When executing merge, within mergesort, what is the worst case time cost on two sequences of length n? Select the tightest bound that holds.

Question

BlackTom AI · Accepted Answer

Merging two sorted sequences of length n requires comparing front elements and moving the smaller into the output until both sequences are exhausted, so the work grows linearly with the total number of elements.

1. Relevant concepts
- Merge step in mergesort combines two sorted lists into one sorted list by repeatedly comparing the current smallest elements and appending the smaller.
- Time complexity for merging two lists of lengths n1 and n2 is Θ(n1 + n2).
- A “tightest bound” means the smallest asymptotic class that always holds (often expressed as Θ(...) but O(...) is acceptable here).

2. Step-by-step reasoning
- Let the two input sequences each have length n, so total elements = 2n.
- Each iteration of the merge loop performs one comparison of the two current heads and then moves one element into the output.
- This loop continues until one input is exhausted; then the remainder of the other list (at most n elements) is copied into the output without further comparisons.

3. Intermediate/counting steps
- In the worst case comparisons can occur nearly every time we append an element until only one element remains: at most 2n − 1 comparisons.
- Total element moves (writes to output) = 2n.
- Thus total primitive operations ≤ a*(2n − 1) + b*(2n) for constants a,b, which simplifies to c·n + constant = Θ(n).

4. Why the correct answer is correct
- Because all costs (comparisons + moves + loop overhead) scale linearly with n, the tight asymptotic bound is Θ(n). Expressed in big-O notation as required, this is O(n).

5. Why other options are incorrect
- O(log n): far too small; merging must touch every element at least once.
- O(1): constant time is impossible because number of elements grows.
- O(n log n): true but not tight; it overestimates the cost (merge is strictly linear, not n log n).

6. Conclusion
- The worst-case time cost to merge two sequences of length n is O(n), which matches the provided answer.

When executing merge, within mergesort, what is the worst case time cost on two sequences of length n? Select the tightest bound that holds.Single choice

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