Question Id : 15010 |
Context : UGC NET Computer Science 26 June 2025 (Paper II)
The tight asymptotic bound for the recurrence:
$T(n) = 2T(n/4) + \sqrt{n}$
🎥 Video solution / Text Solution of this question is given below:
We use the Master Theorem for $T(n) = aT(n/b) + f(n)$
Here, $a = 2$, $b = 4$, so
$n^{\log_b a} = n^{\log_4 2} = n^{1/2} = \sqrt{n}$
Thus, $f(n) = \sqrt{n}$ is of the same order as $n^{\log_b a}$.
Therefore, by Case 2 of Master Theorem,
$T(n) = \Theta(n^{\log_b a} \log n) = \Theta(\sqrt{n} \log n)$
✅ But wait! Let’s check growth dominance carefully:
Actually, when $f(n) = \Theta(n^{\log_b a})$, we multiply by $\log n$, hence the correct bound is
$\boxed{T(n) = \Theta(\sqrt{n} \log n)}$