C. Messenger in MAC
binary search; brute force; constructive algorithms; data structures; dp; greedy; sortings
In the new messenger for the students of the Master's Assistance Center, Keftemerum, an update is planned, in which developers want to optimize the set of messages shown to the user. There are a total of \(n\) messages. Each message is characterized by two integers \(a_i\) and \(b_i\). The time spent reading the set of messages with numbers \(p_1, p_2, \ldots, p_k\) (\(1 \le p_i \le n\), all \(p_i\) are distinct) is calculated by the formula:$$$\(\Large \sum_{i=1}^{k} a_{p_i} + \sum_{i=1}^{k - 1} |b_{p_i} - b_{p_{i+1}}|\)\(Note that the time to read a set of messages consisting of one message with number \)p_1\( is equal to \)a_{p_1}\(. Also, the time to read an empty set of messages is considered to be \)0\(.The user can determine the time \)l\( that he is willing to spend in the messenger. The messenger must inform the user of the maximum possible size of the set of messages, the reading time of which does not exceed \)l\(. Note that the maximum size of the set of messages can be equal to \)0$$$.The developers of the popular messenger failed to implement this function, so they asked you to solve this problem.
Each test consists of multiple test cases. The first line contains a single integer \(t\) (\(1 \leq t \leq 5 \cdot 10^4\)) — the number of test cases. The description of the test cases follows.The first line of each test case contains two integers \(n\) and \(l\) (\(1 \leq n \leq 2000\), \(1 \leq l \leq 10^9\)) — the number of messages and the time the user is willing to spend in the messenger.The \(i\)-th of the next \(n\) lines contains two integers \(a_i\) and \(b_i\) (\(1 \le a_i, b_i \le 10^9\)) — characteristics of the \(i\)-th message.It is guaranteed that the sum of \(n^2\) over all test cases does not exceed \(4 \cdot 10^6\).
For each test case, output a single integer — the maximum possible size of a set of messages, the reading time of which does not exceed \(l\).
In the first test case, you can take a set of three messages with numbers \(p_1 = 3\), \(p_2 = 2\), and \(p_3 = 5\). The time spent reading this set is equal to \(a_3 + a_2 + a_5 + |b_3 - b_2| + |b_2 - b_5| = 2 + 1 + 2 + |4 - 5| + |5 - 3| = 8\).In the second test case, you can take a set of one message with number \(p_1 = 1\). The time spent reading this set is equal to \(a_1 = 4\).In the fifth test case, it can be shown that there is no such non-empty set of messages, the reading time of which does not exceed \(l\).