Enclosed Points

AtCoder

IDabc136_f

Time2000ms

Memory256MB

Difficulty—

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: $N$ $x_1$ $y_1$ $:$ $x_N$ $y_N$ [samples]

Samples

Input #1

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 $13$.

Input #2

Output #2

Input #3

Output #3

7222

Be sure to print the sum modulo $998244353$.

API Response (JSON)

{
  "problem": {
    "name": "Enclosed Points",
    "description": {
      "content": "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 no",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc136_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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.\nFor a no...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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