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 Common Lisp (SBCL 1.1.14) (judge verdict: Accepted)

;; -*- coding: utf-8 -*-
(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-int:defconstant-eqx OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0))
    #'equal)
  #+swank (ql:quickload '(:cl-debug-print :fiveam))
  #-swank (set-dispatch-macro-character #\# #\> (lambda (s c p) (declare (ignore c p)) (read s nil (values) t))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
;;;
;;; 2D range tree with fractional cascading
;;;
;;; build: O(nlog(n))
;;; query: O(log(n))
;;;

;; TODO: map all the points in a given rectangle
;; TODO: introduce abelian group

(defstruct (ynode (:constructor make-ynode (xkeys ykeys lpointers rpointers))
                  (:conc-name %ynode-)
                  (:copier nil))
  (xkeys nil :type (simple-array fixnum (*)))
  (ykeys nil :type (simple-array fixnum (*)))
  (lpointers nil :type (or null (simple-array (integer 0 #.most-positive-fixnum) (*))))
  (rpointers nil :type (or null (simple-array (integer 0 #.most-positive-fixnum) (*)))))

(defstruct (xnode (:constructor make-xnode (xkey ynode left right))
                  (:conc-name %xnode-)
                  (:copier nil))
  (xkey 0 :type fixnum)
  (ynode nil :type ynode)
  (left nil :type (or null xnode))
  (right nil :type (or null xnode)))

(defun %ynode-merge (ynode1 ynode2)
  "Merges two YNODEs non-destructively in O(n)."
  (declare #.OPT)
  (let* ((xkeys1 (%ynode-xkeys ynode1))
         (ykeys1 (%ynode-ykeys ynode1))
         (xkeys2 (%ynode-xkeys ynode2))
         (ykeys2 (%ynode-ykeys ynode2))
         (len1 (length xkeys1))
         (len2 (length xkeys2))
         (new-len (+ len1 len2))
         (new-xkeys (make-array new-len :element-type 'fixnum))
         (new-ykeys (make-array new-len :element-type 'fixnum))
         (lpointers (make-array (+ 1 new-len)
                                :element-type '(integer 0 #.most-positive-fixnum)))
         (rpointers (make-array (+ 1 new-len)
                                :element-type '(integer 0 #.most-positive-fixnum)))
         
         (new-pos 0)
         (pos1 0)
         (pos2 0))
    (declare ((integer 0 #.most-positive-fixnum) len1 len2 new-len new-pos pos1 pos2))
    ;; merge two vectors
    (loop
      (when (= pos1 len1)
        (loop
          for i from pos2 below len2
          do (setf (aref new-xkeys new-pos) (aref xkeys2 i)
                   (aref new-ykeys new-pos) (aref ykeys2 i)
                   (aref lpointers new-pos) pos1
                   (aref rpointers new-pos) i)
             (incf new-pos))
        (return))
      (when (= pos2 len2)
        (loop
          for i from pos1 below len1
          do (setf (aref new-xkeys new-pos) (aref xkeys1 i)
                   (aref new-ykeys new-pos) (aref ykeys1 i)
                   (aref lpointers new-pos) i
                   (aref rpointers new-pos) pos2)
             (incf new-pos))
        (return))
      (if (or (< (aref ykeys1 pos1) (aref ykeys2 pos2))
              (and (= (aref ykeys1 pos1) (aref ykeys2 pos2))
                   (< (aref xkeys1 pos1) (aref xkeys2 pos2))))
          (setf (aref new-xkeys new-pos) (aref xkeys1 pos1)
                (aref new-ykeys new-pos) (aref ykeys1 pos1)
                (aref lpointers new-pos) pos1
                (aref rpointers new-pos) pos2
                pos1 (+ pos1 1))
          (setf (aref new-xkeys new-pos) (aref xkeys2 pos2)
                (aref new-ykeys new-pos) (aref ykeys2 pos2)
                (aref lpointers new-pos) pos1
                (aref rpointers new-pos) pos2
                pos2 (+ pos2 1)))
      (incf new-pos))
    (setf (aref lpointers new-len) len1
          (aref rpointers new-len) len2)
    (make-ynode new-xkeys new-ykeys lpointers rpointers)))

