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 time complexity of the above recursive solution can be written as following.

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



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