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Strong Enough (Hard Version) - Weekly Challenge 1: Approximation | Repovive | Repovive

W1D7B. Strong Enough (Hard Version)

2 pts4s256 MB

You are given a simple directed graph with $n$ vertices and $m$ edges. The vertices are numbered from $1$ to $n$ , and the edges are numbered from $1$ to $m$ in the order in which they appear in the input.

The graph has the following properties:

for every pair of distinct vertices $u$ and $v$ , at most one of the edges $u \to v$ and $v \to u$ exists;
the graph contains no directed cycle with exactly three vertices;
the graph contains a directed Hamiltonian cycle.

Choose some of the given edges so that the directed graph formed by the chosen edges, on all $n$ vertices, is strongly connected.

In the Hard Version, if the number of chosen edges is $k$ , it must satisfy $2k \le 3n$ .

Constraints

$4 \le n \le 2 \times 10^5$
$n \le m \le \min\left(2 \times 10^5,\frac{n(n-1)}{2}\right)$
$1 \le u_i,v_i \le n$
$u_i \ne v_i$
All directed edges are distinct
For every pair of distinct vertices $u$ and $v$ , at most one of $u \to v$ and $v \to u$ exists
The graph contains no directed cycle with exactly three vertices
The graph contains a directed Hamiltonian cycle

Input

n \quad m

\begin{array}{l} u_1 \quad v_1 \\ u_2 \quad v_2 \\ \vdots \\ u_m \quad v_m \end{array}

Output

\begin{array}{l} k \\ e_1 \quad e_2 \quad \cdots \quad e_k \end{array}

Output $k$ followed by $k$ distinct integers $e_1,e_2,\ldots,e_k$ , the indices of the chosen edges, in any order.

Your output must satisfy $2k \le 3n$ , and the graph formed by the chosen edges must be strongly connected.

The integers may be separated by arbitrary whitespace.

Samples

Input

Explanation

The chosen edges form the directed cycle $1 \to 2 \to 3 \to 4 \to 5 \to 6 \to 7 \to 8 \to 1$ . Therefore, the chosen subgraph is strongly connected.

Output

8
1 2 3 9 5 6 7 10

The graph has the following properties:

for every pair of distinct vertices $u$ and $v$ , at most one of the edges $u \to v$ and $v \to u$ exists;
the graph contains no directed cycle with exactly three vertices;
the graph contains a directed Hamiltonian cycle.

Choose some of the given edges so that the directed graph formed by the chosen edges, on all $n$ vertices, is strongly connected.

In the Hard Version, if the number of chosen edges is $k$ , it must satisfy $2k \le 3n$ .

$4 \le n \le 2 \times 10^5$
$n \le m \le \min\left(2 \times 10^5,\frac{n(n-1)}{2}\right)$
$1 \le u_i,v_i \le n$
$u_i \ne v_i$
All directed edges are distinct
For every pair of distinct vertices $u$ and $v$ , at most one of $u \to v$ and $v \to u$ exists
The graph contains no directed cycle with exactly three vertices
The graph contains a directed Hamiltonian cycle

n \quad m

\begin{array}{l} u_1 \quad v_1 \\ u_2 \quad v_2 \\ \vdots \\ u_m \quad v_m \end{array}

\begin{array}{l} k \\ e_1 \quad e_2 \quad \cdots \quad e_k \end{array}

Output $k$ followed by $k$ distinct integers $e_1,e_2,\ldots,e_k$ , the indices of the chosen edges, in any order.

Your output must satisfy $2k \le 3n$ , and the graph formed by the chosen edges must be strongly connected.

The integers may be separated by arbitrary whitespace.

The chosen edges form the directed cycle $1 \to 2 \to 3 \to 4 \to 5 \to 6 \to 7 \to 8 \to 1$ . Therefore, the chosen subgraph is strongly connected.

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W1D7B. Strong Enough (Hard Version)

Constraints

Input

Output

Samples

W1D7B. Strong Enough (Hard Version)

Constraints

Input

Output

Samples