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how to solve T(n)=4T(√n)+3^5n with master theorem
mdboi
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Oct 29, 2022
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how do i apply master theorem to this?
algorithms
recurrence-relation
master-theorem
asymptotic-notation
mdboi
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Kabir5454
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Oct 29, 2022
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Let ,
$n=2^{k}$
So,
$T(2^{k})=4T(2^{\frac{k}{2}})+k^{5}$
Now , Let $T(2^{k})=S(k)$
So,
$S(k)=4S(\frac{k}{2})+k^{5}$
now using master’s theorem , we get ,
$S(k)=\Theta (k^{5})$
$T(2^{k})=\Theta (k^{5})$
$T(n)=\Theta (logn)^{5}$
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let n = 2$^{m}$
T(2$^{m}$) = 4T(2$^{m/2}$)+(log2$^{m}$)$^{5}$
T(2$^{m}$) = 4T(2$^{m/2}$)+m$^{5}$(log2)$^{5}$
let S(k) = T(2$^{m}$)
Therefore S(k) = 4S(m/2) + m$^{5}$
Using the extended master theorem,
a=4,b=2,k=5,p=0
a < b$^{k}$ and p>=0
Therefore S(k) = $\Theta$(n$^{k}$log$^{p}$n) => $\Theta$(m$^{5}$)
but m = logn
Therefore T(n) = $\Theta$(logn)$^{5}$
sreechand
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