P16736 [GKS 2019 #E] Code-Eat Switcher

Umon is a foodie coder. Do you know what two activities that he loves the most? Of course, coding and eating! He always spends the whole day doing only those two activities. However, he thinks that some times of the day are better spent coding, and others are better spent eating. To illustrate this problem, Umon divides his day into $S$ time slots. During the i-th time slot, if Umon codes 100% of the time, he will achieve $C_i$ units of coding. On the other hand, if he eats 100% of the time, he will achieve $E_i$ units of eating. But of course, Umon can also use only a fraction of the time for coding, and the remaining for eating. Formally, he will choose a real number $f$ ($0 \le f \le 1$), code for $f$ of the time, and use the remaining $(1 - f)$ time to eat. This way, he will achieve $f \times C_i$ units of coding and $(1 - f) \times E_i$ units of eating. The total amount of coding Umon achieves for the day is simply the sum of all units of coding he achieved in each of the time slots. The total amount of eating is calculated in a similar way. Umon needs to plan his schedule for the next $D$ days. On the i-th day, he needs to achieve at least a total amount of $A_i$ units of coding and $B_i$ units of eating. For each day, determine whether there is a way for Umon to achieve his target.

The first line of input gives the number of test cases, $T$. $T$ test cases follow. Each test case begins with a line containing two integers $D$ and $S$, the number of days and the number of time slots in a day, respectively. Then $S$ lines follow, each describing a time slot. The i-th line contains two integers $C_i$ and $E_i$, the amount of coding units achieved if Umon codes for 100% of the time slot, and the amount of eating units achieved if he eats for 100% of the time slot, respectively. Then $D$ lines follow, each describing a day. The i-th line contains two integers $A_i$ and $B_i$, the minimal total amount of coding and eating that needs to be achieved on that day.

For each test case, output one line containing Case #x: y, where x is the test case number (starting from 1) and y is a string with D characters, where the i-th character is Y if there exists a schedule that can fulfill the target for the i-th day, otherwise it should be N.

In the first sample case, there are 4 days and 2 time slots for each day. - For day 1, Umon can just eat $100\%$ for both time slots, and therefore achieving a total of 0 units of coding and $8 + 10 = 18$ units of eating, thus reaching the target. - For day 2, Umon can eat $100\%$ of the time for the first time slot, and use $50\%$ of the second time slot for coding and $50\%$ for eating, achieving a total of $0 \times 3 + 0.5 \times 6 = 3$ units of coding, and $1 \times 8 + 0.5 \times 10 = 13$ units of eating, thus reaching the target. - For day 3, it is impossible to get a total of 10 units of coding. - For day 4, there are an infinite amount of ways to achieve the target. One possible strategy is to code $42\%$ (and eat $58\%$) in the first time slot, then code $98.76\%$ (and eat $1.24\%$) in the second time slot. That strategy yields a total of $0.42 \times 3 + 0.9876 \times 6 = 7.1856$ units of coding, and $0.58 \times 8 + 0.0124 \times 10 = 4.764$ units of eating. Thus, the answer should be YYNY. In the second sample case, note that the value of characteristics for the time slots may not necessarily be different from each other. ### Limits $1 \le T \le 100$. $1 \le C_i \le 10^4$, for all $i$. $1 \le E_i \le 10^4$, for all $i$. $0 \le A_i \le 10^8$, for all $i$. $0 \le B_i \le 10^8$, for all $i$. **Test set 1 (Visible)** $1 \le S \le 2$. $1 \le D \le 10$. **Test set 2 (Hidden)** For at least 75% of the test cases: $1 \le S \le 10^3$. $1 \le D \le 10^3$. For all test cases: $1 \le S \le 10^5$. $1 \le D \le 10^5$.