Recr_12
Identify the recurrence relation for the function shown below.

Question

Recr_12
Identify the recurrence relation for the  function shown below.

BlackTom AI · Accepted Answer

Question restatement: Identify the recurrence relation for the function shown below.
Answer options to evaluate:
1) T(n) = { Θ(1) if n < 2; 3 T(n/4) + Θ(log n) otherwise }
2) T(n) = { Θ(1) if n < 2; 3 T(n/4) + Θ(n^2 log n) otherwise }
3) T(n) = { Θ(1) if n < 2; 3 T(n/4) + Θ(n^2) otherwise }
4) T(n) = { Θ(1) if n < 2; 3 T(n/4) + Θ(n) otherwise }
5) T(n) = { Θ(1) if n < 2; 3 T(n/4) + Θ(n log n) otherwise }
6) T(n) = { Θ(1) if n < 2; 3 T(n/4) + Θ(n log n) otherwise }

Option 1 analysis:
- This option uses Θ(log n) as the non-recursive combination cost. The dominant work outside the recursive calls would be proportional to log n. Unless there is a hidden n factor in the combine step, this is unlikely to reflect a typical divide-by-4 recurrence where the combine cost scales with n or a higher order term. Therefore, this option seems to understate the work unless the problem explicitly defines a log n overhead, which is uncommon for a problem with a division by 4.

Option 2 analysis:
- Here the non-recursive cost is Θ(n^2 log n). That would dramatically dominate the recurrence if n grows, making the overall solution potentially Θ(n^2 log n) times a polylog factor, depending on the Master Theorem conditions. This is an unusually large overhead for a simple division-by-4 branching and would not balance with the 3 recursive branches, so this is likely not the intended form.

Option 3 analysis:
- The non-recursive cost is Θ(n^2). Similar to Option 2, the n^2 term is polynomial and quite large, which would lead to a heavy overall cost. In many standard textbook recurrences with a = 3, b = 4, the non-recursive term is often linear or near-linear, not quadratic, unless specifically stated. This makes Option 3 suspect as an incorrect match.

Option 4 analysis:
- The non-recursive cost is Θ(n). This is a plausible overhead for some divide-and-conquer scenarios, since the problem size shrinks by a factor of 4 and the combine step could be linear. However, we must compare with other options to see which best matches the problem’s intended cost growth. Θ(n) is reasonable but may not be the exact intended term unless the problem emphasizes a linear combine.

Option 5 analysis:
- The non-recursive cost is Θ(n log n). This is a common hybrid cost that grows faster than linear but slower than quadratic, and it could arise from sorting-like work or certain log-linear operations. In a recurrence with three subproblems of size n/4, an n log n combine cost is plausible if the combine step does a linearithmic amount of work. This option is a strong candidate if the problem intends a log-linear overhead.

Option 6 analysis:
- The non-recursive cost is Θ(n log n) as in Option 5, with the same structure: T(n) = Θ(1) for n < 2; 3 T(n/4) + Θ(n log n) otherwise. This matches a standard pattern where the combine step scales as n log n, and the recursive term is 3 T(n/4). Given a typical Master Theorem application with a = 3, b = 4, and f(n) = Θ(n log n), this leads to a particular overall growth that is consistent with a non-trivial but common form for such recurrences.

Comparative assessment:
- Options 1, 2, 3, and 4 assign non-recursive costs that are either too small (Θ(log n), Θ(n), Θ(n^2)) or not aligned with a likely problem of this type, making them less plausible for the intended recurrence.
- Options 5 and 6 assign a Θ(n log n) combine cost, which is a natural and common choice for divide-and-conquer recurrences where some merging or processing grows roughly linearly with n and includes a log factor due to sorting, scanning, or similar work.
- Between 5 and 6, the only substantive difference is in formatting of the same cost term; both express the same intended recurrence form: T(n) = Θ(1) for n < 2; 3 T(n/4) + Θ(n log n) otherwise. The mathematical content is equivalent in both, so they describe the same underlying recurrence.

Concluding note on reasoning style:
- When evaluating such options, it helps to align the non-recursive term with the natural costs that arise in a divide-and-conquer setting where the subproblem count is a = 3 and subproblem size is n/b with b = 4. The combination cost often reflects either linear, linearithmic, or higher-order terms depending on the merge/processing step. In this set, Θ(n log n) is the most plausible and commonly encountered non-recursive term for this kind of recurrence, making Options 5 and 6 the ones that fit best with the typical pattern described by the problem statement.

Recr_12 Identify the recurrence relation for the function shown below.

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