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)

// https://atcoder.jp/contests/abc136/tasks/abc136_f

#[allow(unused)]
use std::cmp::{max, min, Ordering};
#[allow(unused)]
use std::collections::{BTreeMap, BTreeSet, HashSet, VecDeque};
use std::io;
use std::io::{BufRead, Read, Write};
use std::str::FromStr;

/* Input macro */

#[allow(unused)]
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
}

#[allow(unused)]
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)*}
    };
}

#[allow(unused)]
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")
    };
}

#[allow(unused, non_snake_case)]
pub fn solve<R: BufRead, W: Write>(mut io: Io<R, W>) {
    let input = io.read_all();
    input! {
        source = input,
        N: usize,
        original_ps: [[isize; 2]; N],
    }
    let MOD = 998244353;

    // y の座標圧縮
    let mut map = BTreeMap::new();

    for p in &original_ps {
        map.insert(p[1], 0);
    }

    for (i, (k, v)) in map.iter_mut().enumerate() {
        *v = i
    }

    let mut ps = original_ps
        .iter()
        .map(|p| [p[0], map[&p[1]] as isize])
        .vec();

    ps.sort_by(|a, b| a[0].cmp(&b[0]));

    let mut lu = vec![0; N];
    let mut ld = vec![0; N];
    let mut ru = vec![0; N];
    let mut rd = vec![0; N];

    {
        let mut bit = fenwick_tree::FenwickTree::new(N, 0);

        for i in 0..ps.len() {
            let p = &ps[i];
            lu[i] = bit.sum(p[1] as usize, N);
            ld[i] = bit.sum_one(p[1] as usize);

            bit.add(p[1] as usize, 1);
        }
    }

    {
        let mut bit = fenwick_tree::FenwickTree::new(N, 0);

        for i in (0..ps.len()).rev() {
            let p = &ps[i];
            ru[i] = bit.sum(p[1] as usize, N);
            rd[i] = bit.sum_one(p[1] as usize);

            bit.add(p[1] as usize, 1);
        }
    }

    let mut res = 0;

    for i in 0..ps.len() {
        let lu_select = (pow_m(2, lu[i], MOD) + MOD - 1) % MOD;
        let lu_any = pow_m(2, lu[i], MOD);
        let ld_select = (pow_m(2, ld[i], MOD) + MOD - 1) % MOD;
        let ld_any = pow_m(2, ld[i], MOD);
        let ru_select = (pow_m(2, ru[i], MOD) + MOD - 1) % MOD;
        let ru_any = pow_m(2, ru[i], MOD);
        let rd_select = (pow_m(2, rd[i], MOD) + MOD - 1) % MOD;
        let rd_any = pow_m(2, rd[i], MOD);

        // その点を選ばない場合
        res += lu_select * ld_any % MOD * ru_any % MOD * rd_select % MOD;
        res %= MOD;
        res += lu_any * ld_select % MOD * ru_select % MOD * rd_any % MOD;
        res %= MOD;
        res -= lu_select * ld_select % MOD * ru_select % MOD * rd_select % MOD;
        res = (res + MOD) % MOD;

        // その点を選ぶ場合
        res += lu_any * ld_any % MOD * ru_any % MOD * rd_any % MOD;
        res %= MOD;
    }

    io.println(res);
}

#[allow(dead_code)]
fn main() {
    let stdin = io::stdin();
    let stdin_lock = stdin.lock();
    let stdout = io::stdout();
    let stdout_lock = stdout.lock();

    let io = Io {
        reader: stdin_lock,
        writer: stdout_lock,
    };

    solve(io);
}

pub trait BinarySearch<T> {
    fn lower_bound(&self, x: &T) -> usize;
    fn upper_bound(&self, x: &T) -> usize;
}
impl<T: Ord> BinarySearch<T> for [T] {
    fn lower_bound(&self, x: &T) -> usize {
        let mut low: isize = -1;
        let mut high: isize = self.len() as isize;
        while high - low > 1 {
            let mid = (low + high) / 2;
            match self[mid as usize].cmp(x) {
                Ordering::Less => {
                    low = mid;
                }
                Ordering::Equal | Ordering::Greater => {
                    high = mid;
                }
            }
        }
        high as usize
    }
    fn upper_bound(&self, x: &T) -> usize {
        let mut low: isize = -1;
        let mut high: isize = self.len() as isize;
        while high - low > 1 {
            let mid = (low + high) / 2;
            match self[mid as usize].cmp(x) {
                Ordering::Less | Ordering::Equal => {
                    low = mid;
                }
                Ordering::Greater => {
                    high = mid;
                }
            }
        }
        high as usize
    }
}

pub mod fenwick_tree {
    use std::ops::{AddAssign, Sub};
    pub struct FenwickTree<T> {
        n: usize,
        data: Vec<T>,
        init: T,
    }
    impl<T: Copy + AddAssign + Sub<Output = T>> FenwickTree<T> {
        pub fn new(size: usize, init: T) -> FenwickTree<T> {
            FenwickTree {
                n: size + 1,
                data: vec![init; size + 1],
                init: init,
            }
        }
        pub fn add(&mut self, k: usize, value: T) {
            let mut x = k;
            while x < self.n {
                self.data[x] += value;
                x |= x + 1;
            }
        }
        #[doc = " Returns a sum of range `[l, r)`"]
        pub fn sum(&self, l: usize, r: usize) -> T {
            self.sum_one(r) - self.sum_one(l)
        }
        #[doc = " Returns a sum of range `[0, k)`"]
        pub fn sum_one(&self, k: usize) -> T {
            assert!(k < self.n, "k={} n={}", k, self.n);
            let mut result = self.init;
            let mut x = k as isize - 1;
            while x >= 0 {
                result += self.data[x as usize];
                x = (x & (x + 1)) - 1;
            }
            result
        }
    }
}

fn pow_m(a: usize, e: usize, m: usize) -> usize {
    if e == 0 {
        return 1;
    }

    if e % 2 == 0 {
        let half = pow_m(a, e / 2, m);
        return half * half % m;
    }

    a * pow_m(a, e - 1, m) % m
}

/* IO helpers */

pub struct Io<R, W>
where
    R: BufRead,
    W: Write,
{
    pub reader: R,
    pub writer: W,
}

#[allow(dead_code)]
impl<R, W> Io<R, W>
where
    R: BufRead,
    W: Write,
{
    fn println<T: std::fmt::Display>(&mut self, value: T) {
        writeln!(&mut self.writer, "{}", value).expect("Unable to write");
    }

    fn print<T: std::fmt::Display>(&mut self, value: T) {
        write!(&mut self.writer, "{}", value).expect("Unable to write");
    }

    fn debug<T: std::fmt::Debug>(&mut self, value: T) {
        writeln!(&mut self.writer, "{:?}", value).expect("Unable to write");
    }

    fn next<T: FromStr>(&mut self) -> T {
        let token = self
            .reader
            .by_ref()
            .bytes()
            .map(|c| c.unwrap() as char)
            .skip_while(|c| c.is_whitespace())
            .take_while(|c| !c.is_whitespace())
            .collect::<String>();
        token.parse::<T>().ok().expect("Unable to parse")
    }

    fn nexts<T: FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.next::<T>()).collect::<Vec<T>>()
    }

    fn read_all(&mut self) -> String {
        let mut res = String::new();
        self.reader.read_to_string(&mut res).unwrap();
        res
    }
}

/* Structure helpers */

trait IteratorExt: Iterator + Sized {
    fn vec(self) -> Vec<Self::Item> {
        self.collect()
    }
}

impl<T: Iterator> IteratorExt for T {}