Telepanting (Codeforces 1552F)
Tags: DP, binary search

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 $$$998\,244\,353$$$.

Input: 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.

Output: 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 $$$998\,244\,353$$$.

Note: 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$$$.Explanation of the second sample: The ant moves as follows (a curvy arrow denotes a teleporting, a straight arrow denotes normal movement with speed $$$1$$$ and the time spent during the movement is written above the arrow). $$$$$$ 0 \stackrel{454971987}{\longrightarrow} 454971987 \leadsto 406874902 \stackrel{48097086}{\longrightarrow} 454971988 $$$$$$ Notice that the total time is $$$454971987+48097086=503069073$$$.Explanation of the third sample: Since all portals are initially off, the ant will not be teleported and will go straight from $$$0$$$ to $$$x_n+1=899754846+1=899754847$$$.

Sample Inputs:
4
3 2 0
6 5 1
7 4 0
8 1 1

1
454971987 406874902 1

5
243385510 42245605 0
644426565 574769163 0
708622105 208990040 0
786625660 616437691 0
899754846 382774619 0

5
200000000 100000000 1
600000000 400000000 0
800000000 300000000 0
900000000 700000000 1
1000000000 500000000 0

Sample Outputs:
23

503069073

899754847

3511295

Solution (Python):
import sys
from bisect import bisect
input = sys.stdin.readline

def solve():
    MOD = 998244353
    n = int(input())
    x = [0] * n
    y = [0] * n
    s = [0] * n
    z = [0] * n
    p = [0] * (n + 1)
    for i in range(n):
        (x[i], y[i], s[i]) = map(int, input().split())
    r = 0
    for i in range(n):
        nn = bisect(x, y[i])
        z[i] = (p[i] - p[nn] + x[i] - y[i]) % MOD
        p[i + 1] = (p[i] + z[i]) % MOD
        if s[i] == 1:
            r = (r + z[i]) % MOD
    print((x[n - 1] + 1 + r) % MOD)
solve()

Solution (Python):
from bisect import bisect

def solve():
    MOD = 998244353
    n = int(input())
    x = [0] * n
    y = [0] * n
    s = [0] * n
    z = [0] * n
    p = [0] * (n + 1)
    r = 0
    for i in range(n):
        (x[i], y[i], s[i]) = map(int, input().split())
    for i in range(n):
        nn = bisect(x, y[i])
        z[i] = (p[i] - p[nn] + x[i] - y[i]) % MOD
        p[i + 1] = (p[i] + z[i]) % MOD
        if s[i] == 1:
            r = (r + z[i]) % MOD
    print((x[n - 1] + 1 + r) % MOD)
solve()

Solution (Python):
from bisect import bisect

def solve():
    MOD = 998244353
    n = int(input())
    x = [0] * n
    y = [0] * n
    s = [0] * n
    z = [0] * n
    p = [0] * (n + 1)
    r = 0
    for i in range(n):
        (x[i], y[i], s[i]) = map(int, input().split())
    for i in range(n):
        nn = bisect(x, y[i])
        z[i] = (p[i] - p[nn] + x[i] - y[i]) % MOD
        p[i + 1] = (p[i] + z[i]) % MOD
        if s[i] == 1:
            r = (r + z[i]) % MOD
    print((x[n - 1] + 1 + r) % MOD)
solve()