Data Structures and Algorithms | Set 30 - GeeksforGeeks

27 Mar, 2017
Following questions have been asked in GATE CS 2013 exam.
1) Which of the following statements is/are TRUE for an undirected graph?P: Number of odd degree vertices is evenQ: Sum of degrees of all vertices is even
A) P OnlyB) Q OnlyC) Both P and QD) Neither P nor Q
Answer (C)Q is true: Since the graph is undirected, every edge increases the sum of degrees by 2.P is true: If we consider sum of degrees and subtract all even degrees, we get an even number (because Q is true). So total number of odd degree vertices must be even.
2) Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is 1/2. What is the expected number of unordered cycles of length three?(A) 1/8(B) 1(C) 7(D) 8
Answer (C)A cycle of length 3 can be formed with 3 vertices. There can be total 8C3 ways to pick 3 vertices from 8. The probability that there is an edge between two vertices is 1/2. So expected number of unordered cycles of length 3 = (8C3)*(1/2)^3 = 7
3) What is the time complexity of Bellman-Ford single-source shortest path algorithm on a complete graph of n vertices?(A) Θ(n2)(B) Θ(n2 Logn)(C) Θ(n3)(D) Θ(n3 Logn)
Answer (C).Time complexity of Bellman-Ford algorithm is Θ(VE) where V is number of vertices and E is number edges (See this). If the graph is complete, the value of E becomes Θ(V2). So overall time complexity becomes Θ(V3)
4) Which of the following statements are TRUE?(1) The problem of determining whether there exists a cycle in an undirected graph is in P.(2) The problem of determining whether there exists a cycle in an undirected graph is in NP.(3) If a problem A is NP-Complete, there exists a non-deterministic polynomial time algorithm to solve A.
(A) 1,2 and 3(B) 1 and 2 only(C) 2 and 3 only(D) 1 and 3 only
Answer (A)1 is true because cycle detection can be done in polynomial time using DFS (See this).2 is true because P is a subset of NP.3 is true because NP complete is also a subset of NP and NP means Non-deterministic Polynomial time solution exists. (See this)
5) Which one of the following is the tightest upper bound that represents the time complexity of inserting an object into a binary search tree of n nodes?(A) O(1)(B) O(log n)(C) O(n)(D) O(n log n)
Answer (C)The worst case occurs for a skewed tree. In a skewed tree, when a new node is inserted as a child of bottommost node, the time for insertion requires traversal of all node. For example, consider the following tree and the case when something smaller than 70 is inserted.
             100
            /
           90
          /
        80
       /
     70   

6) Which one of the following is the tightest upper bound that represents the number of swaps required to sort n numbers using selection sort?(A) O(log n)(B) O(n)(C) O(n log n)(D) O(n^2)
Answer (B)Selection sort requires only O(n) swaps. See this for details.
7) Consider the following operation along with Enqueue and Dequeue operations onqueues, where k is a global parameter
MultiDequeue(Q){
   m = k
   while (Q is not empty and m  > 0) {
      Dequeue(Q)
      m = m - 1
   }
}
What is the worst case time complexity of a sequence of n MultiDequeue() operations on an initially empty queue?(A) Θ(n)(B) Θ(n + k)(C) Θ(nk)(D) Θ(n2)
Answer (A)Since the queue is empty initially, the condition of while loop never becomes true. So the time complexity is Θ(n)
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