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Data Structures and Algorithms | Set 29

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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 Dijkstra?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 

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. 

Please see GATE Corner for all previous year paper/solutions/explanations, syllabus, important dates, notes, etc. 

Please write comments if you find any of the answers/explanations incorrect, or you want to share more information about the topics discussed above.

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Last Updated : 13 Dec, 2022
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