What is time complexity of given C program?

Asked Sep 24 '15 at 12:20

Active Sep 24 '15 at 19:06

Viewed 48 times

int R(int n)
{
    if(n<=1)
        return 1;
    else 
    {
        int sum=0;
        for(int i=1;i<=n;i=i*2)
        for(j=1;j<=i;j=j*2){
            sum=sum+i;
            sum=sum*R(n/2);
        }
    }
    return sum;
}

I try to explain

T(1) = 1

T(2) = 2

T(4) = 6

T(8) = 36

T(16) = 360

T(32) = 5400

so, the growth rate is higher than that of $n^2$.

edited Sep 24 '15 at 19:06

Raphael

72,336
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179
389

asked Sep 24 '15 at 12:20

Mithlesh Upadhyay

Have you checked your numbers with a computer program? – hengxin Sep 24 '15 at 13:51
@hengxin,sir, may be not correct . I am asking : how to find time complexity of program , and I am getting more than O($n^2$) – Mithlesh Upadhyay Sep 24 '15 at 13:54
Your numbers are $\frac{4 (n!)^2}{n 2^n}$ for $n\ge 2$, where n corresponds to $T(2^{n-1})$. – Alex Sep 24 '15 at 13:55
To clarify, $T(m)=\frac{2((1+\log_2 m)!)^2}{(1+\log_2 m)m}$ for $m=2,4,8,...$ – Alex Sep 24 '15 at 14:04
I think the recurrence is $T(m) = (1 + 2 + \ldots + (\lg m + 1)) T(m/2)$. – hengxin Sep 24 '15 at 14:10
@hengxin, sir , yes I also found here , but what is the order ? – Mithlesh Upadhyay Sep 24 '15 at 14:13
You can try to expand $T(m) = (\lg m (\lg m + 2)) / 2 \cdot T(m/2)$. The result may look like the one given by @Alex (I did not do it really). – hengxin Sep 24 '15 at 14:21
A finite sample can never tell you the asymptotics (which you see to be after). – Raphael Sep 24 '15 at 19:07
1

The recurrence is something like $T(m) = \Theta(\log^2 m T(m/2))$, assuming $T(1) > 0$. If $m = 2^k$ and the hidden constant is $C$ (in our case, $C=1/2$) then this gives (up to constants) $T(m) = C^k k^2 (k-1)^2 \cdots 1^2 = m^{\log_2 C} (k!)^2$. Using Stirling's approximation we find that $T(m) = (1/m) \Theta(k(k/e)^{2k}) = \exp \Theta(\log m\log\log m)$. – Yuval Filmus Sep 24 '15 at 19:49
expΘ(logmloglogm)=Θ(exp(logmloglogm)) is rt ? – Mithlesh Upadhyay Sep 25 '15 at 05:23
please post an answer . – Mithlesh Upadhyay Oct 04 '15 at 12:48

What is time complexity of given C program?

0 Answers0