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Consider the following recurrence relation: T(n) = { √n T(√n) + n for n ≥ 1, 1 for n = 1. Which one of the following options is CORRECT?

1. T(n) = Θ(n log log n)
2. T(n) = Θ(n log n)
3. T(n) = Θ(n² log n)
4. T(n) = Θ(n² log log n)

Correct answer: T(n) = Θ(n log log n)

Solution

The recurrence relation indicates that the function T(n) grows based on the size of n and its logarithmic factors, specifically due to the repeated halving of n (as seen in T(√n)). This leads to a complexity that is proportional to n multiplied by the logarithm of the logarithm of n, hence T(n) = Θ(n log log n) is the correct characterization.

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