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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.*;
import java.util.*;
import java.util.function.Function;

public class Main {

    static int N;
    static Cord[] C;

    public static void main(String[] args) {
        FastScanner sc = new FastScanner(System.in);
        N = sc.nextInt();
        C = new Cord[N];
        for (int i = 0; i < N; i++) {
            C[i] = new Cord(sc.nextInt(), sc.nextInt());
        }

        System.out.println(solve());
    }

    static int[] POW2;

    static int solve() {
        POW2 = new int[N+1];
        POW2[0] = 1;
        for (int i = 1; i <= N; i++) {
            POW2[i] = mul(POW2[i-1], 2);
        }

        int[] Y = new int[N];
        for (int i = 0; i < N; i++) {
            Y[i] = C[i].y;
        }
        Zip zip = new Zip(Y);
        Arrays.sort(C, Comparator.comparingInt(c -> c.x));
        // ba
        // cd
        FenwickTree bit = new FenwickTree(N, FenwickTree.SUM);
        for (int i = 0; i < N; i++) {
            Cord c = C[i];

            int lo = (int)bit.query(zip.to_i(c.y));
            int hi = i - lo;
            c.b = hi;
            c.c = lo;
            bit.add(zip.to_i(c.y), 1);
        }
        Arrays.fill(bit.bit, 0);
        for (int i = N-1; i >= 0; i--) {
            Cord c = C[i];

            int lo = (int)bit.query(zip.to_i(c.y));
            int hi = (N-1-i) - lo;
            c.a = hi;
            c.d = lo;
            bit.add(zip.to_i(c.y), 1);
        }

//        for (int i = 0; i < N; i++) {
//            Cord c = C[i];
//            debug(c.x, c.y, c.a, c.b, c.c, c.d);
//        }

        int ans = 0;
        for (int i = 0; i < N; i++) {
            Cord cord = C[i];

            for (int a = 0; a < 2; a++) {
                for (int b = 0; b < 2; b++) {
                    for (int c = 0; c < 2; c++) {
                        for (int d = 0; d < 2; d++) {
                            // 自分がoff
                            if( isOk(a, b, c, d) ) {
                                ans += calc(cord, a, b, c, d);
                                ans %= MOD;
                            }

                            // 自分がon
                            ans += calc(cord, a, b, c, d);
                            ans %= MOD;
                        }
                    }
                }
            }
        }
        return ans;
    }

    // ba
    // cd
    static boolean isOk(int a, int b, int c, int d) {
        boolean up =    a==1 || b==1;
        boolean down =  c==1 || d==1;
        boolean left =  b==1 || c==1;
        boolean right = a==1 || d==1;
        return up && down && left && right;
    }

    static int calc(Cord cord, int a, int b, int c, int d) {
        int ca = count(cord.a, a);
        int cb = count(cord.b, b);
        int cc = count(cord.c, c);
        int cd = count(cord.d, d);
        return mul(mul(ca, cb), mul(cc, cd));
    }

    static int count(int n, int on) {
        if( on == 1 ) {
            return sub(pow2(n), 1);
        } else {
            return 1;
        }
    }

    static int MOD = 998244353;

    static int pow2(int exp) {
        return POW2[exp];
    }

    static int pow(int base, long exp) {
        if (exp == 0) return 1;

        int ans = 1;
        base %= MOD;
        while (exp > 0) {
            if ((exp & 1) == 1) {
                ans = mul(ans, base);
            }

            base = mul(base, base);
            exp = exp >> 1;
        }
        return ans;
    }

    static int sub(int a, int b) {
        int c = a - b;
        if (c < 0) c += MOD;
        return c;
    }

    static int div(int a, int b) {
        return mul(a, pow(b, MOD - 2));
    }

    static int add(int a, int b) {
        int c = a + b;
        if (c >= MOD) c %= MOD;
        return c;
    }

    static int mul(int a, int b) {
        long c = (long) a * b;
        if (c >= MOD) c %= MOD;
        return (int) c;
    }

    static class FermatCombination {
        private final int size;
        private final int[] factorial; // n -> factorial(n)
        private final int[] inverse;   // n -> inverse(factorial(n))

        FermatCombination(int size) {
            this.size = size;
            factorial = new int[size + 1];
            inverse = new int[size + 1];

            init();
        }

        private void init() {
            factorial[0] = 1;
            factorial[1] = 1;
            inverse[0] = 1;
            inverse[1] = 1;
            for (int i = 2; i <= size; i++) {
                factorial[i] = mul(factorial[i - 1], i);
                inverse[i] = pow(factorial[i], MOD - 2);
            }
        }

        int comb(int n, int k) {
            if (n > size) throw new RuntimeException("wtf : size=" + size + " n=" + n);
            return mul(mul(factorial[n], inverse[k]), inverse[n - k]);
        }

