Two Spanning Trees

AtCoder

IDabc251_f

Time2000ms

Memory256MB

Difficulty—

You are given an undirected graph $G$ with $N$ vertices and $M$ edges. $G$ is **simple** (it has no self-loops and multiple edges) and **connected**. For $i = 1, 2, \ldots, M$, the $i$\-th edge is an undirected edge $\lbrace u_i, v_i \rbrace$ connecting Vertices $u_i$ and $v_i$. Construct two spanning trees $T_1$ and $T_2$ of $G$ that satisfy both of the two conditions below. ($T_1$ and $T_2$ do not necessarily have to be different spanning trees.) * $T_1$ satisfies the following. > If we regard $T_1$ as a rooted tree rooted at Vertex $1$, for any edge $\lbrace u, v \rbrace$ of $G$ not contained in $T_1$, one of $u$ and $v$ is an ancestor of the other in $T_1$. * $T_2$ satisfies the following. > If we regard $T_2$ as a rooted tree rooted at Vertex $1$, there is no edge $\lbrace u, v \rbrace$ of $G$ not contained in $T_2$ such that one of $u$ and $v$ is an ancestor of the other in $T_2$. We can show that there always exists $T_1$ and $T_2$ that satisfy the conditions above. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $N-1 \leq M \leq \min\lbrace 2 \times 10^5, N(N-1)/2 \rbrace$ * $1 \leq u_i, v_i \leq N$ * All values in input are integers. * The given graph is simple and connected. ## Input Input is given from Standard Input in the following format: $N$ $M$ $u_1$ $v_1$ $u_2$ $v_2$ $\vdots$ $u_M$ $v_M$ [samples]

Samples

Input #1

Output #1

1 4
4 3
5 3
4 2
6 2
1 5
5 3
1 4
2 1
1 6

In the Sample Output above, $T_1$ is a spanning tree of $G$ containing $5$ edges $\lbrace 1, 4 \rbrace, \lbrace 4, 3 \rbrace, \lbrace 5, 3 \rbrace, \lbrace 4, 2 \rbrace, \lbrace 6, 2 \rbrace$. This $T_1$ satisfies the condition in the Problem Statement. Indeed, for each edge of $G$ not contained in $T_1$:

*   for edge $\lbrace 5, 1 \rbrace$, Vertex $1$ is an ancestor of $5$;
*   for edge $\lbrace 1, 2 \rbrace$, Vertex $1$ is an ancestor of $2$;
*   for edge $\lbrace 1, 6 \rbrace$, Vertex $1$ is an ancestor of $6$.

$T_2$ is another spanning tree of $G$ containing $5$ edges $\lbrace 1, 5 \rbrace, \lbrace 5, 3 \rbrace, \lbrace 1, 4 \rbrace, \lbrace 2, 1 \rbrace, \lbrace 1, 6 \rbrace$. This $T_2$ satisfies the condition in the Problem Statement. Indeed, for each edge of $G$ not contained in $T_2$:

*   for edge $\lbrace 4, 3 \rbrace$, Vertex $4$ is not an ancestor of Vertex $3$, and vice versa;
*   for edge $\lbrace 2, 6 \rbrace$, Vertex $2$ is not an ancestor of Vertex $6$, and vice versa;
*   for edge $\lbrace 4, 2 \rbrace$, Vertex $4$ is not an ancestor of Vertex $2$, and vice versa.

Input #2

Output #2

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

Tree $T$, containing $3$ edges $\lbrace 1, 2\rbrace, \lbrace 1, 3 \rbrace, \lbrace 1, 4 \rbrace$, is the only spanning tree of $G$. Since there are no edges of $G$ that are not contained in $T$, obviously this $T$ satisfies the conditions for both $T_1$ and $T_2$.

API Response (JSON)

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      "content": "You are given an undirected graph $G$ with $N$ vertices and $M$ edges. $G$ is **simple** (it has no self-loops and multiple edges) and **connected**.\nFor $i = 1, 2, \\ldots, M$, the $i$\\-th edge is an ...",
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