A14409. Telepanting
编程题
普及/提高-
知识点
题目描述
An ant moves on the real line with constant speed of $1$ unit per second. It starts at $0$ and always moves to the right (so its position increases by $1$ each second).
There are $n$ portals, the $i$ -th of which is located at position $x_i$ and teleports to position $y_i < x_i$ . Each portal can be either active or inactive. The initial state of the $i$ -th portal is determined by $s_i$ : if $s_i=0$ then the $i$ -th portal is initially inactive, if $s_i=1$ then the $i$ -th portal is initially active. When the ant travels through a portal (i.e., when its position coincides with the position of a portal):
- if the portal is inactive, it becomes active (in this case the path of the ant is not affected);
- if the portal is active, it becomes inactive and the ant is instantly teleported to the position $y_i$ , where it keeps on moving as normal.
How long (from the instant it starts moving) does it take for the ant to reach the position $x_n + 1$ ? It can be shown that this happens in a finite amount of time. Since the answer may be very large, compute it modulo $998\,244\,353$ .
There are $n$ portals, the $i$ -th of which is located at position $x_i$ and teleports to position $y_i < x_i$ . Each portal can be either active or inactive. The initial state of the $i$ -th portal is determined by $s_i$ : if $s_i=0$ then the $i$ -th portal is initially inactive, if $s_i=1$ then the $i$ -th portal is initially active. When the ant travels through a portal (i.e., when its position coincides with the position of a portal):
- if the portal is inactive, it becomes active (in this case the path of the ant is not affected);
- if the portal is active, it becomes inactive and the ant is instantly teleported to the position $y_i$ , where it keeps on moving as normal.
How long (from the instant it starts moving) does it take for the ant to reach the position $x_n + 1$ ? It can be shown that this happens in a finite amount of time. Since the answer may be very large, compute it modulo $998\,244\,353$ .
输入格式
The first line contains the integer $n$ ( $1\le n\le 2\cdot 10^5$ ) — the number of portals.
The $i$ -th of the next $n$ lines contains three integers $x_i$ , $y_i$ and $s_i$ ( $1\le y_i < x_i\le 10^9$ , $s_i\in\{0,1\}$ ) — the position of the $i$ -th portal, the position where the ant is teleported when it travels through the $i$ -th portal (if it is active), and the initial state of the $i$ -th portal.
The positions of the portals are strictly increasing, that is $x_1<x_2<\cdots<x_n$ . It is guaranteed that the $2n$ integers $x_1, \, x_2, \, \dots, \, x_n, \, y_1, \, y_2, \, \dots, \, y_n$ are all distinct.
The $i$ -th of the next $n$ lines contains three integers $x_i$ , $y_i$ and $s_i$ ( $1\le y_i < x_i\le 10^9$ , $s_i\in\{0,1\}$ ) — the position of the $i$ -th portal, the position where the ant is teleported when it travels through the $i$ -th portal (if it is active), and the initial state of the $i$ -th portal.
The positions of the portals are strictly increasing, that is $x_1<x_2<\cdots<x_n$ . It is guaranteed that the $2n$ integers $x_1, \, x_2, \, \dots, \, x_n, \, y_1, \, y_2, \, \dots, \, y_n$ are all distinct.
输出格式
Output the amount of time elapsed, in seconds, from the instant the ant starts moving to the instant it reaches the position $x_n+1$ . Since the answer may be very large, output it modulo $998\,244\,353$ .
输入输出样例
输入 #1
4 3 2 0 6 5 1 7 4 0 8 1 1
输出 #1
23
输入 #2
1 454971987 406874902 1
输出 #2
503069073
输入 #3
5 243385510 42245605 0 644426565 574769163 0 708622105 208990040 0 786625660 616437691 0 899754846 382774619 0
输出 #3
899754847
输入 #4
5 200000000 100000000 1 600000000 400000000 0 800000000 300000000 0 900000000 700000000 1 1000000000 500000000 0
输出 #4
3511295
说明/提示
Explanation of the first sample: The ant moves as follows (a curvy arrow denotes a teleporting, a straight arrow denotes normal movement with speed $1$ and the time spent during the movement is written above the arrow). $$$$ 0 \stackrel{6}{\longrightarrow} 6 \leadsto 5 \stackrel{3}{\longrightarrow} 8 \leadsto 1 \stackrel{2}{\longrightarrow} 3 \leadsto 2 \stackrel{4}{\longrightarrow} 6 \leadsto 5 \stackrel{2}{\longrightarrow} 7 \leadsto 4 \stackrel{2}{\longrightarrow} 6 \leadsto 5 \stackrel{4}{\longrightarrow} 9 $$ Notice that the total time is $6+3+2+4+2+2+4=23$ .</p><p><span class="tex-font-style-bf">Explanation of the second sample:</span> The ant moves as follows (a curvy arrow denotes a teleporting, a straight arrow denotes normal movement with speed $1$ and the time spent during the movement is written above the arrow). $$ 0 \stackrel{454971987}{\longrightarrow} 454971987 \leadsto 406874902 \stackrel{48097086}{\longrightarrow} 454971988 $$ Notice that the total time is $454971987+48097086=503069073$ .</p><p><span class="tex-font-style-bf">Explanation of the third sample:</span> Since all portals are initially off, the ant will not be teleported and will go straight from $0$ to $x\_n+1=899754846+1=899754847$$$.