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 Scala (2.11.7) (judge verdict: Accepted)

object Main {
  import java.io.{BufferedReader, InputStream, InputStreamReader}
  import java.util.StringTokenizer
  import scala.reflect.ClassTag

  def main(args: Array[String]): Unit = {
    val out = new java.io.PrintWriter(System.out)
    new Main(out, new InputReader(System.in)).solve()
    out.flush()
  }

  private val oj = System.getenv("ATCODER_DEBUG") == null
  def DEBUG(f: => Unit): Unit = {
    if (!oj){ f }
  }
  def debug(as: Array[Boolean]): Unit = if (!oj){ debug(as.map(x => if(x) "1" else "0").mkString) }
  def debug(as: Array[Int]): Unit = if (!oj){ debug(as.mkString(" ")) }
  def debug(as: Array[Long]): Unit =if (!oj){ debug(as.mkString(" ")) }
  def debugDim(m: Array[Array[Long]]): Unit = if (!oj){
    REP(m.length) { i =>
      debug(m(i))
    }
  }
  def debugDimFlip(m: Array[Array[Long]]): Unit = if (!oj){
    REP(m(0).length) { j =>
      REP(m.length) { i =>
        System.err.print(m(i)(j))
        System.err.print(" ")
      }
      System.err.println()
    }
  }
  def debug(s: => String): Unit = DEBUG {
    System.err.println(s)
  }
  def debugL(num: => Long): Unit = DEBUG {
    System.err.println(num)
  }

  class InputReader(val stream: InputStream) {
    private val reader = new BufferedReader(new InputStreamReader(stream), 32768)
    private var tokenizer: StringTokenizer = _

    def next(): String = {
      while (tokenizer == null || !tokenizer.hasMoreTokens)
        tokenizer = new StringTokenizer(reader.readLine)
      tokenizer.nextToken
    }

    def nextInt(): Int = Integer.parseInt(next())
    def nextLong(): Long = java.lang.Long.parseLong(next())
    def nextChar(): Char = next().charAt(0)

    def ni(): Int = nextInt()
    def nl(): Long = nextLong()
    def nc(): Char = nextChar()
    def ns(): String = next()
    def ns(n: Int): Array[Char] = ns().toCharArray
    def na(n: Int, offset: Int = 0): Array[Int] = map(n)(_ => ni() + offset)
    def na2(n: Int, offset: Int = 0): (Array[Int], Array[Int]) = {
      val A1, A2 = Array.ofDim[Int](n)
      REP(n) { i =>
        A1(i) = ni() + offset
        A2(i) = ni() + offset
      }
      (A1, A2)
    }
    def nm(n: Int, m: Int): Array[Array[Int]] = {
      val A = Array.ofDim[Int](n, m)
      REP(n) { i =>
        REP(m) { j =>
          A(i)(j) = ni()
        }
      }
      A
    }
    def nal(n: Int): Array[Long] = map(n)(_ => nl())
    def nm_c(n: Int, m: Int): Array[Array[Char]] = map(n) (_ => ns(m))
  }

  def REP(n: Int, offset: Int = 0)(f: Int => Unit): Unit = {
    var i = offset
    val N = n + offset
    while(i < N) { f(i); i += 1 }
  }
  def REP_r(n: Int, offset: Int = 0)(f: Int => Unit): Unit = {
    var i = n - 1 + offset
    while(i >= offset) { f(i); i -= 1 }
  }
  def TO(from: Int, to: Int)(f: Int => Unit): Unit = {
    REP(to - from + 1, from) { i =>
      f(i)
    }
  }
  def map[@specialized A: ClassTag](n: Int, offset: Int = 0)(f: Int => A): Array[A] = {
    val res = Array.ofDim[A](n)
    REP(n)(i => res(i) = f(i + offset))
    res
  }

  def sumL(as: Array[Int]): Long = {
    var s = 0L
    REP(as.length)(i => s += as(i))
    s
  }
  def cumSum(as: Array[Int]): Array[Long] = {
    val cum = Array.ofDim[Long](as.length + 1)
    REP(as.length) { i =>
      cum(i + 1) = cum(i) + as(i)
    }
    cum
  }

  case class Entry(x: Int, y: Int)
}

class Main(out: java.io.PrintWriter, sc: Main.InputReader) {
  import sc._
  import Main._
  import java.util.Arrays.sort

  import scala.collection.mutable
  import math.{abs, max, min}
  import mutable.ArrayBuffer

