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: Time Limit Exceeded)

/*------------------------------------------------------------------*/
/*------------------------- begin template -------------------------*/
/*------------------------------------------------------------------*/

// ref: tanakh <https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8>
// diff: using Parser
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut parser = Parser::from_str($s);
        input_inner!{parser, $($r)*}
    };
    (parser = $parser:ident, $($r:tt)*) => {
        input_inner!{$parser, $($r)*}
    };
    (new_stdin_parser = $parser:ident, $($r:tt)*) => {
        let stdin = std::io::stdin();
        let reader = std::io::BufReader::new(stdin.lock());
        let mut $parser = Parser::new(reader);
        input_inner!{$parser, $($r)*}
    };
    ($($r:tt)*) => {
        input!{new_stdin_parser = parser, $($r)*}
    };
}

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

macro_rules! read_value {
    ($parser:ident, ( $($t:tt),* )) => {
        ( $(read_value!($parser, $t)),* )
    };
    ($parser:ident, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($parser, $t)).collect::<Vec<_>>()
    };
    ($parser:ident, chars) => {
        read_value!($parser, String).chars().collect::<Vec<char>>()
    };
    ($parser:ident, char_) => {
        read_value!($parser, String).chars().collect::<Vec<char>>()[0]
    };
    ($parser:ident, usize1) => {
        read_value!($parser, usize) - 1
    };
    ($parser:ident, i64_) => {
        $parser.fast_i64()
    };
    ($parser:ident, usize_) => {
        $parser.fast_i64() as usize
    };
    ($parser:ident, usize1_) => {
        ($parser.fast_i64() - 1) as usize
    };
    ($parser:ident, $t:ty) => {
        $parser.next::<$t>().expect("Parse error")
    };
}

use std::io;
use std::io::BufRead;
use std::str;

// ref: tatsuya6502 <https://qiita.com/tatsuya6502/items/cd448486f7ef7b5b8c7e>
// ref: wariuni <https://qiita.com/tatsuya6502/items/cd448486f7ef7b5b8c7e#comment-7040a5ae96305e884eb9>
// diff: using std::io::BufRead::fill_buf()
pub struct Parser<R> {
    pub reader: R,
    buf: Vec<u8>,
    pos: usize,
}

impl Parser<io::Empty> {
    pub fn from_str(s: &str) -> Parser<io::Empty> {
        Parser {
            reader: io::empty(),
            buf: s.as_bytes().to_vec(),
            pos: 0,
        }
    }
}

impl<R: BufRead> Parser<R> {
    pub fn new(reader: R) -> Parser<R> {
        Parser {
            reader: reader,
            buf: vec![],
            pos: 0,
        }
    }
    pub fn update_buf(&mut self) {
        self.buf.clear();
        self.pos = 0;
        loop {
            let (len, complete) = {
                let buf2 = self.reader.fill_buf().unwrap();
                self.buf.extend_from_slice(buf2);
                let len = buf2.len();
                (len, buf2[len - 1] <= 0x20)
            };
            self.reader.consume(len);
            if complete {
                break;
            }
        }
    }
    pub fn next<T: str::FromStr>(&mut self) -> Result<T, T::Err> {
        loop {
            let mut begin = self.pos;
            while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                begin += 1;
            }
            let mut end = begin;
            while end < self.buf.len() && (self.buf[end] > 0x20) {
                end += 1;
            }
            if begin != self.buf.len() {
                self.pos = end;
                return unsafe { str::from_utf8_unchecked(&self.buf[begin..end]) }.parse::<T>();
            } else {
                self.update_buf();
            }
        }
    }
    pub fn fast_i64(&mut self) -> i64 {
        loop {
            let mut begin = self.pos;
            while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                begin += 1;
            }
            if begin == self.buf.len() {
                self.update_buf();
                continue;
            }
            let mut res = 0;
            let (is_positive, mut end) = match self.buf[begin] {
                b'+' => (true, begin + 1),
                b'-' => (false, begin + 1),
                _ => (true, begin),
            };
            while end < self.buf.len() && (self.buf[end] > 0x20) {
                res = res * 10 + (self.buf[end] as i64 - '0' as i64);
                end += 1;
            }
            if begin != self.buf.len() {
                self.pos = end;
                return if is_positive { res } else { -res };
            } else {
                self.update_buf();
            }
        }
    }
}

