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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Globalization;
using System.IO;
using System.Linq;

internal partial class Solver {
    public void Run() {
        var n = ni();
        var x = new int[n];
        var y = new int[n];
        for (int i = 0; i < n; i++) {
            x[i] = ni();
            y[i] = ni();
        }
        var d = IndexDictionaryFactory.Create(x.Concat(y));
        const int mod = 998244353;
        ModInt.SetModValue(mod);
        var points = x.Zip(y, (X, Y) => new { X = d.GetIndex(X), Y = d.GetIndex(Y) })
            .OrderBy(p => p.X)
            .ToArray();

        var left = new BinaryIndexTree(d.Count);
        var right = new BinaryIndexTree(d.Count);

        foreach (var p in points) right.Add(p.Y, 1);

        ModInt ans = 0;

        foreach (var p in points) {
            right.Add(p.Y, -1);

            ModInt p00 = ModPow(2, left.Sum(0, p.Y), mod) - 1;
            ModInt p01 = ModPow(2, left.Sum(p.Y + 1, d.Count), mod) - 1;
            ModInt p10 = ModPow(2, right.Sum(0, p.Y), mod) - 1;
            ModInt p11 = ModPow(2, right.Sum(p.Y + 1, d.Count), mod) - 1;

            ModInt pattern = ModPow(2, n - 1, mod);
            pattern += p00 * p11;
            pattern += p01 * p10;
            pattern += p01 * p10 * p11;
            pattern += p00 * p10 * p11;
            pattern += p00 * p01 * p11;
            pattern += p00 * p01 * p10;
            pattern += p00 * p01 * p10 * p11;

            ans += pattern;

            left.Add(p.Y, 1);
        }

        cout.WriteLine(ans);
    }

    public struct ModInt {
        private static int mod = 0;
        private long value;

        public ModInt(long x) {
            value = x;
            Normalize();
        }

        private static long RegularMod(long x, int mod) {
            if (x >= mod) {
                if (x < 2 * mod) {
                    return x - mod;
                }

                return x % mod;
            }
            if (x >= 0) {
                return x;
            }

            x = mod - RegularMod(-x, mod);
            if (x == mod) {
                return 0;
            }

            return x;
        }

        public static void SetModValue(int m) {
            ModInt.mod = m;
        }

        private void Normalize() {
            value = RegularMod(value, mod);
        }

        public override string ToString() {
            return value.ToString();
        }

        public int ToInt() {
            return (int)value;
        }

        public static bool operator ==(ModInt c1, ModInt c2) {
            return c1.value == c2.value;
        }

        public static bool operator !=(ModInt c1, ModInt c2) {
            return !(c1 == c2);
        }

        public static ModInt operator +(ModInt x, ModInt y) {
            return new ModInt(x.value + y.value);
        }
        public static ModInt operator -(ModInt x, ModInt y) {
            return new ModInt(x.value - y.value);
        }
        public static ModInt operator *(ModInt x, ModInt y) {
            return new ModInt(x.value * y.value);
        }
        public static ModInt operator /(ModInt x, ModInt y) {
            return new ModInt(x.value * Inverse(y.value, mod));
        }

        private static long ExtendedGcd(long a, long b, ref long x, ref long y) {
            if (b == 0) {
                x = 1; y = 0;
                return a;
            } else {
                long d = ExtendedGcd(b, a % b, ref y, ref x);
                y -= a / b * x;
                return d;
            }
        }

        private static long Inverse(long a, long mod) {
            long x = 0, y = 0;
            if (ExtendedGcd(a, mod, ref x, ref y) == 1) {
                return (x + mod) % mod;
            } else {
                throw new Exception("Invalid inverse " + a + " " + mod);
            }
        }

        public static implicit operator ModInt(long x) {
            return new ModInt(x);
        }

        public override bool Equals(object obj) {
            if (obj == null) {
                return false;
            }
            return value.Equals(((ModInt)obj).value);
        }

        public override int GetHashCode() {
            return value.GetHashCode();
        }
    }
    public static long ModPow(long x, long n, long mod) {
        long t = x, ret = 1;
        while (n > 0) {
            if ((n & 1) != 0) {
                ret = (t * ret) % mod;
            }
            t = t * t % mod;
            n >>= 1;
        }
        return ret;
    }

    public class BinaryIndexTree {
        private readonly long[] array;
        public BinaryIndexTree(int size) {
            array = new long[size];
        }
        public long Sum(int begin, int end) {
            if (begin != 0) {
                return Sum(0, end) - Sum(0, begin);
            }

            long val = 0;
            end--;
            for (; end >= 0; end = (end & (end + 1)) - 1) {
                val += array[end];
            }

            return val;
        }

        public void Add(int k, long val) {
            for (; k < array.Length; k |= k + 1) {
                array[k] += val;
            }
        }

