# Data Structures and Algorithms | Set 29

Following questions have been asked in GATE 2012 exam.

**1) The recurrence relation capturing the optimal time of the Tower of Hanoi problem with n discs is**

(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)

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

**2) Consider the directed graph shown in the figure below. There are multiple shortest paths between vertices S and T. Which one will be reported by Dijstra?s shortest path algorithm? Assume that, in any iteration, the shortest path to a vertex v is updated only when a strictly shorter path to v is discovered.**

(A) SDT

(B) SBDT

(C) SACDT

(D) SACET

Answer (D)

**3) Suppose a circular queue of capacity (n – 1) elements is implemented with an array of n elements. Assume that the insertion and deletion operation are carried out using REAR and FRONT as array index variables, respectively. Initially, REAR = FRONT = 0. The conditions to detect queue full and queue empty are**

(A) Full: (REAR+1) mod n == FRONT, empty: REAR == FRONT

(B) Full: (REAR+1) mod n == FRONT, empty: (FRONT+1) mod n == REAR

(C) Full: REAR == FRONT, empty: (REAR+1) mod n == FRONT

(D) Full: (FRONT+1) mod n == REAR, empty: REAR == FRONT

Answer (A)

See this for details.

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