Knapsack Queries on a tree

AtCoder

IDtokiomarine2020_d

Time3000ms

Memory256MB

Difficulty—

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: $N$ $V_1$ $W_1$ $:$ $V_N$ $W_N$ $Q$ $v_1$ $L_1$ $:$ $v_Q$ $L_Q$ [samples]

Samples

Input #1

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 $3$.

Input #2

Output #2

API Response (JSON)

{
  "problem": {
    "name": "Knapsack Queries on a tree",
    "description": {
      "content": "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 $\\left[ \\frac{i}{2} \\right]$. Ea",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "tokiomarine2020_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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 $\\left[ \\frac{i}{2} \\right]$.\nEa...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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