C. Mahmoud and a Message

Codeforces

IDCF766C

Time2000ms

Memory256MB

Difficulty—

brute forcedpgreedystrings

English · Original

Chinese · Translation

Formal · Original

Mahmoud wrote a message _s_ of length _n_. He wants to send it as a birthday present to his friend Moaz who likes strings. He wrote it on a magical paper but he was surprised because some characters disappeared while writing the string. That's because this magical paper doesn't allow character number _i_ in the English alphabet to be written on it in a string of length more than _a__i_. For example, if _a_1 = 2 he can't write character '_a_' on this paper in a string of length 3 or more. String "_aa_" is allowed while string "_aaa_" is not. Mahmoud decided to split the message into some non-empty substrings so that he can write every substring on an independent magical paper and fulfill the condition. The sum of their lengths should be _n_ and they shouldn't overlap. For example, if _a_1 = 2 and he wants to send string "_aaa_", he can split it into "_a_" and "_aa_" and use 2 magical papers, or into "_a_", "_a_" and "_a_" and use 3 magical papers. He can't split it into "_aa_" and "_aa_" because the sum of their lengths is greater than _n_. He can split the message into single string if it fulfills the conditions. A substring of string _s_ is a string that consists of some consecutive characters from string _s_, strings "_ab_", "_abc_" and "_b_" are substrings of string "_abc_", while strings "_acb_" and "_ac_" are not. Any string is a substring of itself. While Mahmoud was thinking of how to split the message, Ehab told him that there are many ways to split it. After that Mahmoud asked you three questions: * How many ways are there to split the string into substrings such that every substring fulfills the condition of the magical paper, the sum of their lengths is _n_ and they don't overlap? Compute the answer modulo 109 + 7. * What is the maximum length of a substring that can appear in some valid splitting? * What is the minimum number of substrings the message can be spit in? Two ways are considered different, if the sets of split positions differ. For example, splitting "_aa|a_" and "_a|aa_" are considered different splittings of message "_aaa_". ## Input The first line contains an integer _n_ (1 ≤ _n_ ≤ 103) denoting the length of the message. The second line contains the message _s_ of length _n_ that consists of lowercase English letters. The third line contains 26 integers _a_1, _a_2, ..., _a_26 (1 ≤ _a__x_ ≤ 103) — the maximum lengths of substring each letter can appear in. ## Output Print three lines. In the first line print the number of ways to split the message into substrings and fulfill the conditions mentioned in the problem modulo 109 + 7. In the second line print the length of the longest substring over all the ways. In the third line print the minimum number of substrings over all the ways. [samples] ## Note In the first example the three ways to split the message are: * _a|a|b_ * _aa|b_ * _a|ab_ The longest substrings are "_aa_" and "_ab_" of length 2. The minimum number of substrings is 2 in "_a|ab_" or "_aa|b_". Notice that "_aab_" is not a possible splitting because the letter '_a_' appears in a substring of length 3, while _a_1 = 2.

Mahmoud 写了一条长度为 $n$ 的消息 #cf_span[s]。他想将其作为生日礼物发送给喜欢字符串的朋友 Moaz。他将消息写在一张神奇的纸上，但惊讶地发现有些字符在书写过程中消失了。这是因为这张神奇的纸不允许在长度超过 $a_i$ 的字符串中写入英文字母表中的第 $i$ 个字符。例如，如果 $a_1 = 2$，则他不能在长度为 $3$ 或更长的字符串中写入字符 '_a_'。字符串 "_aa_" 是允许的，而字符串 "_aaa_" 则不允许。 Mahmoud 决定将消息拆分成若干非空子串，以便他可以将每个子串写在独立的神奇纸上，并满足条件。这些子串的长度之和应为 $n$，且不能重叠。例如，如果 $a_1 = 2$ 且他想发送字符串 "_aaa_"，他可以将其拆分为 "_a_" 和 "_aa_"，使用 $2$ 张神奇纸；或者拆分为 "_a_"、"_a_" 和 "_a_"，使用 $3$ 张神奇纸。他不能将其拆分为 "_aa_" 和 "_aa_"，因为它们的长度之和大于 $n$。如果整个消息本身满足条件，他也可以将其作为一个整体字符串发送。字符串 $s$ 的子串是指由 $s$ 中若干连续字符组成的字符串，例如 "_ab_"、"_abc_" 和 "_b_" 都是字符串 "_abc_" 的子串，而 "_acb_" 和 "_ac_" 则不是。任何字符串都是其自身的子串。当 Mahmoud 正在思考如何拆分消息时，Ehab 告诉他有多种拆分方式。之后，Mahmoud 向你提出了三个问题：两种拆分方式被认为是不同的，当且仅当它们的拆分位置集合不同。例如，拆分 "_aa|a_" 和 "_a|aa_" 被视为消息 "_aaa_" 的不同拆分方式。第一行包含一个整数 $n$ ($1 ≤ n ≤ 10^3$)，表示消息的长度。第二行包含长度为 $n$ 的消息 $s$，由小写英文字母组成。第三行包含 $26$ 个整数 $a_1, a_2, ..., a_{26}$ ($1 ≤ a_x ≤ 10^3$) —— 每个字母在子串中允许出现的最大长度。请输出三行。第一行输出满足题目条件的拆分方案数，对 $10^9 + 7$ 取模。第二行输出所有拆分方案中子串的最大长度。第三行输出所有拆分方案中子串的最小数量。在第一个示例中，三种拆分方式为：最长的子串是长度为 $2$ 的 "_aa_" 和 "_ab_"。最少的子串数量是 $2$，出现在 "_a|ab_" 或 "_aa|b_" 中。注意，"_aab_" 不是合法的拆分，因为字母 '_a_' 出现在长度为 $3$ 的子串中，而 $a_1 = 2$。 ## Input 第一行包含一个整数 $n$ ($1 ≤ n ≤ 10^3$)，表示消息的长度。第二行包含长度为 $n$ 的消息 $s$，由小写英文字母组成。第三行包含 $26$ 个整数 $a_1, a_2, ..., a_{26}$ ($1 ≤ a_x ≤ 10^3$) —— 每个字母在子串中允许出现的最大长度。 ## Output 请输出三行。第一行输出满足题目条件的拆分方案数，对 $10^9 + 7$ 取模。第二行输出所有拆分方案中子串的最大长度。第三行输出所有拆分方案中子串的最小数量。 [samples] ## Note 在第一个示例中，三种拆分方式为： _a|a|b_ _aa|b_ _a|ab_ 最长的子串是长度为 $2$ 的 "_aa_" 和 "_ab_"。最少的子串数量是 $2$，出现在 "_a|ab_" 或 "_aa|b_" 中。注意，"_aab_" 不是合法的拆分，因为字母 '_a_' 出现在长度为 $3$ 的子串中，而 $a_1 = 2$。