  // toIntとか+7とかするならvalにしろ
  @inline private def MOD = 998244353

  def solve(): Unit = {
    import java.util
    import java.util.Comparator
    val N = ni()
    val E = Array.ofDim[Entry](N)
    val X = Array.ofDim[Int](N)
    REP(N) { i =>
      E(i) = Entry(ni(), ni())
      X(i) = E(i).x
    }
    sort(X)

    def zip(x: Int): Int = lb(X, x)

    sort(E, new Comparator[Entry] {
      override def compare(o1: Entry, o2: Entry): Int = Integer.compare(o1.y, o2.y)
    })

    val pow2 = Array.ofDim[Long](N)
    pow2(0) = 1
    REP(pow2.length - 1)(i => pow2(i + 1) = pow2(i) * 2 % MOD)

    var ans = 0L
    val bit1 = new BIT(N, 0)(_ + _)
    val downL = Array.ofDim[Int](N)
    REP(N) { i =>
      val xid = zip(E(i).x)
      val l = bit1.sum(xid + 1)
      downL(i) = l
      bit1.add(xid, 1)
    }

    val bit2 = new BIT(N, 0)(_ + _)
    val t2 = new util.TreeSet[Int]()
    val upL = Array.ofDim[Int](N)
    REP_r(N) { i =>
      val xid = zip(E(i).x)
      val l = bit2.sum(xid + 1)
      upL(i) = l
      bit2.add(xid, 1)
    }

    REP(N) { i =>
      val upR = N - 1 - i - upL(i)
      val downR = i - downL(i)
//      ans += downL(i).toLong * upR + downR.toLong * upL(i)

      ans += (MOD + pow2(downL(i)) - 1) % MOD * (MOD + pow2(upR) - 1) % MOD *
        pow2(downR + upL(i)) % MOD

      ans += pow2(downL(i) + upR) *
        (MOD + pow2(downR) - 1) % MOD * (MOD + pow2(upL(i)) - 1) % MOD

      ans = MOD + ans - (MOD + pow2(downL(i)) - 1) % MOD * (MOD + pow2(upR) - 1) % MOD *
        (MOD + pow2(downR) - 1) % MOD * (MOD + pow2(upL(i)) - 1) % MOD

      debug(s"upR:$upR upL${upL(i)} downR:$downR downL:${downL(i)}")

      ans %= MOD
    }

    debug(downL)
    debug(upL)

    val NN = N

    def powMod(x: Int, n: Int, m: Int): Int = {
      def step(x: Long, n: Int, stack: Long): Long = {
        n match {
          case 0 => stack
          case _ => step(x * x % m, n / 2, if (n % 2 == 1) stack * x % m else stack)
        }
      }
      step(x, n, 1).toInt
    }

    val F = Array.ofDim[Long](NN + 1)
    F(0) = 1
    REP(NN) { i =>
      F(i + 1) = F(i) * (i + 1) % MOD
    }
    val I = Array.ofDim[Long](F.length)
    I(NN) = powMod(F(NN).toInt, MOD - 2, MOD)

    // x! = x(x-1)!
    // x!I(x!) ≡ (x-1)!I((x-1)!)
    // I((x-1)!) ≡ I(x!) * x   MODがでかいので(x-1)!はMODと素
    REP_r(NN) { i =>
      I(i) = I(i + 1) * (i + 1) % MOD
    }

    def comb(n: Int, k: Int): Long = {
      if (n < k) 0
      else F(n) * I(n - k) % MOD * I(k) % MOD
    }

    REP(N, 1) { i =>
      ans = (ans + i * comb(N, i)) % MOD
    }

    out.println(ans)
  }

  class BIT(n: Int, zero: Int)(f: (Int, Int) => Int) {
  def this(n: Int) = this(n, 0)(_ + _)

  private val N = {
    val a = Integer.highestOneBit(n)
    if (a == n) a else a << 1
  }
  private val bit = if (zero != 0) Array.fill[Int](N + 1)(zero) else Array.ofDim[Int](N + 1)

  /**
    * 1 index
    * addとindex違うよ
    * cumsumなんかといっしょ
    */
  def sum(i: Int): Int = {
    assert(i <= n)
    var x = i
    var s: Int = zero
    while(x > 0) {
      s = f(s, bit(x))
      x -= x & -x
    }
    s
  }

  /**
    * 0 index
    */
  def add(i: Int, a: Int): Unit = {
    assert(i < n)
    var x = i + 1
    while(x <= N) {
      bit(x) = f(bit(x), a)
      x += x & -x
    }
  }

  def sumAll: Int = sum(n)

  /**
    * [l, r)
    */
  def query(l: Int, r: Int)(sub: (Int, Int) => Int): Int = {
    assert(r > l)
    sub(sum(r), sum(l))
  }

  def lowerBound(W: Int)(sub: (Int, Int) => Int, lt: (Int, Int) => Boolean): Int = {
    var k = N
    var x = 0
    var w = W
    while(k > 0) {
      if (x + k <= N && lt(bit(x + k), w)) {
        w = sub(w, bit(x + k))
        x += k
      }
      k /= 2
    }
    x
  }
}
  def lb(a: Array[Int], x: Int): Int = {
    def step(l: Int, h: Int): Int = {
      if (h - l == 1) h
      else {
        val mid = (l + h) / 2
        if (a(mid) >= x) step(l, mid)
        else step(mid, h)
      }
    }

    step(-1, a.length)
  }
}