K. Crazy queries

Codeforces

IDCF10199K

Time30000ms

Memory1024MB

Difficulty—

English · Original

Formal · Original

Given two strings S and P, and Q queries, for each query you are given a range [L, R]. For each pair of integers i and j chosen with uniform probability among all pairs where 1 ≤ i ≤ L and R ≤ j ≤ |S|, create a new string S2 that is the result of concatenating S[1... i] and S[j... |S|]. You have to find the expected number of occurrences of P in S2. Note that all occurrences of P must be counted even if they overlap. The queries are independent meaning that the string S is the same for all queries. The first line of the input contains a single integer T the number of test cases. Each test case starts with two lines. The first line contains string S and the second line contains P, where 2 ≤ |S| ≤ 105 and 1 ≤ |P| ≤ 100. Each string consists of lowercase English letters. The third line contains an integer Q, where 1 ≤ Q ≤ 5·104. Each of the next Q lines represents a single query with two space-separated integers L and R, where 1 ≤ L < R ≤ |S|. For each test case output Q lines. where each line contains the answer of the corresponding query in the form p/q where p and q are the numerator and denominator of the expected value in its simplest form respectively. ## Input The first line of the input contains a single integer T the number of test cases. Each test case starts with two lines. The first line contains string S and the second line contains P, where 2 ≤ |S| ≤ 105 and 1 ≤ |P| ≤ 100. Each string consists of lowercase English letters. The third line contains an integer Q, where 1 ≤ Q ≤ 5·104. Each of the next Q lines represents a single query with two space-separated integers L and R, where 1 ≤ L < R ≤ |S|. ## Output For each test case output Q lines. where each line contains the answer of the corresponding query in the form p/q where p and q are the numerator and denominator of the expected value in its simplest form respectively. [samples]

**Definitions** Let $ S $ be a string of length $ n = |S| $, and $ P $ be a pattern string of length $ m = |P| $. Let $ Q $ be the number of queries. For each query, we are given integers $ L $ and $ R $ with $ 1 \leq L < R \leq n $. Let $ S[1..i] $ denote the prefix of $ S $ ending at index $ i $, and $ S[j..n] $ denote the suffix of $ S $ starting at index $ j $. Define $ S_2(i,j) = S[1..i] + S[j..n] $ for $ 1 \leq i \leq L $, $ R \leq j \leq n $. Let $ X_{i,j} $ be the number of (possibly overlapping) occurrences of $ P $ in $ S_2(i,j) $. **Constraints** 1. $ 2 \leq |S| \leq 10^5 $, $ 1 \leq |P| \leq 100 $, $ S, P \in \Sigma^* $, $ \Sigma = \{a,\dots,z\} $ 2. $ 1 \leq Q \leq 5 \cdot 10^4 $ 3. For each query: $ 1 \leq L < R \leq |S| $ **Objective** For each query $ (L, R) $, compute the expected value: $$ \mathbb{E}[X] = \frac{1}{L \cdot (n - R + 1)} \sum_{i=1}^{L} \sum_{j=R}^{n} X_{i,j} $$ Express the result as a reduced fraction $ \frac{p}{q} $.

API Response (JSON)

{
  "problem": {
    "name": "K. Crazy queries",
    "description": {
      "content": "Given two strings S and P, and Q queries, for each query you are given a range [L, R]. For each pair of integers i and j chosen with uniform probability among all pairs where 1 ≤ i ≤ L and R ≤ j ≤ |S",
      "description_type": "Markdown"
    },
    "platform": "Codeforces",
    "limit": {
      "time_limit": 30000,
      "memory_limit": 1048576
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "CF10199K"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given two strings S and P, and Q queries, for each query you are given a range [L, R].\n\nFor each pair of integers i and j chosen with uniform probability among all pairs where 1 ≤ i ≤ L and R ≤ j ≤ |S...",
      "is_translate": false,
      "language": "English"
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    {
      "statement_type": "Markdown",
      "content": "**Definitions**  \nLet $ S $ be a string of length $ n = |S| $, and $ P $ be a pattern string of length $ m = |P| $.  \nLet $ Q $ be the number of queries. For each query, we are given integers $ L $ an...",
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}

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