**Definitions:** - Let $ n \in \mathbb{N} $, $ 1 \leq n \leq 10^3 $, be the length of the string $ s \in \Sigma^n $, where $ \Sigma = \{a, b, \dots, z\} $. - Let $ a: \Sigma \to \mathbb{N} $, with $ a[c] \in [1, 10^3] $ for each $ c \in \Sigma $, denote the maximum allowed length of any substring containing character $ c $. **Constraints:** A valid splitting of $ s $ is a sequence of non-overlapping, contiguous substrings $ s_1, s_2, \dots, s_k $ such that: - $ s = s_1 s_2 \cdots s_k $ (concatenation), - Each substring $ s_i $ has length $ \ell_i \geq 1 $, - For each substring $ s_i $ and for every character $ c \in \Sigma $ appearing in $ s_i $, it holds that $ \ell_i \leq a[c] $. **Objective:** Compute the following three values: 1. **Number of valid splittings:** $$ \text{dp}[n] = \sum_{\substack{1 \leq j \leq n \\ \text{substring } s[j:n] \text{ is valid}}} \text{dp}[j-1] $$ with $ \text{dp}[0] = 1 $, modulo $ 10^9 + 7 $. 2. **Maximum length of any substring across all valid splittings:** $$ \max \left\{ \ell \mid \exists \text{ valid splitting containing a substring of length } \ell \right\} $$ 3. **Minimum number of substrings in any valid splitting:** $$ \min \left\{ k \mid \exists \text{ valid splitting into } k \text{ substrings} \right\} $$ **Formal Output Requirements:** - First line: $ \text{dp}[n] \mod (10^9 + 7) $ - Second line: $ \max_{\text{valid splittings}} \max_{i} |s_i| $ - Third line: $ \min_{\text{valid splittings}} k $ **Auxiliary Condition for Validity of Substring $ s[i:j] $:** A substring $ s[i:j] $ (1-indexed, length $ j - i + 1 $) is valid if and only if: $$ \forall c \in \text{set}(s[i:j]), \quad j - i + 1 \leq a[c] $$

Samples

Input #1

3
aab
2 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Output #1

3
2
2

Input #2

10
abcdeabcde
5 5 5 5 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Output #2

401
4
3

API Response (JSON)

{
  "problem": {
    "name": "C. Mahmoud and a Message",
    "description": {
      "content": "Mahmoud wrote a message _s_ of length _n_. He wants to send it as a birthday present to his friend Moaz who likes strings. He wrote it on a magical paper but he was surprised because some characters d",
      "description_type": "Markdown"
    },
    "platform": "Codeforces",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "CF766C"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Mahmoud wrote a message _s_ of length _n_. He wants to send it as a birthday present to his friend Moaz who likes strings. He wrote it on a magical paper but he was surprised because some characters d...",
      "is_translate": false,
      "language": "English"
    },
    {
      "statement_type": "Markdown",
      "content": "Mahmoud 写了一条长度为 $n$ 的消息 #cf_span[s]。他想将其作为生日礼物发送给喜欢字符串的朋友 Moaz。他将消息写在一张神奇的纸上，但惊讶地发现有些字符在书写过程中消失了。这是因为这张神奇的纸不允许在长度超过 $a_i$ 的字符串中写入英文字母表中的第 $i$ 个字符。例如，如果 $a_1 = 2$，则他不能在长度为 $3$ 或更长的字符串中写入字符 '_a_'。字符串 \"_...",
      "is_translate": true,
      "language": "Chinese"
    },
    {
      "statement_type": "Markdown",
      "content": "**Definitions:**\n\n- Let $ n \\in \\mathbb{N} $, $ 1 \\leq n \\leq 10^3 $, be the length of the string $ s \\in \\Sigma^n $, where $ \\Sigma = \\{a, b, \\dots, z\\} $.\n- Let $ a: \\Sigma \\to \\mathbb{N} $, with $ ...",
      "is_translate": false,
      "language": "Formal"
    }
  ]
}

Full JSON Raw Segments