A16452 | Jumping Through Segments

Polycarp is designing a level for a game. The level consists of $n$ segments on the number line, where the $i$ -th segment starts at the point with coordinate $l_i$ and ends at the point with coordinate $r_i$ .

The player starts the level at the point with coordinate $0$ . In one move, they can move to any point that is within a distance of no more than $k$ . After their $i$ -th move, the player must land within the $i$ -th segment, that is, at a coordinate $x$ such that $l_i \le x \le r_i$ . This means:

- After the first move, they must be inside the first segment (from $l_1$ to $r_1$ );
- After the second move, they must be inside the second segment (from $l_2$ to $r_2$ );
- ...
- After the $n$ -th move, they must be inside the $n$ -th segment (from $l_n$ to $r_n$ ).

The level is considered completed if the player reaches the $n$ -th segment, following the rules described above. After some thought, Polycarp realized that it is impossible to complete the level with some values of $k$ .

Polycarp does not want the level to be too easy, so he asks you to determine the minimum integer $k$ with which it is possible to complete the level.

The first line contains a single integer $t$ ( $1 \le t \le 10^4$ )—the number of test cases. Descriptions of the test cases follow.

The first line of each test case contains a single integer $n$ ( $1 \le n \le 2 \cdot 10^5$ )—the number of segments in the level.

The following $n$ lines.

The $i$ -th line contain two integers $l_i$ and $r_i$ ( $0 \le l_i \le r_i \le 10^9$ )—the boundaries of the $i$ -th segment. Segments may intersect.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .

For each test case, output a single integer—the minimum value of $k$ with which it is possible to complete the level.