Question

You have to write your code in C++ (as a cpp file) and prepare a docx file that formally explains the complexity of your implemented algorithm (using recurrence relations). For details, please read the questions carefully.

Question 1:

Use a divide and conquer approach based algorithm that finds the largest item in a given array of size n. Analyze your algorithm and show the complexity by solving its recurrence relation (cpp file for the code and a docx file for the complexity analysis).

Answer 1

C++ CODE (cpp file)

#include <iostream>
#include <bits/stdc++.h>
using namespace std;

// Divide & Conquer solution to find minimum and maximum number in an array
void find_Max(int arr[], int low, int high, int& max)
{
// if array contains only one element
if (low == high) // common comparison
{
if (max < arr[low]) // comparison 1
max = arr[low];

return;
}

// if array contains only two elements
if (high - low == 1) // common comparison
{
if (arr[low] < arr[high]) // comparison 2
{
if (max < arr[high]) // comparison 3
max = arr[high];
}
else
{
if (max < arr[low]) // comparison 4
max = arr[low];
}
return;
}

// find mid element
int mid = (low + high) / 2;

// recur for left sub-array
find_Max(arr, low, mid, max);

// recur for right sub-array
find_Max(arr, mid + 1, high, max);
}

int main()
{
cout<<"Enter the number of elements\n";
int n;
cin>>n;
  
cout<<"\nEnter the elements\n";
int arr[n];
for(int i=0;i<n;i++)
cin>>arr[i];

int max = INT_MIN;
find_Max(arr, 0, n - 1, max);

cout<<"\nThe maximum element in the array is " << max;
   cout<<endl;
return 0;
}

IMAGE OF OUTPUT

COMPLEXITY ANALYSIS (Docx file)

Using master theorem to solve the complexity of the algorithm. We have,

T(n) = 2T(n/2) +1 (recurrence formula for the algorithm used)

Here, a = 2 , b = 2 , k = 0 and c = 1

So, a > bk (condition 3 holds true) - as 2 > 1

Thus,
   => T(n) ~ nlog22   
   => T(n) ~ n

Time complexity of the algorithm is O(n).