[USACO23FEB] Piling Papers G

Luogu

IDLGP9129

Time2000ms

Memory256MB

DifficultyP6

USACO2023数位 DP区间 DP

Farmer John wrote down $N (1 \le N \le 300)$ digits on pieces of paper. For each $i \in [1,N]$, the ith piece of paper contains digit $a_i (1 \le a_i \le 9)$. The cows have two favorite integers $A$ and $B(1 \le A \le B<10^{18})$, and would like you to answer $Q (1 \le Q \le 5⋅10^4)$ queries. For the $i$-th query, the cows will move left to right across papers $l_i \cdots r_i (1 \le l_i \le r_i \le N)$, maintaining an initially empty pile of papers. For each paper, they will either add it to the top of the pile, to the bottom of the pile, or neither. In the end, they will read the papers in the pile from top to bottom, forming an integer. Over all $3 ^ {r_i−l_i+1}$ ways for the cows to make choices during this process, count the number of ways that result in the cows reading an integer in $[A,B]$ inclusive, and output this number modulo $10^9+7$. ## Input The first line contains three space-separated integers $N, A$, and $B$. The second line contains $N$ space-separated digits $a_1,a_2, \cdots ,a_N$. The third line contains an integer $Q$, the number of queries. The next $Q$ lines each contain two space-separated integers $l_i$ and $r_i$. ## Output For each query, a single line containing the answer. [samples] ## Note ### Explanation for Sample 1 For the first query, there are nine ways Bessie can stack papers when reading the interval $[1,2]$: - Bessie can ignore $1$ then ignore $2$, getting $0$. - Bessie can ignore $1$ then add $2$ to the top of the stack, getting $2$. - Bessie can ignore $1$ then add $2$ to the bottom of the stack, getting $2$. - Bessie can add $1$ to the top of the stack then ignore $2$, getting $1$. - Bessie can add $1$ to the top of the stack then add $2$ to the top of the stack, getting $21$. - Bessie can add $1$ to the top of the stack then add $2$ to the bottom of the stack, getting $12$. - Bessie can add $1$ to the bottom of the stack then ignore $2$, getting $1$. - Bessie can add $1$ to the bottom of the stack then add $2$ to the top of the stack, getting $21$. - Bessie can add $1$ to the bottom of the stack then add $2$ to the bottom of the stack, getting $12$. Only the $2$ ways that give $21$ yield a number between $13$ and $327$, so the answer is $2$. ### SCORING - Inputs $2-3$: $B<100$ - Inputs $4-5$: $A=B$ - Inputs $6-13$: No additional constraints.

Samples

Input #1

Output #1

2
18
34

API Response (JSON)

{
  "problem": {
    "name": "[USACO23FEB] Piling Papers G",
    "description": {
      "content": "Farmer John wrote down $N (1 \\le N \\le 300)$ digits on pieces of paper. For each $i \\in [1,N]$, the ith piece of paper contains digit $a_i (1 \\le a_i \\le 9)$. The cows have two favorite integers $A$ ",
      "description_type": "Markdown"
    },
    "platform": "Luogu",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": {
      "LuoguStyle": "P6"
    },
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "LGP9129"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Farmer John wrote down $N (1 \\le N \\le 300)$ digits on pieces of paper. For each $i \\in [1,N]$, the ith piece of paper contains digit $a_i (1 \\le a_i \\le 9)$.\n\nThe cows have two favorite integers $A$ ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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