Constructing a Harmonious Sequence

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Unit 7 Session 2 (Click for link to problem statements)

Problem Highlights

• 💡 Difficulty: Hard
• ⏰ Time to complete: 30 mins
• 🛠️ Topics: Divide and Conquer, Recursion, Permutations

1: U-nderstand

Understand what the interviewer is asking for by using test cases and questions about the problem.

• Established a set (2-3) of test cases to verify their own solution later.
• Established a set (1-2) of edge cases to verify their solution handles complexities.
• Have fully understood the problem and have no clarifying questions.
• Have you verified any Time/Space Constraints for this problem?

• What should be returned if n is 1?
  • Return [1] since there's only one note and no other possible sequence.
• How should the sequence be structured to avoid the midpoint condition?
  • The sequence must be generated recursively by separating odd and even indexed notes, ensuring the conditions are met.
• Can there be multiple valid sequences for a given n?
  • Yes, but the solution will generate one valid sequence using the described method.

HAPPY CASE
Input: n = 4
Output: [1, 3, 2, 4]
Explanation: The sequence [1, 3, 2, 4] is a harmonious sequence because it is a permutation of [1, 2, 3, 4] and satisfies the harmonious condition.

Input: n = 5
Output: [1, 3, 5, 2, 4]
Explanation: The sequence [1, 3, 5, 2, 4] is a harmonious sequence because it is a permutation of [1, 2, 3, 4, 5] and satisfies the harmonious condition.

EDGE CASE
Input: n = 1
Output: [1]
Explanation: There's only one note, so the sequence is trivially harmonious.

Input: n = 2
Output: [1, 2]
Explanation: The sequence [1, 2] is harmonious because it trivially satisfies the conditions.

2: M-atch

Match what this problem looks like to known categories of problems, e.g., Linked List or Dynamic Programming, and strategies or patterns in those categories.

For constructing sequences with specific constraints, we can consider the following approaches:

• Divide and Conquer: Use a recursive approach to build the sequence by splitting the problem into smaller subproblems, ensuring that the conditions are met.
• Recursion: The sequence can be generated recursively by first creating subsequences for odd and even indices, then merging them together in a way that maintains the conditions.

3: P-lan

Plan the solution with appropriate visualizations and pseudocode.

Plan

1) Base Case: If n is 1, return [1]. 2) Recursive Step:

• Split the sequence into two subsequences: one for odd indices and one for even indices.
• Recursively generate these subsequences.
• Scale the subsequences to fit the current size n and combine them. 3) Merge: Combine the scaled subsequences to form the final sequence.

Recursive Implementation

Pseudocode:

1) Define a helper function `construct_sequence(n)`:
   * If `n == 1`, return `[1]`.
   * Generate the odd-indexed subsequence by recursively calling `construct_sequence` with `(n + 1) // 2`.
   * Generate the even-indexed subsequence by recursively calling `construct_sequence` with `n // 2`.
   * Scale the odd sequence: `[2 * x * 1 for x in odd_sequence]`.
   * Scale the even sequence: `[2 * x for x in even_sequence]`.
   * Combine the two sequences and return the result.
2) The main function `harmonious_sequence(n)` will return `construct_sequence(n)`.

4: I-mplement

Implement the code to solve the algorithm.

def harmonious_sequence(n):
    def construct_sequence(n):
        if n == 1:
            return [1]
        
        # Generate the odd-indexed and even-indexed subsequences
        odd_sequence = construct_sequence((n + 1) // 2)
        even_sequence = construct_sequence(n // 2)
        
        # Scale and combine the sequences
        return [2 * x * 1 for x in odd_sequence] + [2 * x for x in even_sequence]
    
    return construct_sequence(n)

5: R-eview

Review the code by running specific example(s) and recording values (watchlist) of your code's variables along the way.

• Trace through your code with the input n = 4:
  • The sequence should be [1, 3, 2, 4], ensuring that no note is the exact midpoint between two others.

6: E-valuate

Evaluate the performance of your algorithm and state any strong/weak or future potential work.

Assume N represents the input integer.

• Time Complexity: O(N log N) because the sequence is generated recursively by splitting the problem into smaller subproblems.
• Space Complexity: O(N log N) due to the recursive call stack and the space required to store the sequence at each level.