Algorithms | Analysis of Algorithms | Question 3

The recurrence relation capturing the optimal time of the Tower of Hanoi problem with n discs is. (GATE CS 2012)

(A)

T(n) = 2T(n – 2) + 2

(B)

T(n) = 2T(n – 1) + n

(C)

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

(D)

T(n) = 2T(n – 1) + 1

Answer: (D)

Explanation:

Following are the steps to follow to solve Tower of Hanoi problem recursively.

Let the three pegs be A, B and C. The goal is to move n pegs from A to C.
To move n discs from peg A to peg C:
    move n-1 discs from A to B. This leaves disc n alone on peg A
    move disc n from A to C
    move n-1 discs from B to C so they sit on disc n

The recurrence function T(n) for the time complexity of the above recursive solution can be written as follows. 
T(n) = 2T(n-1) + 1

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