use std::marker::PhantomData;
use std::ops::*;

pub trait ModP {
    fn as_u64() -> u64;
}

#[allow(non_camel_case_types)]
#[derive(Clone, Copy, Debug)]
pub struct Mod_1_000_000_007();
impl ModP for Mod_1_000_000_007 {
    fn as_u64() -> u64 {
        1_000_000_007
    }
}

#[allow(non_camel_case_types)]
#[derive(Clone, Copy, Debug)]
pub struct Mod_998_244_353();
impl ModP for Mod_998_244_353 {
    fn as_u64() -> u64 {
        998_244_353
    }
}

#[derive(Clone, Copy, PartialEq, Debug)]
pub struct Modular<P: ModP + Copy>(u32, PhantomData<P>);

impl<P: ModP + Copy> Modular<P> {
    pub fn new(x: u32) -> Modular<P> {
        Modular((x as u64 % P::as_u64()) as u32, PhantomData)
    }
    pub fn pow(self, n: u64) -> Modular<P> {
        if n == 0 {
            return Modular(1, PhantomData);
        }
        let t = self.pow(n / 2);
        if n % 2 == 0 {
            t * t
        } else {
            t * t * self
        }
    }
    pub fn as_u32(self) -> u32 {
        self.0
    }
}

impl<P: ModP + Copy> Add for Modular<P> {
    type Output = Self;
    fn add(self, rhs: Self) -> Self::Output {
        let t = (self.0 + rhs.0) as u64;
        let u = if t < P::as_u64() { t } else { t - P::as_u64() };
        Modular(u as u32, PhantomData)
    }
}

impl<P: ModP + Copy> AddAssign for Modular<P> {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}

impl<P: ModP + Copy> Sub for Modular<P> {
    type Output = Self;
    fn sub(self, rhs: Self) -> Self::Output {
        let t = if self.0 >= rhs.0 {
            self.0 as u64 - rhs.0 as u64
        } else {
            self.0 as u64 + P::as_u64() - rhs.0 as u64
        };
        assert!(t < P::as_u64());
        Modular(t as u32, PhantomData)
    }
}

impl<P: ModP + Copy> SubAssign for Modular<P> {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}

impl<P: ModP + Copy> Neg for Modular<P> {
    type Output = Modular<P>;
    fn neg(self) -> Modular<P> {
        Modular((P::as_u64() - self.0 as u64) as u32, PhantomData)
    }
}

impl<P: ModP + Copy> Mul for Modular<P> {
    type Output = Self;
    fn mul(self, rhs: Self) -> Self::Output {
        Modular(
            ((self.0 as u64 % P::as_u64()) * (rhs.0 as u64 % P::as_u64()) % P::as_u64()) as u32,
            PhantomData,
        )
    }
}

impl<P: ModP + Copy> MulAssign for Modular<P> {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}

impl<P: ModP + Copy> Div for Modular<P> {
    type Output = Self;
    fn div(self, rhs: Self) -> Self::Output {
        if rhs.0 == 0 {
            loop {}
        }
        assert!(rhs.0 != 0);
        self * rhs.pow(P::as_u64() - 2)
    }
}

pub fn gen_fact_table<P: ModP + Copy>(n: usize) -> Vec<Modular<P>> {
    let mut res = vec![Modular::<P>::new(1); n + 1];
    for i in 0..n {
        res[i + 1] = res[i] * Modular::<P>::new((i + 1) as u32);
    }
    res
}

pub fn gen_inv_table<P: ModP + Copy>(n: usize) -> Vec<Modular<P>> {
    let mut res = vec![Modular::<P>::new(0); n + 1];
    res[1] = Modular::<P>::new(1);
    for i in 2..n + 1 {
        res[i] =
            -res[P::as_u64() as usize % i] * Modular::<P>::new((P::as_u64() / i as u64) as u32);
    }
    res
}

