406725: GYM102511 E Dead-End Detector

The council of your home town has decided to improve road sign placement, especially for dead ends. They have given you a road map, and you must determine where to put up signs to mark the dead ends. They want you to use as few signs as possible.

The road map is a collection of locations connected by two-way streets. The following rule describes how to obtain a complete placement of dead-end signs. Consider a street $$$S$$$ connecting a location $$$x$$$ with another location. The $$$x$$$-entrance of $$$S$$$ gets a dead-end sign if, after entering $$$S$$$ from $$$x$$$, it is not possible to come back to $$$x$$$ without making a U-turn. A U-turn is a 180-degree turn immediately reversing the direction.

To save costs, you have decided not to install redundant dead-end signs, as specified by the following rule. Consider a street $$$S$$$ with a dead-end sign at its $$$x$$$-entrance and another street $$$T$$$ with a dead-end sign at its $$$y$$$-entrance. If, after entering $$$S$$$ from $$$x$$$, it is possible to go to $$$y$$$ and enter $$$T$$$ without making a U-turn, the dead-end sign at the $$$y$$$-entrance of $$$T$$$ is redundant. See Figure E.1 for examples.

Figure E.1: Illustration of sample inputs, indicating where non-redundant dead-end signs are placed.

The first line of input contains two integers $$$n$$$ and $$$m$$$, where $$$n$$$ ($$$1 \le n \le 5 \cdot 10^5$$$ ) is the number of locations and $$$m$$$ ($$$0 \le m \le 5 \cdot 10^5$$$) is the number of streets. Each of the following $$$m$$$ lines contains two integers $$$v$$$ and $$$w$$$ ($$$1 \le v < w \le n$$$) indicating that there is a two-way street connecting locations $$$v$$$ and $$$w$$$. All location pairs in the input are distinct.

On the first line, output $$$k$$$, the number of dead-end signs installed. On each of the next $$$k$$$ lines, output two integers $$$v$$$ and $$$w$$$ marking that a dead-end sign should be installed at the $$$v$$$-entrance of a street connecting locations $$$v$$$ and $$$w$$$. The lines describing dead-end signs must be sorted in ascending order of $$$v$$$-locations, breaking ties in ascending order of $$$w$$$-locations.

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

2
4 5
6 5

8 8
1 2
1 3
2 3
3 4
1 5
1 6
6 7
6 8

3
1 5
1 6
3 4