Rolling Hash

AtCoder

IDarc171_d

Time2000ms

Memory256MB

Difficulty—

You are given non-negative integers $P$ and $B$. Here, $P$ is prime, and $1 \leq B \leq P-1$. For a sequence of non-negative integers $X=(x_1,x_2,\dots,x_n)$, the hash value $\mathrm{hash}(X)$ is defined as follows. $\displaystyle \mathrm{hash}(X) = \left(\sum_{i=1}^n x_i B^{n-i}\right) \bmod P$ You are given $M$ pairs of integers $(L_1, R_1), (L_2, R_2), \dots, (L_M, R_M)$. Is there a sequence of non-negative integers $A=(A_1, A_2, \dots, A_N)$ of length $N$ that satisfies the condition below? * For all $i$ $(1 \leq i \leq M)$, the following condition holds: * Let $s$ be the sequence $(A_{L_i}, A_{L_i + 1}, \dots, A_{R_i})$ obtained by taking the $L_i$\-th to the $R_i$\-th elements of $A$. Then, $\mathrm{hash}(s) \neq 0$. ## Constraints * $2 \leq P \leq 10^9$ * $P$ is prime. * $1 \leq B \leq P - 1$ * $1 \leq N \leq 16$ * $1 \leq M \leq \frac{N(N+1)}{2}$ * $1 \leq L_i \leq R_i \leq N$ * $(L_i, R_i) \neq (L_j, R_j)$ if $i \neq j$. * All input values are integers. ## Input The input is given from Standard Input in the following format: $P$ $B$ $N$ $M$ $L_1$ $R_1$ $L_2$ $R_2$ $\vdots$ $L_M$ $R_M$ [samples]

Samples

Input #1

Output #1

Yes

The sequence $A = (2, 0, 4)$ satisfies the condition because $\mathrm{hash}((A_1)) = 2, \mathrm{hash}((A_1, A_2)) = 1, \mathrm{hash}((A_2, A_3)) = 1$.

Input #2

Output #2

No

No sequence satisfies the condition.

Input #3

998244353 986061415 6 11
1 5
2 2
2 5
2 6
3 4
3 5
3 6
4 4
4 5
4 6
5 6

Output #3

Yes

API Response (JSON)

{
  "problem": {
    "name": "Rolling Hash",
    "description": {
      "content": "You are given non-negative integers $P$ and $B$. Here, $P$ is prime, and $1 \\leq B \\leq P-1$.   For a sequence of non-negative integers $X=(x_1,x_2,\\dots,x_n)$, the hash value $\\mathrm{hash}(X)$ is de",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc171_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given non-negative integers $P$ and $B$. Here, $P$ is prime, and $1 \\leq B \\leq P-1$.  \nFor a sequence of non-negative integers $X=(x_1,x_2,\\dots,x_n)$, the hash value $\\mathrm{hash}(X)$ is de...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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