Recr_Ms_1
For the best case : Identify the recurrence relation merge_sort function described below.

Question

Recr_Ms_1
For the best case : Identify the recurrence relation   merge_sort function described below.

BlackTom AI · Accepted Answer

Question restatement: We are asked to identify the recurrence relation that describes the best-case time complexity for the described merge_sort function.

Option A: T(n) = { Θ(1) if n < 2 ; T(n-1) + Θ(1) otherwise }.
- This would model a process that reduces n by 1 at each step with constant work per step. It yields O(n) time, which is not consistent with the typical divide-and-conquer behavior of merge sort. Therefore this option is not appropriate for the best-case recurrence of merge sort.

Option B: T(n) = { Θ(1) if n < 2 ; T(n-1) + Θ(n) otherwise }.
- Here, we still decrement by 1, but the non-leaf work is Θ(n) at each step. This would accumulate to a higher-than-necessary cost and does not reflect the standard merge sort pattern of dividing the problem. It is not a correct model for merge sort.

Option C: T(n) = { Θ(1) if n < 2 ; 2 T(n/2) + Θ(n) otherwise }.
- This matches the classic divide-and-conquer recurrence for merge sort: two recursive calls on halves (2 T(n/2)) plus the merge step cost Θ(n). For the best case, the merge still costs Θ(n) because merging two sorted halves takes linear time in the number of elements being merged. This is the standard recurrence used to analyze merge sort. This option is plausible as the merge step does not shrink to sublinear work in the best case; the Θ(n) merge cost remains.

Option D: T(n) = { Θ(1) if n < 2 ; 2 T(n/2) + Θ(n log n) }.
- The 2 T(n/2) term reflects the same two recursive calls, but the extra work is Θ(n log n) rather than Θ(n). In merge sort, the merge step is linear, not n log n, so this overestimates the non-recursive work. This is not the correct recurrence for the standard merge sort time, even in the best case.

Option E: T(n) = { Θ(1) if n < 2 ; 2 T(n-1) + Θ(1) otherwise }.
- This represents a binary branching with linear depth and constant per-step work, which is not how merge sort operates (merge sort divides into halves, not subtracts 1). It does not reflect the typical divide-and-conquer structure, so it is not appropriate here.

Summary: Among the given choices, the one that aligns with the standard best-case recurrence for merge sort is the option that uses two recursive calls on halves plus a linear merge cost, i.e., 2 T(n/2) + Θ(n). The other options either model linear (or sublinear) growth incorrectly, fail to capture the divide-and-conquer paradigm, or inflate the merge cost beyond linear, which is inconsistent with merge sort behavior.

Recr_Ms_1 For the best case : Identify the recurrence relation merge_sort function described below.

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