Hulksmash Big O Analysis

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, Iteration, Modulus Operations

1: U-nderstand

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

Questions:

1. What is the time complexity of hulk_smash() based on the input size (n)?
2. What is the space complexity of the function?
3. How would a dictionary-based lookup for divisibility checks affect the time complexity?

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2: M-atch

Match this problem to known complexity analysis concepts and scenarios.

Key Observations:

1. Iterative Processing:

   • The function iterates through (1) to (n), performing a constant number of operations for each element.

2. Modulus Operations:

   • Each modulus operation ((i % 3), (i % 5)) is a constant-time operation ((O(1))).

3. Output List Construction:

   • A list of size (n) is created, requiring linear space.

4. Dictionary-Based Optimization:

   • Using a dictionary for logic centralization does not affect the overall iteration count or complexity.

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3: P-lan

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

Time Complexity:

1. Determine the cost of iterating through (1) to (n).
2. Evaluate the impact of performing modulus operations during each iteration.

Space Complexity:

1. Identify the memory required for storing the output list.
2. Assess whether auxiliary space (e.g., temporary variables or data structures) is used.

Optimization Discussion:

1. Analyze how using a dictionary impacts readability and runtime.
2. Determine whether the dictionary-based approach affects asymptotic complexity.

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4: I-mplement

Implement the analysis with clear justifications.

1. Time Complexity of hulk_smash()

• Overall Complexity: The time complexity of hulk_smash() is O(n).
• Reasoning:
  • The function iterates over (n) elements.
  • For each element, it performs a constant number of modulus operations ((i % 3), (i % 5), and possibly (i % 15)).
  • Each modulus operation is (O(1)), so the total complexity is (O(n)).

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2. Space Complexity of hulk_smash()

• Overall Complexity: The space complexity of hulk_smash() is O(n).
• Reasoning:
  • The function constructs a list answer of size (n), which requires (O(n)) space.
  • No other data structures or additional memory are used, so the auxiliary space is constant ((O(1))).

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3. Optimization Discussion

Using Dictionary-Based Lookup:

• Time Complexity:

  • Dictionary lookups for divisibility logic would still involve modulus operations ((O(1))).
  • The total iteration count over (n) elements dominates the complexity, which remains (O(n)).

• Advantages:

  • Centralizes the logic for handling multiple divisibility conditions.
  • Improves code readability and maintainability.

• Disadvantages:

  • Slight overhead for dictionary setup and lookups, though this does not affect asymptotic complexity.

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5: R-eview

Review the scenarios and validate with examples.

1. Input: n = 5

   • Expected Output: ["1", "2", "Fizz", "4", "Buzz"]
   • Observed Complexity: (O(n)) with (O(n)) space.

2. Input: n = 15

   • Expected Output: ["1", "2", "Fizz", "4", "Buzz", "Fizz", ..., "FizzBuzz"]
   • Observed Complexity: (O(n)), with (n) modulus checks and a list of size (n).

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6: E-valuate

Evaluate the performance of hulk_smash() and trade-offs between iterative logic and dictionary-based lookup.

Summary of Complexity:

1. Time Complexity:
   • (O(n)), as the function iterates through (n) elements, performing constant-time operations for each.
2. Space Complexity:
   • (O(n)), as the output list stores (n) elements.

Trade-offs:

1. Current Approach:

   • Simple and efficient for small to moderate (n).
   • Logic is directly written into the iteration loop.

2. Dictionary-Based Lookup:

   • Better for extensibility and maintaining complex divisibility conditions.
   • Slightly more overhead for dictionary setup, but complexity remains (O(n)).