Problem:
Score : 600 points

Problem Statement

We have a set S of N points in a two-dimensional plane. The coordinates of the i-th point are (x_i, y_i). The N points have distinct x-coordinates and distinct y-coordinates.

For a non-empty subset T of S, let f(T) be the number of points contained in the smallest rectangle, whose sides are parallel to the coordinate axes, that contains all the points in T. More formally, we define f(T) as follows:

f(T) := (the number of integers i (1 \leq i \leq N) such that a \leq x_i \leq b and c \leq y_i \leq d, where a, b, c, and d are the minimum x-coordinate, the maximum x-coordinate, the minimum y-coordinate, and the maximum y-coordinate of the points in T)

Find the sum of f(T) over all non-empty subset T of S. Since it can be enormous, print the sum modulo 998244353.

Constraints

1 \leq N \leq 2 \times 10^5

-10^9 \leq x_i, y_i \leq 10^9

x_i \neq x_j (i \neq j)

y_i \neq y_j (i \neq j)

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
:
x_N y_N

Output

Print the sum of f(T) over all non-empty subset T of S, modulo 998244353.

Sample Input 1

3
-1 3
2 1
3 -2

Sample Output 1

13

Let the first, second, and third points be P_1, P_2, and P_3, respectively. S = \{P_1, P_2, P_3\} has seven non-empty subsets, and f has the following values for each of them:

f(\{P_1\}) = 1

f(\{P_2\}) = 1

f(\{P_3\}) = 1

f(\{P_1, P_2\}) = 2

f(\{P_2, P_3\}) = 2

f(\{P_3, P_1\}) = 3

f(\{P_1, P_2, P_3\}) = 3

The sum of these is 13.

Sample Input 2

4
1 4
2 1
3 3
4 2

Sample Output 2

34

Sample Input 3

10
19 -11
-3 -12
5 3
3 -15
8 -14
-9 -20
10 -9
0 2
-7 17
6 -6

Sample Output 3

7222

Be sure to print the sum modulo 998244353.

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

#[allow(unused_imports)]
use std::cmp;
use std::io::Write;

// Input macro from https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8
macro_rules! input {
  (source = $s:expr, $($r:tt)*) => {
    let mut iter = $s.split_whitespace();
    input_inner!{iter, $($r)*}
  };
  ($($r:tt)*) => {
    let s = {
      use std::io::Read;
      let mut s = String::new();
      std::io::stdin().read_to_string(&mut s).unwrap();
      s
    };
    let mut iter = s.split_whitespace();
    input_inner!{iter, $($r)*}
  };
}

macro_rules! input_inner {
  ($iter:expr) => {};
  ($iter:expr, ) => {};

  ($iter:expr, $var:ident : $t:tt $($r:tt)*) => {
    let $var = read_value!($iter, $t);
    input_inner!{$iter $($r)*}
  };
}

macro_rules! read_value {
  ($iter:expr, ( $($t:tt),* )) => {
    ( $(read_value!($iter, $t)),* )
  };

  ($iter:expr, [ $t:tt ; $len:expr ]) => {
    (0..$len).map(|_| read_value!($iter, $t)).collect::<Vec<_>>()
  };

  ($iter:expr, chars) => {
    read_value!($iter, String).chars().collect::<Vec<char>>()
  };

  ($iter:expr, usize1) => {
    read_value!($iter, usize) - 1
  };

  ($iter:expr, $t:ty) => {
    $iter.next().unwrap().parse::<$t>().expect("Parse error")
  };
}
// 1-indexed
// example: Bit::new(n, |x, y| *x += *y, 0)
struct Bit<T, F> {
    n: usize,
    dat: Vec<T>,
    op: F,
    unit: T,
}

impl<T: Clone, F: Fn(&mut T, &T)> Bit<T, F> {
    #[allow(dead_code)]
    fn new(n: usize, op: F, unit: T) -> Self {
        Bit {
            n: n,
            dat: vec![unit.clone(); n + 1],
            op: op,
            unit: unit,
        }
    }
    #[allow(dead_code)]
    fn from_vec(mut v: Vec<T>, op: F, unit: T) -> Self {
        let n = v.len();
        let mut dat = vec![unit.clone()];
        dat.append(&mut v);
        for i in 1..n {
            let j = i as i32;
            let b = (j & -j) as usize;
            let x = dat[i].clone();
            op(&mut dat[i + b], &x);
        }
        Bit {
            n: n,
            dat: dat,
            op: op,
            unit: unit,
        }
    }
    fn operate(&mut self, k: usize, a: &T) {
        let mut k = k;
        while k <= self.n {
            (self.op)(&mut self.dat[k], &a);
            let l = k as i32;
            k += (l & -l) as usize;
        }
    }
    fn accum(&self, k: usize) -> T {
        let mut k = k;
        let mut sum = self.unit.clone();
        while k > 0 {
            (self.op)(&mut sum, &self.dat[k]);
            let l = k as i32;
            k -= (l & -l) as usize;
        }
        sum
    }
}

const MOD: i64 = 998244353;

