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.

In the first test case, all $b_i$ 's are in $[0,1]$ . Because of the constraints $1=y_2^-\le b_2-b_1\le y_2^+=1$ , $b_2-b_1$ must be $1$ . So $b_2=1$ and $b_1=0$ must hold. Then by $1=y_3^-\le b_3-b_2\le y_3^+=1$ , $b_3$ equals $2$ . This contradicts the constraint of $b_3\le 1$ . So no solution exists.

In the second test case, we can let all $b_i$ 's be $0$ .

In the third test case, one possible solution is $b_1=0$ , $b_2=1/3$ , $b_3=2/3$ , $b_4=1$ .