        // 重複組み合わせ
        // k種類からn個選ぶ場合の数
        int hcomb(int k, int n) {
            return comb(k + n - 1, n);
        }

        int group(int n, int g, int k) {
            // C(n, g) * C(n-g, g)... / k!
            // n! / (n-gk)! / g! ^ k / k!
            int ret = factorial[n];             // n!
            ret = mul(ret, inverse[n - g * k]);   // 1 / (n-gk)!
            ret = mul(ret, pow(inverse[g], k)); // 1 / (g! ^ k)
            ret = mul(ret, inverse[k]);         // 1 / k!
            return ret;
        }
    }

    static class Cord {
        int x, y;
        int a, b, c, d;

        public Cord(int x, int y) {
            this.x = x;
            this.y = y;
        }
    }

    static class FenwickTree {

        interface Monoid {
            long identity();
            long apply(long a, long b);
            long inverse(long a);
        }

        static Monoid SUM = new Monoid() {
            public long identity() { return 0; }
            public long apply(long a, long b) { return a+b; }
            public long inverse(long a) { return -a; }
        };

        static Monoid MAX = new Monoid() {
            public long identity() { return 0; }
            public long apply(long a, long b) { return Math.max(a, b); }
            public long inverse(long a) { throw new RuntimeException("no inverse"); }
        };


        int size;
        long[] bit;
        Monoid m;
        long identity;

        FenwickTree(int size, Monoid m) {
            this.size = 1;
            while( this.size < size ) this.size *= 2;
            this.bit = new long[this.size+1];
            this.identity = m.identity();
            if( this.identity != 0 ) {
                Arrays.fill(this.bit, this.identity);
            }
            this.m = m;
        }

        void add(int i, long v) {
            i++; // 0 index -> 1 index

            while( i <= size ) {
                bit[i] = m.apply(bit[i], v);
                i += i & -i;
            }
        }

        // [0, r)
        long query(int r) {
            // 0 index -> 1 index -> -1
            long ret = identity;
            while(r > 0) {
                ret = m.apply(ret, bit[r]);
                r -= r & -r;
            }
            return ret;
        }

        long query(int l, int r) {
            return query(r) + m.inverse(query(l));
        }

        int lowerBound(int v) {
            if( bit[size] < v ) return size;

            int x = 0;
            for (int k = size/2; k > 0; k /= 2 ) {
                if( bit[x + k] < v ) {
                    v -= bit[x+k];
                    x += k;
                }
            }
            return x;
        }
    }

    static class Zip {

        Map<Integer, Integer> toIndex;
        int[] toValue;

        Zip(int[] values) {
            toValue = Arrays.copyOf(values, values.length);
            Arrays.sort(toValue);
            toIndex = new HashMap<>();
            for (int i = 0; i < values.length; i++) {
                toIndex.put(toValue[i], i);
            }
        }

        Zip(Set<Integer> set) {
            toValue = new int[set.size()];
            int idx = 0;
            for (Integer each : set) {
                toValue[idx++] = each;
            }
            Arrays.sort(toValue);
            toIndex = new HashMap<>();
            for (int i = 0; i < set.size(); i++) {
                toIndex.put(toValue[i], i);
            }
        }

        int to_i(int value) {
            return toIndex.get(value);
        }

        int to_v(int index) {
            return toValue[index];
        }
    }

    @SuppressWarnings("unused")
    static class FastScanner {
        private BufferedReader reader;
        private StringTokenizer tokenizer;

        FastScanner(InputStream in) {
            reader = new BufferedReader(new InputStreamReader(in));
            tokenizer = null;
        }

        String next() {
            if (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken();
        }

        String nextLine() {
            if (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    return reader.readLine();
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken("\n");
        }

        long nextLong() {
            return Long.parseLong(next());
        }

        int nextInt() {
            return Integer.parseInt(next());
        }

        int[] nextIntArray(int n) {
            int[] a = new int[n];
            for (int i = 0; i < n; i++)
                a[i] = nextInt();
            return a;
        }

        int[] nextIntArray(int n, int delta) {
            int[] a = new int[n];
            for (int i = 0; i < n; i++)
                a[i] = nextInt() + delta;
            return a;
        }

        long[] nextLongArray(int n) {
            long[] a = new long[n];
            for (int i = 0; i < n; i++)
                a[i] = nextLong();
            return a;
        }
    }

    static <A> void writeLines(A[] as, Function<A, String> f) {
        PrintWriter pw = new PrintWriter(System.out);
        for (A a : as) {
            pw.println(f.apply(a));
        }
        pw.flush();
    }

    static void writeLines(int[] as) {
        PrintWriter pw = new PrintWriter(System.out);
        for (int a : as) pw.println(a);
        pw.flush();
    }

    static void writeLines(long[] as) {
        PrintWriter pw = new PrintWriter(System.out);
        for (long a : as) pw.println(a);
        pw.flush();
    }

    static int max(int... as) {
        int max = Integer.MIN_VALUE;
        for (int a : as) max = Math.max(a, max);
        return max;
    }

    static int min(int... as) {
        int min = Integer.MAX_VALUE;
        for (int a : as) min = Math.min(a, min);
        return min;
    }

    static void debug(Object... args) {
        StringJoiner j = new StringJoiner(" ");
        for (Object arg : args) {
            if (arg instanceof int[]) j.add(Arrays.toString((int[]) arg));
            else if (arg instanceof long[]) j.add(Arrays.toString((long[]) arg));
            else if (arg instanceof double[]) j.add(Arrays.toString((double[]) arg));
            else if (arg instanceof Object[]) j.add(Arrays.toString((Object[]) arg));
            else j.add(arg.toString());
        }
        System.err.println(j.toString());
    }
}