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 Java (OpenJDK 11.0.6) (judge verdict: Time Limit Exceeded)


import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStream;
import java.io.InputStreamReader;
import java.util.HashMap;
import java.util.Map;
import java.util.StringTokenizer;

public class Main {
    private static Item[] items;
    private static final Map<Integer, Map<Integer, Integer>> tree = new HashMap<>();

    public static void main(final String[] args) {
        final FastScanner scanner = new FastScanner(System.in);
        final int n = scanner.nextInt();
        items = new Item[n + 1];
        for (int i = 1; i <= n; i++) {
            items[i] = new Item(scanner.nextInt(), scanner.nextInt());
        }

        final Map<Integer, Integer> initial = new HashMap<>();
        initial.put(0, 0);
        initial.put(items[1].weight, items[1].value);
        tree.put(1, initial);

        final int q = scanner.nextInt();
        for (int i = 0; i < q; i++) {
            final int v = scanner.nextInt();
            final int l = scanner.nextInt();
            System.out.println(entry(v, l));
        }
    }

    private static int entry(final int current, final int l) {
        if (tree.containsKey(current)) {
            return tree.get(current).entrySet().stream()
                .filter(e -> e.getKey() <= l)
                .mapToInt(Map.Entry::getValue)
                .max()
                .orElseThrow();
        }

        final int parent = current / 2;
        if (!tree.containsKey(parent)) {
            recursive(parent);
        }

        final Map<Integer, Integer> parentMap = tree.get(parent);
        final Map<Integer, Integer> currentMap = copyMap(parentMap);
        final Item item = items[current];
        for (final Map.Entry<Integer, Integer> entry : parentMap.entrySet()) {
            if (entry.getKey() + item.weight > 100_000) {
                continue;
            }

            currentMap.compute(entry.getKey() + item.weight, (k, v) -> v == null ? entry.getValue() + item.value : Math.max(entry.getValue() + item.value, v));
        }

        tree.put(current, currentMap);
        return currentMap.entrySet().stream()
            .filter(e -> e.getKey() <= l)
            .mapToInt(Map.Entry::getValue)
            .max()
            .orElseThrow();
    }

    private static Map<Integer, Integer> copyMap(final Map<Integer, Integer> map) {
        final Map<Integer, Integer> copy = new HashMap<>();
        for (final Map.Entry<Integer, Integer> entry : map.entrySet()) {
            copy.put(entry.getKey(), entry.getValue());
        }
        return copy;
    }

    private static void recursive(final int current) {
        if (tree.containsKey(current)) {
            return;
        }

        final int parent = current / 2;
        if (!tree.containsKey(parent)) {
            recursive(parent);
        }

        final Map<Integer, Integer> parentMap = tree.get(parent);
        final Map<Integer, Integer> currentMap = copyMap(parentMap);
        final Item item = items[current];
        for (final Map.Entry<Integer, Integer> entry : parentMap.entrySet()) {
            if (entry.getKey() + item.weight > 100_000) {
                continue;
            }

            currentMap.compute(entry.getKey() + item.weight, (k, v) -> v == null ? entry.getValue() + item.value : Math.max(entry.getValue() + item.value, v));
        }

        tree.put(current, currentMap);
    }

    private static class Item {
        final int value;
        final int weight;

        Item(final int value, final int weight) {
            this.value = value;
            this.weight = weight;
        }
    }

    private static class FastScanner {
        private final BufferedReader reader;
        private StringTokenizer tokenizer;

        FastScanner(final InputStream in) {
            reader = new BufferedReader(new InputStreamReader(in));
        }

        String next() {
            if (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (final IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken();
        }

        int nextInt() {
            return Integer.parseInt(next());
        }
    }
}