102627: [AtCoder]ABC262 Ex - Max Limited Sequence

Problem Statement

Find the number, modulo $998244353$, of integer sequences $A = (A_1, \dots, A_N)$ of length $N$ that satisfy all of the following conditions:

• $0 \leq A_i \leq M$ for all $i$ such that $1 \leq i \leq N$.
• The maximum value of $A_{L_j}, \dots, A_{R_j}$ is $X_j$ for all $j$ such that $1 \leq j \leq Q$.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq M \lt 998244353$
• $1 \leq Q \leq 2 \times 10^5$
• $1 \leq L_i \leq R_i \leq N \, (1 \leq i \leq Q)$
• $1 \leq X_i \leq M \, (1 \leq i \leq Q)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $Q$
$L_1$ $R_1$ $X_1$
$\vdots$
$L_Q$ $R_Q$ $X_Q$

Output

Print the answer.

Sample Input 1

3 3 2
1 2 2
2 3 3

Sample Output 1

5

$A = (0, 2, 3), (1, 2, 3), (2, 0, 3), (2, 1, 3), (2, 2, 3)$ satisfy the conditions.

Sample Input 2

1 1 1
1 1 1

Sample Output 2

1

Sample Input 3

6 40000000 3
1 4 30000000
2 6 20000000
3 5 10000000

Sample Output 3

135282163

问题描述

找到满足以下所有条件的整数序列$A = (A_1, \dots, A_N)$的个数（对$998244353$取模）：

• 对于所有满足$1 \leq i \leq N$的$i$，$0 \leq A_i \leq M$。
• 对于所有满足$1 \leq j \leq Q$的$j$，$A_{L_j}, \dots, A_{R_j}$的最大值为$X_j$。

约束条件

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq M \lt 998244353$
• $1 \leq Q \leq 2 \times 10^5$
• $1 \leq L_i \leq R_i \leq N \, (1 \leq i \leq Q)$
• $1 \leq X_i \leq M \, (1 \leq i \leq Q)$
• 输入中的所有值都是整数。

输入

输入将从标准输入中按照以下格式给出：

$N$ $M$ $Q$
$L_1$ $R_1$ $X_1$
$\vdots$
$L_Q$ $R_Q$ $X_Q$

输出

打印答案。

样例输入1

3 3 2
1 2 2
2 3 3

样例输出1

5

满足条件的$A$为$(0, 2, 3), (1, 2, 3), (2, 0, 3), (2, 1, 3), (2, 2, 3)$。

样例输入2

1 1 1
1 1 1

样例输出2

1

样例输入3

6 40000000 3
1 4 30000000
2 6 20000000
3 5 10000000

样例输出3

135282163