Many Xor Optimization Problems

AtCoder

IDarc139_f

Time8000ms

Memory256MB

Difficulty—

PCT made the following problem. > **Xor Optimization Problem**You are given a sequence of non-negative integers of length $N$: $A_1,A_2,...,A_N$. When it is allowed to choose any number of elements in $A$, what is the maximum possible $\mathrm{XOR}$ of the chosen values? Nyaan thought it was too easy and revised it to the following. > **Many Xor Optimization Problems**There are $2^{NM}$ sequences of length $N$ consisting of integers between $0$ and $2^M-1$. Find the sum, modulo $998244353$, of the answers to **Xor Optimization Problem** for all those sequences. Solve **Many Xor Optimization Problems**. What is bitwise $\mathrm{XOR}$?The bitwise $\mathrm{XOR}$ of non-negative integers $A$ and $B$, $A \oplus B$, is defined as follows: * When $A \oplus B$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if exactly one of $A$ and $B$ is $1$, and $0$ otherwise. For example, we have $3 \oplus 5 = 6$ (in base two: $011 \oplus 101 = 110$). Generally, the bitwise $\mathrm{XOR}$ of $k$ non-negative integers $p_1, p_2, p_3, \dots, p_k$ is defined as $(\dots ((p_1 \oplus p_2) \oplus p_3) \oplus \dots \oplus p_k)$. We can prove that this value does not depend on the order of $p_1, p_2, p_3, \dots, p_k$. ## Constraints * $1 \le N,M \le 250000$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

2 1

Output #1

3

We are to solve **Xor Optimization Problem** for all sequences of length $2$ consisting of integers between $0$ and $1$.

*   The answer for $A=(0,0)$ is $0$.
*   The answer for $A=(0,1)$ is $1$.
*   The answer for $A=(1,0)$ is $1$.
*   The answer for $A=(1,1)$ is $1$.

Thus, the final answer is $0+1+1+1=3$.

Input #2

3 4

Output #2

Input #3

1234 5678

Output #3

495502261

API Response (JSON)

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