Training Material

## Greedy Algorithms→Teleporters (IOI 2008)

You are participating in a competition that involves crossing Egypt from west to east along a straight line segment. Initially you are located at the westmost point of the segment. It is a rule of the competition that you must always move along the segment, and always eastward.

There are N teleporters on the segment. A teleporter has two endpoints. Whenever you reach one of the endpoints, the teleporter immediately teleports you to the other endpoint. (Note that, depending on which endpoint of the teleporter you reach, teleportation can transport you either eastward or westward of your current position.) After being teleported, you must continue to move eastward along the segment; you can never avoid a teleporter endpoint that is on your way. There will never be two teleporter endpoints at the same position. Endpoints will be strictly between the start and the end of the segment.

Every time you get teleported, you earn 1 point. The objective of the competition is to earn as many points as possible. In order to maximize the points you earn, you are allowed to add up to M new teleporters to the segment before you start your journey. You also earn points for using the new teleporters.

You can set the endpoints of the new teleporters wherever you want (even at non‐integer coordinates) as long as they do not occupy a position already occupied by another endpoint. That is, the positions of the endpoints of all teleporters must be unique. Also, endpoints of new teleporters must lie strictly between the start and the end of the segment.

Note that it is guaranteed that no matter how you add the teleporters, you can always reach the end of the segment.

Constraints:

• 1≤N≤106
• 1≤M≤106
• Distance between endpoints of segment≤2×106

Solution