Solve the recurrence relation:$T(n) = \sqrt{n} T \left(\sqrt n \right) + n$

$$T(n) = \sqrt{n} T \left(\sqrt n \right) + n$$

Master method does not apply here. Recursion tree goes a long way. Iteration method would be preferable.

The answer is $Θ (n \log \log n)$.

Can anyone arrive at the solution.

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Let $n = m^{2^k}$. We then get that $$T(m^{2^k}) = m^{2^{k-1}} T (m^{2^{k-1}}) + m^{2^{k}}$$ \begin{align} f_m(k) & = m^{2^{k-1}} f_m(k-1) + m^{2^k} = m^{2^{k-1}}(m^{2^{k-2}} f_m(k-2) + m^{2^{k-1}}) + m^{2^k}\\ & = 2 m^{2^k} + m^{3 \cdot 2^{k-2}} f_m(k-2) \end{align} $$m^{3 \cdot 2^{k-2}} f_m(k-2) = m^{3 \cdot 2^{k-2}} (m^{2^{k-3}} f_m(k-3) + m^{2^{k-2}}) = m^{2^k} + m^{7 \cdot 2^{k-3}} f_m(k-3)$$ Hence, $$f_m(k) = 2 m^{2^k} + m^{3 \cdot 2^{k-2}} f_m(k-2) = 3m^{2^k} + m^{7 \cdot 2^{k-3}} f_m(k-3)$$ In general, it is not hard to see that $$f_m(k) = \ell m^{2^k} + m^{(2^{\ell}-1)2^{k-\ell}} f_m(k-\ell)$$ $\ell$ can go up to $k$, to give us $$f_m(k) = km^{2^k} + m^{(2^{k}-1)} f_m(0) = km^{2^k} + m^{(2^{k}-1)} m^{2^0} = (k+1) m^{2^k}$$ This gives us $$f_m(k) = (k+1) m^{2^k} = n \left(\log_2(\log_m n) + 1\right) = \mathcal{O}(n \log_2(\log_2 n))$$ since $$n=m^{2^k} \implies \log_m(n) = 2^k \implies \log_2(\log_m(n)) = k$$

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Thanks Marvis. But how did log appear suddenly. Are there some more steps in between. – Vishnu Vivek Nov 17 '12 at 18:56
@VISHNUVIVEK I have added some more details. Hope it is clear now. – user17762 Nov 17 '12 at 19:12
I appreciate your effort Marvis. I have a question for you. Is it possible to determine the base-case for a recurrence problem if it is not given in the question. Some say that it is not possible to solve recurrence problems if the base-case is not given. In this problem, we have to assume that the base case is T(2)=2. How did you solve it without knowing it. In the answer by Amr, he has left it at the end as T(2). Thus if we substitute T(2)=2, we almost arrive at the answer. – Vishnu Vivek Nov 17 '12 at 22:47
@VISHNUVIVEK The base case is $\mathcal{O}(1)$ and hence it typically won't affect the overall order. For instance, as Amr has shown that $$T(m) = m \left(\log_2 \log_2 (m) + \dfrac{T(2)}2 \right)$$ $T(2)$ is just a constant say $T(2) = k$. Hence, the overall cost is $$T(m) = m \left(\log_2 \log_2 (m) + \dfrac{k}2 \right) = \mathcal{O}(m \log_2 \log_2 m)$$ – user17762 Nov 17 '12 at 22:50
Oh! I almost forgot the fact that it was a constant term. Great. Thanks buddy. – Vishnu Vivek Nov 17 '12 at 22:58

Let $n=2^{2^u}$, thus we get: $$T(2^{2^u})=2^{2^{u-1}}T(2^{2^{u-1}})+2^{2^{u}}$$ Now divide both sides by $2^{2^u}$ to get (Note that $2^{2^{u-1}}/2^{2^u}=2^{-2^{u-1}}$) $$2^{-2^u}T(2^{2^u})=2^{-2^{u-1}}T(2^{2^{u-1}})+1$$ $$2^{-2^u}T(2^{2^u})-2^{-2^{u-1}}T(2^{2^{u-1}})=1$$ By summing from 1 to n we get: $$2^{-2^n}T(2^{2^n})-2^{-1}T(2)=n$$ therefore: $$T(2^{2^n})=2^{2^n}(2^{-1}T(2)+n)$$

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Thanks Amr.. can u pls arrive till the answer which I've prescribed. – Vishnu Vivek Nov 17 '12 at 18:52
Actually I don't know the definitions of big O and theta (which you have). That's why I stopped here. I know they are used a lot in CS, but my field is not CS – Amr Nov 17 '12 at 18:53
The other answer has the big O notation. I think you might want to see this answer. – Amr Nov 17 '12 at 18:55
Actually your answer is quite understandable. I have a question for you. Is it possible to determine the base-case for a recurrence problem if it is not given in the question. – Vishnu Vivek Nov 17 '12 at 19:04

Yes!!, the Master Theorem can be applied. I'm going to show you

T(n) = $\sqrt{n}$ T($\sqrt{n}$) + n = $\sqrt{n}$ T($\sqrt{n}$) + O(n)

