AND Segments

3000ms

262144K

Description:

You are given three integers $$$n$$$, $$$k$$$, $$$m$$$ and $$$m$$$ conditions $$$(l_1, r_1, x_1), (l_2, r_2, x_2), \dots, (l_m, r_m, x_m)$$$.

Calculate the number of distinct arrays $$$a$$$, consisting of $$$n$$$ integers such that:

$$$0 \le a_i < 2^k$$$ for each $$$1 \le i \le n$$$;
bitwise AND of numbers $$$a[l_i] \& a[l_i + 1] \& \dots \& a[r_i] = x_i$$$ for each $$$1 \le i \le m$$$.

Two arrays $$$a$$$ and $$$b$$$ are considered different if there exists such a position $$$i$$$ that $$$a_i \neq b_i$$$.

The number can be pretty large so print it modulo $$$998244353$$$.

Input:

The first line contains three integers $$$n$$$, $$$k$$$ and $$$m$$$ ($$$1 \le n \le 5 \cdot 10^5$$$, $$$1 \le k \le 30$$$, $$$0 \le m \le 5 \cdot 10^5$$$) — the length of the array $$$a$$$, the value such that all numbers in $$$a$$$ should be smaller than $$$2^k$$$ and the number of conditions, respectively.

Each of the next $$$m$$$ lines contains the description of a condition $$$l_i$$$, $$$r_i$$$ and $$$x_i$$$ ($$$1 \le l_i \le r_i \le n$$$, $$$0 \le x_i < 2^k$$$) — the borders of the condition segment and the required bitwise AND value on it.

Output:

Print a single integer — the number of distinct arrays $$$a$$$ that satisfy all the above conditions modulo $$$998244353$$$.

Sample Input:

4 3 2
1 3 3
3 4 6

Sample Output:

Sample Input:

Sample Output:

Note:

You can recall what is a bitwise AND operation here.

In the first example, the answer is the following arrays: $$$[3, 3, 7, 6]$$$, $$$[3, 7, 7, 6]$$$ and $$$[7, 3, 7, 6]$$$.

Informação

Provedor Codeforces

Código CF1327F