Minimization of Divide

AtCoder

IDagc075_c

Time2000ms

Memory256MB

Difficulty—

You are given two length-$N$ sequences of non-negative integers: $A=(A_1,A_2,\dots,A_N)$ and $B=(B_1,B_2,\dots,B_N)$. For a permutation $P=(P_1,P_2,\dots,P_N)$ of $(1,2,\dots,N)$, define the cost of $P$ as $\sum_{i=1}^{N} \left \lfloor \frac{A_{P_i}}{2^{B_i}} \right \rfloor$. Find the number, modulo $998244353$, of permutations $P$ that achieve the minimum cost. You are given $T$ test cases; solve each of them. ## Constraints * $1 \le T \le 2 \times 10^5$ * $1 \le N \le 2 \times 10^5$ * $1 \le A_i \le 10^9$ * $0 \le B_i \le 30$ * The sum of $N$ over all test cases is at most $2 \times 10^5$. ## Input The input is given from Standard Input in the following format: $T$ $\mathrm{case}_1$ $\mathrm{case}_2$ $\vdots$ $\mathrm{case}_T$ Each test case is given in the following format: $N$ $A_1\ A_2\ \dots\ A_N$ $B_1\ B_2\ \dots\ B_N$ [samples]

Samples

Input #1

7
2
5 9
0 1
3
3 4 10
0 1 2
4
1 2 3 4
0 0 1 1
5
1000000000 1000000000 1000000000 1000000000 1000000000
0 0 0 0 0
11
19 68 97 62 99 52 57 19 43 80 96
5 3 3 2 3 4 3 2 3 4 3
14
86 49 83 31 5 43 7 46 98 36 60 4 69 59
3 1 4 4 4 0 1 3 0 4 3 5 1 2
10
907139744 237517128 852012347 350549430 62876062 196019710 263351472 184393437 281593038 753973502
23 12 1 26 29 24 0 7 10 4

Output #1

1
2
4
120
8640
15552
1

For the first test case, the cost for each $P$ is as follows:

*   For $P=(1,2)$, $\left \lfloor \frac{A_{1}}{2^{B_1}} \right \rfloor + \left \lfloor \frac{A_{2}}{2^{B_2}} \right \rfloor = \left \lfloor \frac{5}{1} \right \rfloor + \left \lfloor \frac{9}{2} \right \rfloor = 9$
*   For $P=(2,1)$, $\left \lfloor \frac{A_{2}}{2^{B_1}} \right \rfloor + \left \lfloor \frac{A_{1}}{2^{B_2}} \right \rfloor = \left \lfloor \frac{9}{1} \right \rfloor + \left \lfloor \frac{5}{2} \right \rfloor = 11$

Thus, the minimum cost is $9$, and there is one permutation $P = (1,2)$ that achieves it.
For the second test case, $P = (1,2,3),(2,1,3)$ achieve the minimum cost of $7$.
For the third test case, $P = (1,2,3,4),(2,1,3,4),(1,2,4,3),(2,1,4,3)$ achieve the minimum cost of $6$.
For the fourth test case, all permutations of $(1,2,3,4,5)$ achieve the minimum cost of $5000000000$.

API Response (JSON)

{
  "problem": {
    "name": "Minimization of Divide",
    "description": {
      "content": "You are given two length-$N$ sequences of non-negative integers: $A=(A_1,A_2,\\dots,A_N)$ and $B=(B_1,B_2,\\dots,B_N)$. For a permutation $P=(P_1,P_2,\\dots,P_N)$ of $(1,2,\\dots,N)$, define the cost of $",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc075_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given two length-$N$ sequences of non-negative integers: $A=(A_1,A_2,\\dots,A_N)$ and $B=(B_1,B_2,\\dots,B_N)$.\nFor a permutation $P=(P_1,P_2,\\dots,P_N)$ of $(1,2,\\dots,N)$, define the cost of $...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments