Algorithms | Analysis of Algorithms (Recurrences) | Question 6
The running time of an algorithm is represented by the following recurrence relation:
if n <= 3 then T(n) = n
else T(n) = T(n/3) + cn
Which one of the following represents the time complexity of the algorithm?
(B) (n log n)
(D) (n^2log n)
T(n) = cn + T(n/3)
= cn + cn/3 + T(n/9)
= cn + cn/3 + cn/9 + T(n/27)
Taking the sum of infinite GP series. The value of T(n) will
be less than this sum.
T(n) <= cn(1/(1-1/3))
or we can say
cn <= T(n) <= 3cn/2
Therefore T(n) = (n)
This can also be solved using Master Theorem for solving recurrences. The given expression lies in Case 3 of the theorem.
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