Median of Medians

AtCoder

IDarc101_b

Time2000ms

Memory256MB

Difficulty—

We will define the **median** of a sequence $b$ of length $M$, as follows: * Let $b'$ be the sequence obtained by sorting $b$ in non-decreasing order. Then, the value of the $(M / 2 + 1)$\-th element of $b'$ is the median of $b$. Here, $/$ is integer division, rounding down. For example, the median of $(10, 30, 20)$ is $20$; the median of $(10, 30, 20, 40)$ is $30$; the median of $(10, 10, 10, 20, 30)$ is $10$. Snuke comes up with the following problem. You are given a sequence $a$ of length $N$. For each pair $(l, r)$ ($1 \leq l \leq r \leq N$), let $m_{l, r}$ be the median of the contiguous subsequence $(a_l, a_{l + 1}, ..., a_r)$ of $a$. We will list $m_{l, r}$ for all pairs $(l, r)$ to create a new sequence $m$. Find the median of $m$. ## Constraints * $1 \leq N \leq 10^5$ * $a_i$ is an integer. * $1 \leq a_i \leq 10^9$ ## Input Input is given from Standard Input in the following format: $N$ $a_1$ $a_2$ $...$ $a_N$ [samples]

Samples

Input #1

3
10 30 20

Output #1

30

The median of each contiguous subsequence of $a$ is as follows:

*   The median of $(10)$ is $10$.
*   The median of $(30)$ is $30$.
*   The median of $(20)$ is $20$.
*   The median of $(10, 30)$ is $30$.
*   The median of $(30, 20)$ is $30$.
*   The median of $(10, 30, 20)$ is $20$.

Thus, $m = (10, 30, 20, 30, 30, 20)$ and the median of $m$ is $30$.

Input #2

1
10

Output #2

Input #3

10
5 9 5 9 8 9 3 5 4 3

Output #3

API Response (JSON)

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    "name": "Median of Medians",
    "description": {
      "content": "We will define the **median** of a sequence $b$ of length $M$, as follows: *   Let $b'$ be the sequence obtained by sorting $b$ in non-decreasing order. Then, the value of the $(M / 2 + 1)$\\-th eleme",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc101_b"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We will define the **median** of a sequence $b$ of length $M$, as follows:\n\n*   Let $b'$ be the sequence obtained by sorting $b$ in non-decreasing order. Then, the value of the $(M / 2 + 1)$\\-th eleme...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments