Red and Blue Graph

AtCoder

IDabc262_e

Time2000ms

Memory256MB

Difficulty—

You are given a simple undirected graph with $N$ vertices and $M$ edges. The vertices are numbered $1, \dots, N$, and the $i$\-th $(1 \leq i \leq M)$ edge connects Vertices $U_i$ and $V_i$. There are $2^N$ ways to paint each vertex red or blue. Find the number, modulo $998244353$, of such ways that satisfy all of the following conditions: * There are exactly $K$ vertices painted red. * There is an even number of edges connecting vertices painted in different colors. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $1 \leq M \leq 2 \times 10^5$ * $0 \leq K \leq N$ * $1 \leq U_i \lt V_i \leq N \, (1 \leq i \leq M)$ * $(U_i, V_i) \neq (U_j, V_j) \, (i \neq j)$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $K$ $U_1$ $V_1$ $\vdots$ $U_M$ $V_M$ [samples]

Samples

Input #1

Output #1

2

The following two ways satisfy the conditions.

*   Paint Vertices $1$ and $2$ red and Vertices $3$ and $4$ blue.
*   Paint Vertices $3$ and $4$ red and Vertices $1$ and $2$ blue.

In either of the ways above, the $2$\-nd and $3$\-rd edges connect vertices painted in different colors.

Input #2

Output #2

API Response (JSON)

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      "content": "You are given a simple undirected graph with $N$ vertices and $M$ edges. The vertices are numbered $1, \\dots, N$, and the $i$\\-th $(1 \\leq i \\leq M)$ edge connects Vertices $U_i$ and $V_i$. There are ",
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    "limit": {
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      "memory_limit": 262144
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    "sign": "abc262_e"
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  "statements": [
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      "statement_type": "Markdown",
      "content": "You are given a simple undirected graph with $N$ vertices and $M$ edges. The vertices are numbered $1, \\dots, N$, and the $i$\\-th $(1 \\leq i \\leq M)$ edge connects Vertices $U_i$ and $V_i$.\nThere are ...",
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