如何计算此函数的增长率：T(n) = 2T(n^(1/2)) + 2(n^(1/2))

Question

I need to calculate the growth rate of this function for my homework:

T(n) = 2T( n^(1/2) ) + 2( n^(1/2) )

To put it another way:

T(n) = 2T( sqrt(n) ) + 2( sqrt(n) )

Changing variables might help (something like n = 2^m )

The answer I found is log(n)*log(log(n)) , but I know that is incorrect.

Answer 1

n = 2^m is indeed the correct variable substitution to use. Define a function S(m) :

S(m) = T(n) = T(2^m)

T(sqrt(n)) = T(2^[m/2]) = S(m/2)

S(m) = 2S(m/2) + 2^[m/2+1]

Expansion:

S(m) =  4*S(m/4)  + 2*2^[m/4+1]  + 2^[m/2+1]
     =  8*S(m/8)  + 4*2^[m/8+1]  + 2^[m/4+2] + 2^[m/2+1]
     = 16*S(m/16) + 8*2^[m/16+1] + 2^[m/8+3] + 2^[m/4+2] + 2^[m/2+1]
     = ...

2^[m/2] will dominate all of the other terms, so:

S(m) = O(2^[m/2])

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*                   *
* T(n) = O(sqrt(n)) *
*                   *
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The above can also be derived using the Master Theorem (Case 3).