Recr_1
What is the correct recurrence relation for the power function provided below, which calculates xn using a recursive divide-and-conquer approach?

Question

BlackTom AI · Accepted Answer

The problem asks for the correct recurrence relation for a power function that uses a recursive divide-and-conquer approach to compute xn.
Option 1: T(n) = { Θ(1) if n = 0; 2T(n-1) + Θ(1) otherwise }. This represents a linear, unbalanced recurrence that reduces by 1 each call and multiplies the work by 2 each step, which does not reflect a divide-and-conquer scheme and would yield exponential time, not the intended logarithmic or sublinear behavior. It is therefore incorrect for a divide-and-conquer power function.
Option 2: T(n) = { Θ(1) if n = 0; T(n/2) + Θ(1) otherwise }. This matches a standard divide-and-conquer pattern: split the problem in half (n/2) and combine with constant extra work. This is typical for exponentiation by squaring or similar recursive strategies, assuming constant-time work to combine results.
Option 3: T(n) = { Θ(1) if n = 0; T(n-1) + Θ(1) otherwise }. This is a linear recurrence that reduces n by 1 at each step with constant work per step, leading to Θ(n) time. While correct for a naive linear approach, it does not embody divide-and-conquer and is therefore not appropriate for a recursive halving strategy.
Option 4: T(n) = { Θ(1) if n = 0; T(n-1) + Θ(n) otherwise }. Here the non-constant Θ(n) term in the combine step dominates, yielding Θ(n^2) overall time in a simple summation, and it also uses a linear reduction by 1 rather than halving. This contradicts the expected divide-and-conquer pattern and is incorrect for a typical power computation via recursion.
Option 5: T(n) = { Θ(1) if n = 0; T(n/2) + Θ(n) otherwise }. This uses a halving recurrence but adds Θ(n) work at each level, which leads to a total cost of Θ(n log n) if you sum the geometric levels, or potentially higher depending on how Θ(n) scales with levels. For many divide-and-conquer exponentiation schemes, the combine step is constant work, so this option overestimates the cost and is not aligned with a typical constant-work combine.
Option 6: T(n) = { Θ(1) if n = 0; 2T(n/2) + Θ(1) otherwise }. This is a perfectly valid divide-and-conquer recurrence for some algorithms, yielding Θ(n) time if the recurrence resolves to the master theorem form aT(n/b) + f(n) with a = 2, b = 2, and f(n) = Θ(1). However, for the specific problem statement described—an approach that computes xn using a recursive divide-and-conquer method—the provided target recurrence in the prompt is T(n) = { Θ(1) if n = 0; T(n/2) + Θ(1) otherwise }. Since option 6 introduces an extra factor of 2 in the recursive term, it represents a different algorithmic cost model and does not match the given correct form.
Overall, the option that exactly matches the stated correct recurrence relation is Option 2, which captures the halving of the problem with constant extra work per level. The remaining options either use a linear reduction, include non-constant combine costs, or introduce an extra multiplicative factor that changes the growth rate, making them inconsistent with the described divide-and-conquer power computation.

Recr_1 What is the correct recurrence relation for the power function provided below, which calculates xn using a recursive divide-and-conquer approach?

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