        /// <summary>return min pos where sum(A[0] .. A[pos]) >= w</summary>
        public int LowerBound(long w) {
            if (w <= 0) {
                return 0;
            }

            int x = 0, k = 1, N = array.Length;
            while ((k << 1) <= N) {
                k <<= 1;
            }

            for (; k > 0; k >>= 1) {
                if (x + k - 1 < N && array[x + k - 1] < w) {
                    w -= array[x + k - 1];
                    x += k;
                }
            }
            return (x == N) ? -1 : x;
        }
    }
    public class IndexDictionary<T> {
        private readonly Dictionary<T, int> _toIndex;
        private readonly T[] _array;

        public IndexDictionary(IEnumerable<T> source) : this(source, Comparer<T>.Default) { }

        public IndexDictionary(IEnumerable<T> source, IComparer<T> comparer) {
            _array = source.Distinct().ToArray();
            Array.Sort(_array, comparer);
            _toIndex = new Dictionary<T, int>(_array.Length);
            int index = -1;
            foreach (var x in _array) {
                _toIndex[x] = ++index;
            }
        }

        public int Count { get { return _array.Length; } }

        public T this[int index] { get { return _array[index]; } }

        public int GetIndex(T value) { return _toIndex[value]; }
    }

    public class IndexDictionaryFactory {
        public static IndexDictionary<T> Create<T>(IEnumerable<T> source) {
            return new IndexDictionary<T>(source);
        }
    }
}

// PREWRITEN CODE BEGINS FROM HERE
internal partial class Solver : Scanner {
    public static void Main(string[] args) {
#if LOCAL
        new Solver(Console.In, Console.Out).Run();
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver(Console.In, Console.Out).Run();
        Console.Out.Flush();
#endif
    }

#pragma warning disable IDE0052
    private readonly TextReader cin;
    private readonly TextWriter cout;
#pragma warning restore IDE0052

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        cin = reader;
        cout = writer;
    }
    public Solver(string input, TextWriter writer)
        : this(new StringReader(input), writer) {
    }

#pragma warning disable IDE1006
#pragma warning disable IDE0051
    private int ni() { return NextInt(); }
    private int[] ni(int n) { return NextIntArray(n); }
    private long nl() { return NextLong(); }
    private long[] nl(int n) { return NextLongArray(n); }
    private double nd() { return NextDouble(); }
    private double[] nd(int n) { return NextDoubleArray(n); }
    private string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
#pragma warning restore IDE1006
#pragma warning restore IDE0051
}

namespace cslib.Extensions {
    public static class LinqPadExtension {
        public static T Dump<T>(this T obj) {
#if LOCAL
            return LINQPad.Extensions.Dump(obj);
#else
            return obj;
#endif
        }
    }
}
public class Scanner {
    private readonly TextReader Reader;
    private readonly Queue<string> TokenQueue = new Queue<string>();
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

    public Scanner()
        : this(Console.In) {
    }

    public Scanner(TextReader reader) {
        Reader = reader;
    }

    public int NextInt() { return int.Parse(Next(), ci); }
    public long NextLong() { return long.Parse(Next(), ci); }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        string[] array = new string[size];
        for (int i = 0; i < size; i++) {
            array[i] = Next();
        }

        return array;
    }
    public int[] NextIntArray(int size) {
        int[] array = new int[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextInt();
        }

        return array;
    }

    public long[] NextLongArray(int size) {
        long[] array = new long[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextLong();
        }

        return array;
    }

    public double[] NextDoubleArray(int size) {
        double[] array = new double[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextDouble();
        }

        return array;
    }

    public string Next() {
        if (TokenQueue.Count == 0) {
            if (!StockTokens()) {
                throw new InvalidOperationException();
            }
        }
        return TokenQueue.Dequeue();
    }

    public bool HasNext() {
        if (TokenQueue.Count > 0) {
            return true;
        }

        return StockTokens();
    }

    private static readonly char[] _separator = new[] { ' ', '\t' };
    private bool StockTokens() {
        while (true) {
            string line = Reader.ReadLine();
            if (line == null) {
                return false;
            }

            string[] tokens = line.Split(_separator, StringSplitOptions.RemoveEmptyEntries);
            if (tokens.Length == 0) {
                continue;
            }

            foreach (string token in tokens) {
                TokenQueue.Enqueue(token);
            }

            return true;
        }
    }
}