Problem:
Score : 700 points

Problem Statement

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

Output

Print the median of m.

Sample Input 1

3
10 30 20

Sample 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.

Sample Input 2

1
10

Sample Output 2

10

Sample Input 3

10
5 9 5 9 8 9 3 5 4 3

Sample Output 3

8

Solution in Rust (1.15.1) (judge verdict: Accepted)

use std::io::*;
use std::str::FromStr;

#[allow(dead_code)]
fn read<T: FromStr>() -> T {
    let stdin = stdin();
    let stdin = stdin.lock();
    let token: String = stdin
        .bytes()
        .map(|c| c.expect("failed to read char") as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().expect("failed to parse token")
}

fn calc(seq: &Vec<i64>, v: i64) -> usize {
    let n = seq.len();
    let mut total_cnts = vec![0, 0];
    let mut cnts = vec![vec![]; 2];
    let mut ret = 0;
    for i in 0..n {
        if seq[i] < v {
            let cnt1;
            if let Some(c) = cnts[1].pop() {
                total_cnts[1] -= c;
                cnt1 = 1 + c;
            } else {
                cnt1 = 1;
            }
            total_cnts[0] += cnt1;
            cnts[0].push(cnt1);
        } else {
            let cnt0;
            if let Some(c) = cnts[0].pop() {
                total_cnts[0] -= c;
                cnt0 = c;
            } else {
                cnt0 = 0;
            }
            if cnts[1].len() == 0 {
                cnts[1].push(1);
            } else {
                let l = cnts[1].len();
                cnts[1][l - 1] += 1;
            }
            total_cnts[1] += 1 + cnt0;
            cnts[1].push(cnt0);
        }
        // println!("{:?}", total_cnts);
        // println!("^ {:?}", cnts[1]);
        // println!("v {:?}", cnts[0]);
        ret += total_cnts[1];
    }
    return ret;
}

fn main() {
    let n = read::<usize>();
    let mut seq = vec![0; n];
    for i in 0..n {
        seq[i] = read::<i64>();
    }
    let ans = solve(&seq);
    println!("{}", ans);
    // let ans2 = naive(&seq);
}

fn solve(seq: &Vec<i64>) -> i64 {
    let n = seq.len();
    let m = n * (n + 1) / 2;
    let mut sort_seq = seq.clone();
    sort_seq.sort();
    let mut l = 0;
    let mut r = n;
    while r - l > 1 {
        let index = (r + l) / 2;
        let v = sort_seq[index];
        let target = if m % 2 == 0 { m / 2 } else { m / 2 + 1 };
        if calc(seq, v) >= target {
            l = index;
        } else {
            r = index;
        }
    }
    return sort_seq[l];
}

fn naive(seq: &Vec<i64>) -> i64 {
    let n = seq.len();
    let mut meds = vec![];
    for l in 0..n {
        for r in l..n {
            let mut items = vec![];
            for i in l..(r + 1) {
                items.push(seq[i]);
            }
            items.sort();
            meds.push(items[items.len() / 2]);
        }
    }
    meds.sort();
    // println!("{:?}", meds);
    return meds[meds.len() / 2];
}

#[allow(dead_code)]
struct XorShift {
    x: u32,
    y: u32,
    z: u32,
    w: u32,
}
#[allow(dead_code)]
impl XorShift {
    fn new() -> XorShift {
        XorShift {
            x: 0x34fb2383,
            y: 0x327328fa,
            z: 0xabd4b54a,
            w: 0xa9dba8d1,
        }
    }
    fn seed(s: u32) -> XorShift {
        let mut rnd = XorShift::new();
        rnd.w = s;
        rnd
    }
    fn xor128(&mut self) -> u32 {
        let t = self.x ^ (self.x << 11);
        self.x = self.y;
        self.y = self.z;
        self.z = self.w;
        self.w = (self.w ^ (self.w >> 19)) ^ (t ^ (t >> 8));
        self.w
    }
    fn next_i32(&mut self, r: i32) -> i32 {
        (self.xor128() % r as u32) as i32
    }
    fn next_i32_lr(&mut self, l: i32, r: i32) -> i32 {
        self.next_i32(r - l + 1) + l
    }
    fn next_i64(&mut self, r: i64) -> i64 {
        ((((self.xor128() as u64) << 32) + self.xor128() as u64) % r as u64) as i64
    }
    fn next_i64_lr(&mut self, l: i64, r: i64) -> i64 {
        self.next_i64(r - l + 1) + l
    }
    fn next_f64(&mut self, r: f64) -> f64 {
        (self.xor128() as f64) / (0xffffffffu32 as f64) * r
    }
    fn next_f64_lr(&mut self, l: f64, r: f64) -> f64 {
        self.next_f64(r - l) + l
    }
}

#[test]
fn test() {
    let mut rnd = XorShift::new();
    for _iter in 0..100 {
        let n = rnd.next_i32_lr(1, 8) as usize;
        let mut seq = vec![0; n];
        for i in 0..n {
            seq[i] = rnd.next_i64_lr(1, 10);
        }
        let v1 = solve(&seq);
        let v2 = naive(&seq);
        if v1 != v2 {
            println!("{}", n);
            for i in 0..n {
                print!("{} ", seq[i]);
            }
            println!("");
            println!("solve={} naive={}", v1, v2);
            assert!(v1 == v2);
        }
    }
}