CS 601_Mid-Term 2.docx - 1. Suppose you are given an array...

1. Suppose you are given an array a [1, …, n] of distinct sorted (in
decreasing order) integers that has been circularly shifted x
positions to the right. For example, [10, 5, 3, 99, 55, 47, 38, 25, 12]
is a sorted array that has been circularly shifted x = 3 positions,
while [5, 3, 99, 55, 47, 38, 25, 12, 10] has been shifted x = 2
positions.
1) Write a ? (n) algorithm to find the largest element in the array.
Justify why your algorithm is ? (n). Provide enough details.
2) Can you improve your algorithm in part 1 to something more
efficient? If yes, write the algorithm and analyze that in detail.
If not, argue why ? (n) is the most efficient algorithm you could
design for this question.
Implement your most-efficient algorithm using ONLY C++/Java. Have
your complete program files submitted together with your solution
file. You need to read the array values from an input file called
input.txt. Here is the required format:
9
10
5
3
99
55
47
38
25
12
where 9 is the size of the array. The rest of numbers are the array
values, i.e., a = [10, 5, 3, 99, 55, 47, 38, 25, 12].

Let I, j = 0
For (int i; i<n; i++): n being the number of elements in array a
{
If a[i] > a[j]
{
J = I
}
}
Return a[j]
 Time Complexity for above Algorithm:
 All the elements are traversed and compared linearly in this
algorithm.
 This gives us the linear search algorithm with time complexity of
O (n).
 The for loop would execute n+1 number of times.
 The condition i++ would be executed n number of times.
 If condition would be checked n number of times.
 Rest all the operations would give us constant time execution O
(1).
 Summing up n + 1 + n + n = 3n + 1. Ignoring the constants in
the result would yield us 3n.
 This would make the complexity of C*n = 3 * n = O (n)
 Hence time complexity for the above algorithm would be
O (n)
.
2. Algorithm to find largest element in a sorted list with better approach:
1) A divide and conquer solution can be used to answer this
question in log n time.
2) It would help us to split the array into two parts and only one
part of the array would have to be traversed to find the answer.