File size: 2,048 Bytes
3e66834 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 |
This is the hard version of the problem. In this version, $n \leq 10^6$. You can only make hacks if both versions of the problem are solved.
Orangutans are powerful beings—so powerful that they only need $1$ unit of time to destroy every vulnerable planet in the universe!
There are $n$ planets in the universe. Each planet has an interval of vulnerability $[l, r]$, during which it will be exposed to destruction by orangutans. Orangutans can also expand the interval of vulnerability of any planet by $1$ unit.
Specifically, suppose the expansion is performed on planet $p$ with interval of vulnerability $[l_p, r_p]$. Then, the resulting interval of vulnerability may be either $[l_p - 1, r_p]$ or $[l_p, r_p + 1]$.
Given a set of planets, orangutans can destroy all planets if the intervals of vulnerability of all planets in the set intersect at least one common point. Let the score of such a set denote the minimum number of expansions that must be performed.
Orangutans are interested in the sum of scores of all non-empty subsets of the planets in the universe. As the answer can be large, output it modulo $998\,244\,353$.
The first line contains an integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases.
The first line of each test case contains an integer $n$ ($1 \leq n \leq 10^6$) — the number of planets in the universe.
The following $n$ lines contain two integers $l_i$ and $r_i$ ($1 \leq l_i \leq r_i \leq n$) — the initial interval of vulnerability of the $i$-th planet.
It is guaranteed that the sum of $n$ does not exceed $10^6$ over all test cases.
For each test case, output an integer — the sum of scores to destroy all non- empty subsets of the planets in the universe, modulo $998\,244\,353$.
In the first testcase, there are seven non-empty subsets of planets we must consider:
* For each of the subsets $\\{[1,1]\\}, \\{[2,3]\\}, \\{[3,3]\\}$, the score is $0$. * For the subset $\\{[2,3], [3,3]\\}$, the score is $0$, because the point $3$ is already contained in both planets' interv |