A14184 | Fly Around the World

After hearing the story of Dr. Zhang, Wowo decides to plan his own flight around the world.

He already chose $n$ checkpoints in the world map. Due to the landform and the clouds, he cannot fly too high or too low. Formally, let $b_i$ be the height of Wowo's aircraft at checkpoint $i$ , $x_i^-\le b_i\le x_i^+$ should be satisfied for all integers $i$ between $1$ and $n$ , where $x_i^-$ and $x_i^+$ are given integers.

The angle of Wowo's aircraft is also limited. For example, it cannot make a $90$ -degree climb. Formally, $y_i^-\le b_i-b_{i-1}\le y_i^+$ should be satisfied for all integers $i$ between $2$ and $n$ , where $y_i^-$ and $y_i^+$ are given integers.

The final limitation is the speed of angling up or angling down. An aircraft should change its angle slowly for safety concerns. Formally, $z_i^- \le (b_i - b_{i-1}) - (b_{i-1} - b_{i-2}) \le z_i^+$ should be satisfied for all integers $i$ between $3$ and $n$ , where $z_i^-$ and $z_i^+$ are given integers.

Taking all these into consideration, Wowo finds that the heights at checkpoints are too hard for him to choose. Please help Wowo decide whether there exists a sequence of real numbers $b_1, \ldots, b_n$ satisfying all the contraints above.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 66\,666$ ). Description of the test cases follows.

The first line of each test case contains a single integer $n$ ( $3 \le n \le 100\,000$ ).

The $i$ -th of the next $n$ lines contains two integers $x_i^-$ , $x_i^+$ ( $-10^8\le x_i^-\le x_i^+\le 10^8$ ) denoting the lower and upper bound of $b_i$ .

The $i$ -th of the next $n-1$ lines contains two integers $y_{i+1}^-$ , $y_{i+1}^+$ ( $-10^8\le y_{i+1}^-\le y_{i+1}^+\le 10^8$ ) denoting the lower and upper bound of $b_{i+1}-b_i$ .

The $i$ -th of the next $n-2$ lines contains two integers $z_{i+2}^-$ , $z_{i+2}^+$ ( $-10^8\le z_{i+2}^-\le z_{i+2}^+\le 10^8$ ) denoting the lower and upper bound of $(b_{i+2}-b_{i+1}) - (b_{i+1}-b_i)$ .

It is guaranteed that the sum of $n$ over all test cases does not exceed $200\,000$ .

It is guaranteed that relaxing every constraint by $10^{-6}$ (i.e., decrease $x_i^-, y_i^-, z_i^-$ by $10^{-6}$ and increase $x_i^+, y_i^+, z_i^+$ by $10^{-6}$ ) will not change the answer.

For each test case, output YES if a sequence $b_1,\ldots, b_n$ satisfying the constraints exists and NO otherwise. The sequence $b_1,\ldots, b_n$ is not required.