P4612 [COI 2012] SETNJA

Mirko is going to deliver concert tickets to his friends.

Each friend’s home can be represented on a 2D grid. When Mirko walks, he can move in $8$ directions, always staying on integer coordinates. In each step, he moves one grid unit up, down, left, right, or along one of the $4$ diagonal directions. Each friend’s home is a point $(x, y)$ on the plane. When Mirko arrives to deliver a ticket, the friend can also walk out to meet him, and can also move in $8$ directions. Therefore, Mirko and the friend can meet at a position up to $P$ steps away from the friend’s home. The value of $P$ may be different for each friend. Mirko’s starting position and ending position are both unknown. Find the minimum number of steps Mirko needs to move in order to deliver all tickets.

The first line contains an integer $N$, the number of Mirko’s friends $(2 \le N \le 200{,}000)$. The next $N$ lines each contain three numbers $x, y, P\ (0 \le x, y, P \le 200{,}000)$. The friends are given in the order in which Mirko delivers the tickets.

Output one integer: the minimum number of steps Mirko must walk.

For all testdata, it is guaranteed that $2 \le N \le 200{,}000$ and $0 \le x, y, P \le 200{,}000$. Translated by ChatGPT 5