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

fn main() {
    input! {
        new_stdin_parser = parser,
    }
    let res = solve(&mut parser);
    print!("{}", res);
}

/*------------------------------------------------------------------*/
/*-------------------------- end template --------------------------*/
/*------------------------------------------------------------------*/

use std::rc::Rc;
enum KDNode {
    Leaf {
        x: i64,
        y: i64,
    },
    XNode {
        left: Rc<KDNode>,
        right: Rc<KDNode>,
        x: i64, // right
        n: usize,
        lu: (i64, i64),
        rd: (i64, i64),
    },
    YNode {
        left: Rc<KDNode>,
        right: Rc<KDNode>,
        y: i64, // right
        n: usize,
        lu: (i64, i64),
        rd: (i64, i64),
    },
}

impl KDNode {
    fn new(xs: &Vec<(i64, i64)>) -> Rc<KDNode> {
        KDNode::new_x(xs)
    }
    fn new_x(xs: &Vec<(i64, i64)>) -> Rc<KDNode> {
        let n = xs.len();
        if n == 1 {
            return Rc::new(KDNode::Leaf {
                x: xs[0].0,
                y: xs[0].1,
            });
        }
        let mut xs = xs.clone();
        xs.sort_by(|&x, &y| x.0.cmp(&y.0));
        let left = KDNode::new_y(&xs[0..n / 2].to_vec());
        let right = KDNode::new_y(&xs[n / 2..n].to_vec());
        let l = xs.iter().map(|&(x, _)| x).min().unwrap();
        let r = xs.iter().map(|&(x, _)| x).max().unwrap() + 1;
        let u = xs.iter().map(|&(_, y)| y).min().unwrap();
        let d = xs.iter().map(|&(_, y)| y).max().unwrap() + 1;
        Rc::new(KDNode::XNode {
            left: left,
            right: right,
            x: xs[n / 2].0,
            n: n,
            lu: (l, u),
            rd: (r, d),
        })
    }
    fn new_y(xs: &Vec<(i64, i64)>) -> Rc<KDNode> {
        let n = xs.len();
        if n == 1 {
            return Rc::new(KDNode::Leaf {
                x: xs[0].0,
                y: xs[0].1,
            });
        }
        let mut xs = xs.clone();
        xs.sort_by(|&x, &y| x.1.cmp(&y.1));
        let left = KDNode::new_x(&xs[0..n / 2].to_vec());
        let right = KDNode::new_x(&xs[n / 2..n].to_vec());
        let l = xs.iter().map(|&(x, _)| x).min().unwrap();
        let r = xs.iter().map(|&(x, _)| x).max().unwrap() + 1;
        let u = xs.iter().map(|&(_, y)| y).min().unwrap();
        let d = xs.iter().map(|&(_, y)| y).max().unwrap() + 1;
        Rc::new(KDNode::YNode {
            left: left,
            right: right,
            y: xs[n / 2].1,
            n: n,
            lu: (l, u),
            rd: (r, d),
        })
    }
    fn count(node: &Rc<KDNode>, lu: (i64, i64), rd: (i64, i64)) -> usize {
        let box_lu = match node.as_ref() {
            &KDNode::Leaf { x, y } => (x, y),
            &KDNode::XNode {
                left: _,
                right: _,
                x: _,
                n: _,
                lu,
                rd: _,
            } => lu,
            &KDNode::YNode {
                left: _,
                right: _,
                y: _,
                n: _,
                lu,
                rd: _,
            } => lu,
        };
        let box_rd = match node.as_ref() {
            &KDNode::Leaf { x, y } => (x + 1, y + 1),
            &KDNode::XNode {
                left: _,
                right: _,
                x: _,
                n: _,
                lu: _,
                rd,
            } => rd,
            &KDNode::YNode {
                left: _,
                right: _,
                y: _,
                n: _,
                lu: _,
                rd,
            } => rd,
        };
        KDNode::count_sub(node, lu, rd, box_lu, box_rd)
    }
    fn count_sub(
        node: &Rc<KDNode>,
        lu: (i64, i64),
        rd: (i64, i64),
        box_lu: (i64, i64),
        box_rd: (i64, i64),
    ) -> usize {
        if lu.0 <= box_lu.0 && lu.1 <= box_lu.1 && box_rd.0 <= rd.0 && box_rd.1 <= rd.1 {
            return match node.as_ref() {
                &KDNode::Leaf { x: _, y: _ } => 1,
                &KDNode::XNode {
                    left: _,
                    right: _,
                    x: _,
                    n,
                    lu: _,
                    rd: _,
                } => n,
                &KDNode::YNode {
                    left: _,
                    right: _,
                    y: _,
                    n,
                    lu: _,
                    rd: _,
                } => n,
            };
        }
        if rd.0 <= box_lu.0 || rd.1 <= box_lu.1 {
            return 0;
        }
        if box_rd.0 <= lu.0 || box_rd.1 <= lu.1 {
            return 0;
        }
        match node.as_ref() {
            &KDNode::Leaf { x, y } => {
                if lu.0 <= x && x < rd.0 && lu.1 <= y && y < rd.1 {
                    1
                } else {
                    0
                }
            }
            &KDNode::XNode {
                ref left,
                ref right,
                x,
                n,
                lu: _,
                rd: _,
            } => {
                let mut res = 0;
                if lu.0 <= x {
                    res += KDNode::count_sub(left, lu, rd, box_lu, (x, box_rd.1));
                }
                if x <= rd.0 {
                    res += KDNode::count_sub(right, lu, rd, (x, box_lu.1), box_rd);
                }
                res
            }
            &KDNode::YNode {
                ref left,
                ref right,
                y,
                n,
                lu: _,
                rd: _,
            } => {
                let mut res = 0;
                if lu.1 <= y {
                    res += KDNode::count_sub(left, lu, rd, box_lu, (box_rd.0, y));
                }
                if y <= rd.1 {
                    res += KDNode::count_sub(right, lu, rd, (box_lu.0, y), box_rd);
                }
                res
            }
        }
    }
}

