CF1227A.Math Problem

Your math teacher gave you the following problem:

There are $n$ segments on the $x$-axis, $[l_1; r_1], [l_2; r_2], \ldots, [l_n; r_n]$. The segment $[l; r]$ includes the bounds, i.e. it is a set of such $x$ that $l \le x \le r$. The length of the segment $[l; r]$ is equal to $r - l$.

Two segments $[a; b]$ and $[c; d]$ have a common point (intersect) if there exists $x$ that $a \leq x \leq b$ and $c \leq x \leq d$. For example, $[2; 5]$ and $[3; 10]$ have a common point, but $[5; 6]$ and $[1; 4]$ don't have.

You should add one segment, which has at least one common point with each of the given segments and as short as possible (i.e. has minimal length). The required segment can degenerate to be a point (i.e a segment with length zero). The added segment may or may not be among the given $n$ segments.

In other words, you need to find a segment $[a; b]$, such that $[a; b]$ and every $[l_i; r_i]$ have a common point for each $i$, and $b-a$ is minimal.

你的数学老师给你布置了如下题目：

在 $x$ 轴上有 $n$ 条线段：$[l_1; r_1], [l_2; r_2], \ldots, [l_n; r_n]$。线段 $[l; r]$ 包含端点，即它是满足 $l \le x \le r$ 的所有 $x$ 构成的集合。线段 $[l; r]$ 的长度为 $r - l$。

两条线段 $[a; b]$ 和 $[c; d]$ 有公共点（相交），当且仅当存在某个 $x$，使得 $a \leq x \leq b$ 且 $c \leq x \leq d$。例如，$[2; 5]$ 与 $[3; 10]$ 有公共点，但 $[5; 6]$ 与 $[1; 4]$ 没有公共点。

你需要添加一条新线段，使其与给定的每一条线段都至少有一个公共点，并且该线段尽可能短（即长度最小）。所添加的线段可以退化为一个点（即长度为零的线段）。添加的线段可以是给定的 $n$ 条线段之一，也可以不是。

换言之，你需要找出一条线段 $[a; b]$，使得对每个 $i$，$[a; b]$ 与 $[l_i; r_i]$ 均有公共点，且 $b-a$ 最小。

The first line contains integer number $t$ ($1 \le t \le 100$) — the number of test cases in the input. Then $t$ test cases follow.

The first line of each test case contains one integer $n$ ($1 \le n \le 10^{5}$) — the number of segments. The following $n$ lines contain segment descriptions: the $i$-th of them contains two integers $l_i,r_i$ ($1 \le l_i \le r_i \le 10^{9}$).

The sum of all values $n$ over all the test cases in the input doesn't exceed $10^5$.

第一行包含一个整数 $t$（$1 \le t \le 100$）—— 表示输入中测试用例的数量。随后是 $t$ 个测试用例。

每个测试用例的第一行包含一个整数 $n$（$1 \le n \le 10^{5}$）—— 表示线段的数量。接下来的 $n$ 行描述这些线段：其中第 $i$ 行包含两个整数 $l_i, r_i$（$1 \le l_i \le r_i \le 10^{9}$）。

输入中所有测试用例的 $n$ 值之和不超过 $10^5$。

For each test case, output one integer — the smallest possible length of the segment which has at least one common point with all given segments.

对于每个测试用例，输出一个整数——与所有给定线段均至少有一个公共点的线段的最小可能长度。

In the first test case of the example, we can choose the segment $[5;7]$ as the answer. It is the shortest segment that has at least one common point with all given segments.

在示例的第一个测试用例中，我们可以选择区间 $[5;7]$ 作为答案。它是与所有给定区间均至少有一个公共点的最短区间。