Let n = $2^k$, $\sqrt{n}$ = $2^{k/2}$, and k = $\log{n}$

T($2^k$) = $2^{k/2}$T($2^{k/2}$) + $2^k$ / dividing by $2^k$

$\frac{T(2^k)}{2^k}$ = $\frac{2^{k/2}T(2^{k/2})}{2^k}$ + 1 (we know that $\frac{2^{k/2}}{2^k}$ = $\frac{1}{2^{k/2}}$ )

$\frac{T(2^k)}{2^k}$ = $\frac{T(2^{k/2})}{2^{k/2}}$ + 1

Let y(k) = $\frac{T(2^k)}{2^k}$, then

y(k) = y($\frac{k}{2}$) + 1

Now, we apply the Master Theorem (T(n) = aT($\frac{n}{b}$) + $n^d$), where a=1, b=2, and d=0, a=$b^d$ = 1

y(k) = $k^d\log{k}$ = $\log{k}$ (because d=0)

But, we also know that T($2^k$) = $2^ky(k)$, then

T($2^k$) = $2^k\log{k}$ = T(n) = $n\log{\log{n}}$ (because n=$2^k$ and k = $\log{n}$)

Finally: T(n) = $\Theta(n\log{\log{n}})$

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Looks correct and generalizable to T(n) = sqrt(n)*T(sqrt(n)) + sqrt(n). – Eugene K Sep 24 '15 at 17:15

Solution with Detailed Explanation:

Master theorem cannot be applied here because for applying the Master theorem the number of sub-problems generated must be constant (a).

 T(n)=√n*T(√n)+n

Let, m=lg n; n=2^m


Therefore,

 T(2^m)=2^(m/2)* T(2^(m/2)) + 2^m .....(i)

Let, S(m)=T(2^m)


Therefore changing the equation (i) we get,

   S(m)=2^(m/2) * S(m/2) + 2^m .....(ii) //Level 1

S(m/2)=2^(m/4) * S(m/4) + 2^(m/2) .....(iii)

S(m/4)=2^(m/8) * S(m/8) + 2^(m/4) .....(iv)

....

....

So on and so forth.


By putting value of S(m/4) from equation (iv) in equation (iii) we get,

     S(m/2)=2^(m/4) * [2^(m/8) * S(m/8) + 2^(m/4)] + 2^(m/2)  //Replaced value is inside [ ]

=[(2^(m/4)*2^(m/8)) * S(m/8) + 2^(m/4) * 2^(m/4)] + 2^(m/2)

=[(2^(m/4)*2^(m/8)) * S(m/8) + 2^(m/2)] + 2^(m/2) .....(v) //Level 2


By putting value of equation (v) in equation (ii) we get,

    S(m)=2^(m/2) * [[(2^(m/4)*2^(m/8)) * S(m/8) + 2^(m/2)] + 2^(m/2)] + 2^m

=[(2^(m/2)*2^(m/4)*2^(m/8)) * S(m/8) + 2^(m/2) *2^(m/2)] + 2^(m/2) * 2^(m/2)] + 2^m

=[(2^(m/2)*2^(m/4)*2^(m/8)) * S(m/8) + 2^m] + 2^m] + 2^m  ......(vi) //Level 3


So, if you follow the expressions you will find in each level a factor of (2^m) is added and so (2^(m/(2^i))) is multiplied to S(m/(2^i)).

Note: You can verify it by comparing equation (ii) and equation (v) or equation (v) and equation (vi).

So it turns out,

 S(m)=[(2^(m/(2^1))*2^(m/(2^2))*2^(m/(2^3))*....*2^(m/(2^i)))* S(m/(2^i))]

+[2^m + 2^m + ..... i terms]


Note: You may directly reach to the above relation just after equation (ii) or (iii) depending on your expertise.

Now, if you follow the first part in [ ] you can make it,

  2^((m/2)[1+(1/2)+(1/4)+...+(1/(2^(i-1))])  .....(vii)


The part in [ ] in the above line is a GP series with first term a=1 and r=1/2 with i terms. As this is a monotonically increasing series, if we make it for infinite terms, still it holds the upper bound.

Therefore, the sum of the series ie. the section inside [ ] has become,

 1/(1-(1/2)) =2


So now putting the value in equation (vii), we get,

 2^((m/2)*2) =2^m


Therefore,

 S(m) <= 2^m * S(m/(2^i) + i * 2^m  //As we took the series sum for infinite terms


No if we consider S(1) as the base case and S(1)=1, then

 m/(2^i) =1; and i=lg m; ..............(viii)


put the value of i in the equation (viii) to get

 S(m) <= 2^m * S(1) + lg m * 2^m

<= 2^m + lg m * 2^m     //As, S(1)=1


 S(lg n) <= n + lg (lg n) * n

T(n) <= n + n * lg (lg n)


Therefore,

 T(n) = O(n lg(lg n))


Now, for the approximation we made for the GP series in equation (vii) is not relevant for calculation that lower bound as,

 n = O(n lg(lg n))


So, we can conclude,

 T(n) = 0(n lg(lg n)) //"Theta of n lg of lg of n"

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