Words Containing Character Big O Analysis

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JCSU Unit 4 Problem Set 2 (Click for link to problem statements)

Problem Highlights

• 💡 Difficulty: Easy
• ⏰ Time to complete: 10-15 mins
• 🛠️ Topics: Complexity Analysis, String Search, Python Built-ins

1: U-nderstand

Understand what the interviewer is asking for by analyzing the behavior and efficiency of the function words_with_char().

Questions:

1. What is the time complexity of words_with_char() when analyzing each word in a list for a character?
2. What is the space complexity, considering the input, output, and any auxiliary space?
3. How does using Python’s in operator affect the function’s efficiency compared to manual iteration?

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2: M-atch

Match this problem to known complexity analysis concepts and scenarios.

Key Observations:

1. Iterative Search Through Strings:

   • Each word is checked for the presence of a character using either manual iteration or the in operator.
   • Both approaches involve a linear search through each word.

2. Output List Construction:

   • Indices of matching words are stored in a new list, which requires additional space proportional to the number of matches.

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3: P-lan

Plan the analysis by breaking down the function’s behavior step by step.

Time Complexity:

1. Analyze the number of iterations over the list of words.
2. Evaluate the cost of searching for the character in each word.

Space Complexity:

1. Identify auxiliary space usage, such as the list of matching indices.
2. Assess whether any additional data structures are used.

Optimization Discussion:

1. Compare the efficiency of Python’s in operator versus manual iteration over characters.

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4: I-mplement

Implement the analysis with clear justifications.

1. Time Complexity of words_with_char()

• Overall Complexity: The time complexity of words_with_char() is O(n * m).
• Reasoning:
  • The function iterates through each of the (n) words in the list.
  • For each word, it performs a linear search for the character x. In the worst case, all (m) characters in the word are examined.
  • The total complexity is the product of the number of words and the average word length, resulting in (O(n * m)).

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2. Space Complexity of words_with_char()

• Overall Complexity: The space complexity is O(k), where (k) is the number of words containing the character x.
• Reasoning:
  • The function uses a list to store the indices of words that match the condition.
  • The size of the list depends on the number of matches, which can be at most (n) (if every word matches).
  • No other data structures or additional memory are used, so the complexity is proportional to the output size.

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3. Optimization Discussion

• Using Python’s in Operator:

  • The in operator performs a linear search through the characters of a string.
  • This has the same (O(m)) complexity as manually iterating over each character.
  • Python’s in operator is internally optimized and generally faster for most practical use cases but does not change the asymptotic complexity.

• Comparison:

  • Manual Iteration: Explicitly iterates over each character; slightly more control but can be verbose.
  • in Operator: Cleaner and often faster due to internal optimizations.
  • Both approaches have the same (O(n * m)) time complexity.

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5: R-eview

Review the scenarios and validate with examples.

1. Input: words = ["apple", "banana", "cherry"], x = "a"

   • Expected Output: [0, 1] (indices of words containing "a").
   • Observed Complexity: (O(n * m)).

2. Input: words = ["xyz", "123", "abc"], x = "z"

   • Expected Output: [0] (only the first word contains "z").
   • Observed Complexity: (O(n * m)).

3. Input: words = ["dog", "cat", "bat"], x = "e"

   • Expected Output: [] (no matches).
   • Observed Complexity: (O(n * m)).

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6: E-valuate

Evaluate the performance of words_with_char() and the trade-offs between manual iteration and the in operator.

Summary of Complexity:

1. Time Complexity:

   • (O(n * m)), where (n) is the number of words and (m) is the average word length.
   • Remains the same for both manual iteration and the in operator.

2. Space Complexity:

   • (O(k)), where (k) is the number of matches.

Trade-offs:

1. Manual Iteration:

   • Slightly more control over character processing.
   • Can be more verbose and prone to errors.

2. in Operator:

   • Cleaner and often faster due to Python’s internal optimizations.
   • Preferred for readability and simplicity.