#[derive(Copy, Clone, Hash, Debug)]
struct ModInt {
  x: i64
}
impl ModInt {
  fn new(x: i64) -> Self {
    ModInt {
      x: x % MOD
    }
  }
  fn pow(self, e: i64) -> Self {
    if e == 0 {
      Self::new(1)
    } else if e % 2 == 1 {
      self.pow(e - 1) * self
    } else {
      let y = self.pow(e / 2);
      y * y
    }
  }
  #[allow(dead_code)]
  fn inv(self) -> Self {
    self.pow(MOD - 2)
  }
}
impl<T: Into<ModInt>> std::ops::Add<T> for ModInt {
  type Output = Self;
  fn add(self, other: T) -> Self {
    let other = other.into();
    let sum = self.x + other.x;
    ModInt {
      x: if sum >= MOD { sum - MOD } else { sum }
    }
  }
}
impl<T: Into<ModInt>> std::ops::Sub<T> for ModInt {
  type Output = Self;
  fn sub(self, other: T) -> Self {
    let other = other.into();
    let sum = self.x - other.x;
    ModInt {
      x: if sum < 0 { sum + MOD } else { sum }
    }
  }
}
impl<T: Into<ModInt>> std::ops::Mul<T> for ModInt {
  type Output = Self;
  fn mul(self, other: T) -> Self {
    let other = other.into();
    ModInt {
      x: self.x * other.x % MOD
    }
  }
}
impl<T: Into<ModInt>> std::ops::Div<T> for ModInt {
  type Output = Self;
  fn div(self, other: T) -> Self {
    let other = other.into();
    self * other.inv()
  }
}
impl From<i64> for ModInt {
  fn from(x: i64) -> Self {
    ModInt::new(x)
  }
}
impl std::fmt::Display for ModInt {
  fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
    self.x.fmt(f)
  }
}
impl<T: Into<ModInt>> std::ops::AddAssign<T> for ModInt {
  fn add_assign(&mut self, other: T) {
    *self = *self + other;
  }
}
impl<T: Into<ModInt>> std::ops::SubAssign<T> for ModInt {
  fn sub_assign(&mut self, other: T) {
    *self = *self - other;
  }
}
impl<T: Into<ModInt>> std::ops::MulAssign<T> for ModInt {
  fn mul_assign(&mut self, other: T) {
    *self = *self * other;
  }
}
impl<T: Into<ModInt>> std::ops::DivAssign<T> for ModInt {
  fn div_assign(&mut self, other: T) {
    *self = *self / other;
  }
}

fn main() {
  let out = std::io::stdout();
  let mut out = std::io::BufWriter::new(out.lock());
  macro_rules! puts {
    ($($format:tt)*) => (write!(out,$($format)*).unwrap());
  }
  input! {
    n: usize,
    ps: [(i64, i64); n],
  }
  let mut xs: Vec<_> = ps.iter().map(|&(x, _)| x).collect();
  xs.sort();
  use std::collections::BTreeMap;
  let x_compress: BTreeMap<_, _> = xs.iter().enumerate().map(|(i, &x)| (x, i)).collect();
  let mut ps = ps;
  ps.sort_by_key(|&(_, y)| y);
  let left = {
    let mut left = vec![(0, 0); n];
    let mut bit = Bit::new(n, |x, y| *x += *y, 0);
    for (i, &(x, _)) in ps.iter().enumerate() {
      let xc = *x_compress.get(&x).unwrap() + 1;
      let bottom = bit.accum(xc);
      left[i] = (i - bottom, bottom);
      bit.operate(xc, &1);
    }
    left
  };
  let right = {
    let mut right = vec![(0, 0); n];
    let mut bit = Bit::new(n, |x, y| *x += *y, 0);
    for (i, &(x, _)) in ps.iter().enumerate().rev() {
      let xc = *x_compress.get(&x).unwrap() + 1;
      let bottom = bit.accum(xc);
      right[i] = (n - 1 - i - bottom, bottom);
      bit.operate(xc, &1);
    }
    right
  };
  let mut ans = ModInt::new(0);
  for (&(tl, bl), &(tr, br)) in left.iter().zip(right.iter()) {
    let ptl = ModInt::new(2).pow(tl as i64) - 1;
    let pbl = ModInt::new(2).pow(bl as i64) - 1;
    let ptr = ModInt::new(2).pow(tr as i64) - 1;
    let pbr = ModInt::new(2).pow(br as i64) - 1;
    // 0
    ans += 1;
    // 1
    ans += ptl + pbl + ptr + pbr;
    // 2 - straight
    ans += ptl * pbl + pbl * pbr + pbr * ptr + ptr * ptl;
    // 2 - cross
    ans += (ptl * pbr + pbl * ptr) * 2;
    // 3
    ans += ptl * pbl * ptr       * 2;
    ans += ptl * pbl       * pbr * 2;
    ans += ptl       * ptr * pbr * 2;
    ans +=       pbl * ptr * pbr * 2;
    // 4
    ans += ptl * pbl * ptr * pbr * 2;
  }

  puts!("{}\n", ans);
}