Match, Mod, Minimize

AtCoder

IDarc201_d

Time2000ms

Memory256MB

Difficulty—

You are given sequences of non-negative integers $A=(A_1,A_2,\dots ,A_N),B=(B_1,B_2,\dots ,B_N)$ of length $N$ and a positive integer $M$. When the elements of $A$ can be rearranged freely, find the minimum possible value of $\max_{1\le i\le N} ((A_i+B_i) \bmod M)$. $T$ test cases are given, so find the answer for each. ## Constraints * $1 \le T \le 10^5$ * $1 \le N \le 3 \times 10^5$ * $1 \le M \le 10^9$ * $0 \le A_i,B_i \lt M$ * The sum of $N$ over all test cases is at most $3 \times 10^5$. * All input values are integers. ## Input The input is given from Standard Input in the following format: $T$ $\text{case}_1$ $\text{case}_2$ $\vdots$ $\text{case}_T$ Each test case is given in the following format: $N$ $M$ $A_1$ $A_2$ $\ldots$ $A_N$ $B_1$ $B_2$ $\ldots$ $B_N$ [samples]

Samples

Input #1

3
3 6
3 1 4
2 0 1
1 1000000000
999999999
999999999
10 201
144 150 176 154 110 187 38 136 111 46
96 109 73 63 85 1 156 7 13 171

Output #1

3
999999998
111

For the first test case, if we rearrange $A$ to $4,3,1$, then $(A_i+B_i) \bmod M$ becomes $0,3,2$ respectively, and the $\max$ of these is $3$.

API Response (JSON)

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      "content": "You are given sequences of non-negative integers $A=(A_1,A_2,\\dots ,A_N),B=(B_1,B_2,\\dots ,B_N)$ of length $N$ and a positive integer $M$.\nWhen the elements of $A$ can be rearranged freely, find the m...",
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