Many Lamps

题目描述

There is a simple graph with $N$ vertices numbered $1$ to $N$ and $M$ edges numbered $1$ to $M$. Edge $i$ connects vertices $u_i$ and $v_i$.
Each vertex has one lamp on it. Initially, all the lamps are off. Determine whether it is possible to turn exactly $K$ lamps on by performing the following operation between $0$ and $M$ times, inclusive.

Choose one edge. Let $u$ and $v$ be the endpoints of the edge. Toggle the states of the lamps on $u$ and $v$. That is, if the lamp is on, turn it off, and vice versa.

If it is possible to turn exactly $K$ lamps on, print a sequence of operations that achieves this state.

输入格式

The input is given from Standard Input in the following format: $N$ $M$ $K$ $u_1$ $v_1$ $u_2$ $v_2$ $\vdots$ $u_M$ $v_M$

输出格式

If it is impossible to turn exactly $K$ lamps on, print $No$.
Otherwise, first print $Yes$, and then print a sequence of operations in the following format: $X$ $e_1$ $e_2$ $\dots$ $e_X$

Here, $X$ is the number of operations, and $e_i$ is the number of the edge chosen in the $i$-th operation. These must satisfy the following:

$0 \leq X \leq M$ $1 \leq e_i \leq M$

If multiple sequences of operations satisfy the conditions, any of them will be considered correct.

5 5 4
1 2
1 3
2 4
3 5
1 5

Yes
3
3 4 5

5 5 5
1 2
1 3
2 4
3 5
1 5

No

10 10 6
2 5
2 6
3 5
3 8
4 6
4 8
5 9
6 7
6 10
7 9

Yes
3
10 9 6

提示

$1 \leq N \leq 2 \times 10^5$ $0 \leq M \leq \min\left( 2 \times 10^5, \frac{N(N-1)}{2} \right)$ $0 \leq K \leq N$ 1 \leq u_i < v_i \leq N The given graph is simple. All input values are integers.If we operate according to the sample output, it will go as follows:

Choose edge $3$. Turn on the lamps on vertex $2$ and vertex $4$. Choose edge $4$. Turn on the lamps on vertex $3$ and vertex $5$. Choose edge $5$. Turn on the lamp on vertex $1$ and turn off the lamp on vertex $5$.

After completing all operations, the lamps on vertices $1$, $2$, $3$, and $4$ are on. Therefore, this sequence of operations satisfies the conditions. Other possible sequences of operations that satisfy the conditions include $X = 4, (e_1,e_2,e_3,e_4) = (3,4,3,1)$. (It is allowed to choose the same edge more than once.)

标签: AtCoder|abc345F

来源

AtCoder|abc345F