] + W[i]
            p += 1
            i >>= 1
        best = 0
        for n in range(1 << p):
            if weights[n] > lim:
                continue
            x = dp[i][lim - weights[n]] + values[n]
            best = max(best, x)
        return best

    Q = len(v)
    for i in range(Q):
        x = solve(v[i], L[i], values, weights, dp)
        print(x)

N = int(readline())
stdin = np.array(read().split(), np.int32)
V = np.zeros(N + 1, np.int32)
W = np.zeros(N + 1, np.int32)
V[1:] = stdin[:N + N:2]
W[1:] = stdin[1:N + N:2]
query = stdin[N + N + 1:]
v, L = query[::2], query[1::2]

main(N, V, W, v, L)