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QUESTION: 1

A binary search tree is generated by inserting in order the following integers: 50, 15, 62, 5, 20, 58, 91,3, 8, 37,60, 24. The number of nodes in the left subtree and right subtree of the root respectively is

Solution:

The binary search tree is as follows:

The number of nodes in the left subtree and right subtree of the root respectively is (7, 4).

QUESTION: 2

Which one of the following algorithm design techniques is used in finding all pairs of shortest distances in a graph?

Solution:

Bellman-Ford algorithm is used to find all pairs shortest distances in a graph and it is dynamic programming technique.

QUESTION: 3

A B-tree of order 4 is built from scratch by 10 successive insertions. What is the maximum number of node splitting operations that may take place?

Solution:

A B-tree is similar to 2-3 tree. Consider a B-tree of order 4.

A B-tree of order m contains n records and if each contains b records on the average then the tree has about [ n / b ] leaves, if we split k nodes along the path from leaves then

in given problem n = 10, b = 3, m = 4

so,

QUESTION: 4

A problem in NP is NP-complete if

Solution:

3-SAT being an NPC problem, reducing NP problem to 3-SAT would mean that NP problem is NPC

QUESTION: 5

For problems X and Y, Y is NP-complete and X reduces to Y in polynomial time. Which of the following is TRUE?

Solution:

X is reducible to NPC.

Hence X is also NPC

QUESTION: 6

Let π_{A } be a problem that belongs to the class NP. Then which one of the following is TRUE?

Solution:

it is given that π_{A} ∈ NP

∴ If π_{A} is NP-hard, and since it is given that π_{A} ∈ NP , this means that π_{A} is NP-complete

∴ choice (c) is correct.

Notice that choice (a) is incorrect since as P ∈ NP, some NP problems are actually P and hence polynomial time algorithm can exist for these.

Choice (b) is incorrect since, If π_{A} can be solved deterministicaily in polynomial time, it does not generate that P=NP, unless of-course it is additionally known that π_{A} is NP-complete.

Choice (d) is incorrect since,

All problems belonging to P or NP have to be decidable.

QUESTION: 7

The maximum number of edges in a n-nodes undirected graph without self loops is

Solution:

The maximum number of edges in a n-node undirected graph without self loop i.e., complete graph.

n-node each having degree such each edge so total number of edges =

QUESTION: 8

The average number of key comparisons required for a successful search for sequential search on n items is

Solution:

Number of comparison in worst case = n

Number of comparison in best case = 1 So, average number of comparison

QUESTION: 9

A complete binary tree with n non-leaf nodes contains

Solution:

In complete binary tree.

Number of internal nodes = No. of leaf node - 1

So total number of nodes = Internal nodes + Leaf node

QUESTION: 10

Algorithm design technique used in quicksort algorithm is

Solution:

Quick sort algorithm is based on divide an conquer approach.

Since we conquer the array by dividing it one the basis of pivot elements till the sorted array is obtained

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