# 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?

(A) (n)

(B) (n log n)

(C) (n^2)

(D) (n^2log n)

**(A)** A

**(B)** B

**(C)** C

**(D)** D

**Answer:** **(A)** **Explanation:**

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)) <= 3cn/2 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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