Large DP Table

AtCoder

IDarc169_f

Time2000ms

Memory256MB

Difficulty—

You are given integer sequences of length $N$: $A=(A_1,A_2,\cdots,A_N)$, $B=(B_1,B_2,\cdots,B_N)$, $X=(X_1,X_2,\cdots,X_N)$, and $Y=(Y_1,Y_2,\cdots,Y_N)$. Here, $A$ and $B$ satisfy the following properties: * $A_1=1$. * $B_1=2$. * $(A_1,A_2,\cdots,A_N,B_1,B_2,\cdots,B_N)$ is a permutation of $(1,2,\cdots,2N)$. Define the integers $d_{i,j}$ ($1 \leq i,j \leq N$) as follows: * $d_{1,1}=0$. * If $(i,j) \neq (1,1)$ and $A_i<B_j$, then $d_{i,j}=d_{i,j-1}+X_i$. * If $(i,j) \neq (1,1)$ and $A_i>B_j$, then $d_{i,j}=d_{i-1,j}+Y_j$. Find $\sum_{1 \leq i \leq N}\sum_{1 \leq j \leq N}d_{i,j}$, modulo $998244353$. ## Constraints * $2 \leq N \leq 250000$ * $A_1=1$ * $B_1=2$ * $(A_1,A_2,\cdots,A_N,B_1,B_2,\cdots,B_N)$ is a permutation of $(1,2,\cdots,2N)$. * $1 \leq X_i \leq 10^9$ * $1 \leq Y_i \leq 10^9$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\cdots$ $A_N$ $B_1$ $B_2$ $\cdots$ $B_N$ $X_1$ $X_2$ $\cdots$ $X_N$ $Y_1$ $Y_2$ $\cdots$ $Y_N$ [samples]

Samples

Input #1

Output #1

8

The values of $d_{i,j}$ are as follows:

*   $d_{1,1}=0$
*   $d_{1,2}=d_{1,1}+X_1=0+2=2$
*   $d_{2,1}=d_{1,1}+Y_1=0+1=1$
*   $d_{2,2}=d_{1,2}+Y_2=2+3=5$

Thus, the answer is $0+2+1+5=8$.

Input #2

3
1 3 5
2 6 4
1 10 100
1000 10000 100000

Output #2

Input #3

3
1 6 5
2 4 3
1 10 100
1000 10000 100000

Output #3

Input #4

10
1 17 4 7 16 18 9 3 12 6
2 19 20 14 5 11 13 8 15 10
744280520 249168130 239276621 320064892 910500852 164832983 245532751 198319687 715892722 967824729
769431650 80707350 459924868 257261830 777045524 583882654 950300099 438099970 322288793 532405020

Output #4

746075419

API Response (JSON)

{
  "problem": {
    "name": "Large DP Table",
    "description": {
      "content": "You are given integer sequences of length $N$: $A=(A_1,A_2,\\cdots,A_N)$, $B=(B_1,B_2,\\cdots,B_N)$, $X=(X_1,X_2,\\cdots,X_N)$, and $Y=(Y_1,Y_2,\\cdots,Y_N)$. Here, $A$ and $B$ satisfy the following prope",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc169_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given integer sequences of length $N$: $A=(A_1,A_2,\\cdots,A_N)$, $B=(B_1,B_2,\\cdots,B_N)$, $X=(X_1,X_2,\\cdots,X_N)$, and $Y=(Y_1,Y_2,\\cdots,Y_N)$. Here, $A$ and $B$ satisfy the following prope...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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