(declaim (inline make-range-tree))
(defun make-range-tree (points &key (xkey #'car) (ykey #'cdr))
  "points := vector of poins

Makes a range tree from the points. These points must be sorted
w.r.t. lexicographical order and must not contain duplicate points. (Duplicate
coordinates are allowed.) E.g. (-1, 3), (-1, 4), (-1, 7) (0, 1) (0, 3) (2,
-1) (2, 1))."
  (declare ((simple-array (cons int32 int32) (*)) points))
  (when (zerop (length points))
    (return-from make-range-tree nil))
  (let ((pointers-for-leaf
          (make-array 2
                      :element-type '(integer 0 #.most-positive-fixnum)
                      :initial-element 0)))
    (labels
        ((build (l r)
           (declare ((integer 0 #.most-positive-fixnum) l r))
           (if (= (- r l) 1)
               (let* ((point (aref points l))
                      (x (funcall xkey point))
                      (y (funcall ykey point))
                      (xkeys (make-array 1 :element-type 'fixnum))
                      (ykeys (make-array 1 :element-type 'fixnum)))
                 (setf (aref xkeys 0) x
                       (aref ykeys 0) y)
                 (make-xnode x (make-ynode xkeys ykeys
                                           pointers-for-leaf
                                           pointers-for-leaf)
                             nil nil))
               (let* ((mid (ash (+ l r) -1))
                      (left (build l mid))
                      (right (build mid r)))
                 (make-xnode (funcall xkey (aref points mid))
                             (%ynode-merge (%xnode-ynode left)
                                           (%xnode-ynode right))
                             left right)))))
      (build 0 (length points)))))

(defconstant +neg-inf+ most-negative-fixnum)
(defconstant +pos-inf+ most-positive-fixnum)

(declaim (inline xleaf-p))
(defun xleaf-p (xnode)
  (and (null (%xnode-left xnode)) (null (%xnode-right xnode))))

(defun rt-count (range-tree x1 y1 x2 y2)
  "Returns the number of the nodes within the rectangle [x1, x2)*[y1, y2). A
part or all of these coordinates can be NIL; then they are regarded as the
negative or positive infinity."
  (declare #.OPT
           ((or null fixnum) x1 y1 x2 y2))
  (setq x1 (or x1 +neg-inf+)
        x2 (or x2 +pos-inf+)
        y1 (or y1 +neg-inf+)
        y2 (or y2 +pos-inf+))
  (unless range-tree
    (return-from rt-count 0))
  (let* ((ynode (%xnode-ynode range-tree))
         (xkeys (%ynode-xkeys ynode))
         (ykeys (%ynode-ykeys ynode)))
    (labels ((bisect-left (y)
               (declare (fixnum y))
               (let ((left 0)
                     (ok (length xkeys)))
                 (declare ((integer 0 #.most-positive-fixnum) left ok))
                 (loop
                   (let ((mid (ash (+ left ok) -1)))
                     (if (= mid left)
                         (if (< (aref ykeys left) y)
                             (return ok)
                             (return left))
                         (if (< (aref ykeys mid) y)
                             (setq left mid)
                             (setq ok mid)))))))
             (recur (xnode x1 x2 start end)
               (declare ((or null xnode) xnode)
                        (fixnum x1 x2)
                        ;; KLUDGE: declaring ftype is not sufficient for the
                        ;; optimization on SBCL 1.1.14.
                        #+sbcl (values (integer 0 #.most-positive-fixnum)))
               (cond ((null xnode) 0)
                     ((and (= x1 +neg-inf+) (= x2 +pos-inf+))
                      (- end start))
                     (t
                      (let* ((xkey (%xnode-xkey xnode))
                             (ynode (%xnode-ynode xnode))
                             (lpointers (%ynode-lpointers ynode))
                             (rpointers (%ynode-rpointers ynode)))
                        (if (<= x1 xkey)
                            (if (< xkey x2)
                                ;; XKEY is in [X1, X2)
                                (if (xleaf-p xnode)
                                    (- end start)
                                    (+ (recur (%xnode-left xnode)
                                              x1 +pos-inf+
                                              (aref lpointers start)
                                              (aref lpointers end))
                                       (recur (%xnode-right xnode)
                                              +neg-inf+ x2
                                              (aref rpointers start)
                                              (aref rpointers end))))
                                ;; XKEY is in [X2, +inf)
                                (recur (%xnode-left xnode)
                                       x1 x2
                                       (aref lpointers start)
                                       (aref lpointers end)))
                            ;; XKEY is in (-inf, X1)
                            (recur (%xnode-right xnode)
                                   x1 x2
                                   (aref rpointers start)
                                   (aref rpointers end))))))))
      (let ((start (bisect-left y1))
            (end (bisect-left y2)))
        (recur range-tree x1 x2 start end)))))

(declaim (ftype (function * (values fixnum &optional)) read-fixnum))
(defun read-fixnum (&optional (in *standard-input*))
  (declare #.OPT)
  (macrolet ((%read-byte ()
               `(the (unsigned-byte 8)
                     #+swank (char-code (read-char in nil #\Nul))
                     #-swank (sb-impl::ansi-stream-read-byte in nil #.(char-code #\Nul) nil))))
    (let* ((minus nil)
           (result (loop (let ((byte (%read-byte)))
                           (cond ((<= 48 byte 57)
                                  (return (- byte 48)))
                                 ((zerop byte) ; #\Nul
                                  (error "Read EOF or #\Nul."))
                                 ((= byte #.(char-code #\-))
                                  (setf minus t)))))))
      (declare ((integer 0 #.most-positive-fixnum) result))
      (loop
        (let* ((byte (%read-byte)))
          (if (<= 48 byte 57)
              (setq result (+ (- byte 48) (* 10 (the (integer 0 #.(floor most-positive-fixnum 10)) result))))
              (return (if minus (- result) result))))))))

(defmacro dbg (&rest forms)
  #+swank
  (if (= (length forms) 1)
      `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
      `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

(defconstant +mod+ 998244353)

;; Body
;;;
;;; Arithmetic operations with static modulus
;;;

(defmacro define-mod-operations (&optional (divisor 1000000007))
  `(progn
     (defun mod* (&rest args)
       (reduce (lambda (x y) (mod (* x y) ,divisor)) args))

     (sb-c:define-source-transform mod* (&rest args)
       (if (null args)
           1
           (reduce (lambda (x y) `(mod (* ,x ,y) ,',divisor)) args)))

     (defun mod+ (&rest args)
       (reduce (lambda (x y) (mod (+ x y) ,divisor)) args))

     (sb-c:define-source-transform mod+ (&rest args)
       (if (null args)
           0
           (reduce (lambda (x y) `(mod (+ ,x ,y) ,',divisor)) args)))

     (define-modify-macro incfmod (delta divisor)
       (lambda (x y divisor) (mod (+ x y) divisor)))

     (define-modify-macro decfmod (delta divisor)
       (lambda (x y divisor) (mod (- x y) divisor)))))

(define-mod-operations +mod+)

(defun main ()
  (declare #.OPT
           (inline sort)
           (muffle-conditions style-warning))
  (let* ((n (read))
         (powers (make-array 200001 :element-type 'uint32))
         (points (make-array n :element-type '(cons int32 int32)))
         (res 0))
    (declare (uint32 n)
             (uint62 res)
             ((simple-array uint32 (*)) powers))
    ;; construct table of 2^n
    (setf (aref powers 0) 1)
    (loop for i from 1 below (length powers)
          do (setf (aref powers i)
                   (mod* 2 (aref powers (- i 1)))))
    (dotimes (i n)
      (setf (aref points i) (cons (read-fixnum) (read-fixnum))))
    (setq points
          (sort points #'< :key (lambda (x) (the int32 (car x)))))
    (let ((rtree (make-range-tree points)))
      ;; L R U D
      (incfmod res (mod* (- n 4) (- (aref powers n) 1)) +mod+)
      ;; LU LD RU RD
      (dotimes (i n)
        (let* ((point (aref points i))
               (x (car point))
               (y (cdr point))
               (ld (rt-count rtree nil nil x y))
               (lu (rt-count rtree nil (+ y 1) x nil))
               (rd (rt-count rtree (+ x 1) (+ y 1) nil nil))
               (ru (rt-count rtree (+ x 1) nil nil y)))
          (declare (int32 x y))
          (incf res (+ (aref powers ld) (aref powers lu) (aref powers rd) (aref powers ru))))))
    (println (mod res +mod+))))

#-swank (main)