# Data Structures and Algorithms | Set 9

1 In a heap with n elements with the smallest element at the root, the 7th smallest element can be found in time (GATE CS 2003)
a) Θ(n log n)
b) Θ(n)
c) Θ(log n)
d) Θ(1)

The 7th smallest element must be in first 7 levels. Total number of nodes in any Binary Heap in first 7 levels is at most 1 + 2 + 4 + 8 + 16 + 32 + 64 which is a constant. Therefore we can always find 7th smallest element in Θ(1) time.

2. Suppose the numbers 7, 5, 1, 8, 3, 6, 0, 9, 4, 2 are inserted in that order into an initially empty binary search tree. The binary search tree uses the usual ordering on natural numbers. What is the in-order traversal sequence of the resultant tree? (GATE CS 2003)

a) 7 5 1 0 3 2 4 6 8 9
b) 0 2 4 3 1 6 5 9 8 7
c) 0 1 2 3 4 5 6 7 8 9
d) 9 8 6 4 2 3 0 1 5 7

In-order traversal of a BST gives elements in increasing order. So answer c is correct without any doubt.

3. Let S be a stack of size n >= 1. Starting with the empty stack, suppose we push the first n natural numbers in sequence, and then perform n pop operations. Assume that Push and Pop operation take X seconds each, and Y seconds elapse between the end of one such stack operation and the start of the next operation. For m >= 1, define the stack-life of m as the time elapsed from the end of Push(m) to the start of the pop operation that removes m from S. The average stack-life of an element of this stack is (GATE CS 2003)

a) n(X+ Y)
b) 3Y + 2X
c) n(X + Y)-X
d) Y + 2X