type Mod = Modular<Mod_998_244_353>;

use std::fmt::Write;
fn solve<R: BufRead>(parser: &mut Parser<R>) -> String {
    let mut buf = String::new();
    input! {
        parser = parser,
        n: usize,
        ps: [(i64,i64); n],
    }
    let tree = KDNode::new(&ps);
    let inf = 1i64 << 60;
    let mut res = Mod::new(0);
    let one = Mod::new(1);
    let two = Mod::new(2);
    for i in 0..n {
        let lu = KDNode::count(&tree, (-inf, -inf), (ps[i].0, ps[i].1));
        let rd = KDNode::count(&tree, (ps[i].0 + 1, ps[i].1 + 1), (inf, inf));
        res += two.pow((n - lu - rd - 1) as u64)
            * (two.pow((lu + rd) as u64) - two.pow(lu as u64) - two.pow(rd as u64) + Mod::new(1));
        let a = lu;
        let b = rd;
        // eprintln!("lu = {}, rd = {}", lu, rd);
        let lu = KDNode::count(&tree, (ps[i].0 + 1, -inf), (inf, ps[i].1));
        let rd = KDNode::count(&tree, (-inf, ps[i].1 + 1), (ps[i].0, inf));
        res += two.pow((n - lu - rd - 1) as u64)
            * (two.pow((lu + rd) as u64) - two.pow(lu as u64) - two.pow(rd as u64) + Mod::new(1));
        let c = lu;
        let d = rd;
        // eprintln!("lu = {}, rd = {}", lu, rd);
        res += two.pow((n - 1) as u64);
        res -= (two.pow(a as u64) - one)
            * (two.pow(b as u64) - one)
            * (two.pow(c as u64) - one)
            * (two.pow(d as u64) - one);
    }
    writeln!(buf, "{}", res.as_u32());
    buf
}