Count of ways to convert given Array such that array maximum is not present in the first half

Given an array arr[] of even size N. The task is to count the number of ways of converting arr[] such that the first half of the array does not contain the maximum number. 

Examples:

Input: arr[] = {2, 2, 5, 2, 2, 2}
Output: 3
Explanation: Following are the ways where the maximum element 5 is not present in the first half of the array.
[2, 2, 2, 5, 2, 2]  when x=1 (shifted to the right by 1)
[2, 2, 2, 2, 5, 2]  when x=2 (shifted to the right by 2)
[2, 2, 2, 2, 2, 5]  when x=3 (shifted to the right by 3)
[5, 2, 2, 2, 2, 2]  when x=4 NOT A VALID CASE

Input: arr[] = {3, 3, 6, 3, 3, 6}
Output: 0
Explanation: No matter how many shifts we perform, the maximum number 6 is always present in the first array.

Naive Approach: Do right shifts in arr[] and check for each case according to the given condition. Count all the possible ways and print it.

Time Complexity: O(N * N) //since two nested loops are used the time taken by the algorithm to complete all operation is quadratic.
Auxiliary Space: O(1) since no extra array is used so the space taken by the algorithm is constant

Efficient Approach: This problem is implementation based. Follow the steps below to solve the given problem.

• Take two halves of the array arr[].
• Find and save the maximum value in the vector.
• Take a variable to store the maximum value of arr[].
• Since the maximum value can occur more than once in the array, so save the position of the maximum value in front and last.
• If the position of the maximum value is in such a way that it's less than half the size of the array, there won't be any way possible where the front half of the array wouldn't have a value this large.
• And if that is not the case, then the number of ways possible would be N/2 - (last position - first position).

Below is the implementation of the above approach.

// C++ program for above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find the number of ways to
// achieve the required array
void countWays(vector<int>& arr)
{
    int last_pos = -1;
    int front_pos = -1;
    int N = arr.size();
    int maxi = INT_MIN;
    for (int i = 0; i < N; i++) {
        maxi = max(maxi, arr[i]);
    }
    for (int i = 0; i < N; i++) {
        if (arr[i] == maxi) {
            front_pos = i;
            break;
        }
    }
    for (int i = N - 1; i >= 0; i--) {
        if (arr[i] == maxi) {
            last_pos = i;
            break;
        }
    }

    if (N / 2 >= (last_pos - front_pos))
        cout << (N / 2 - (last_pos - front_pos));
    else
        cout << "0";
}

// Driver Code
int main()
{
    vector<int> arr = { 2, 2, 5, 2, 2, 2 };

    // Function Call
    countWays(arr);
    return 0;
}

// Java program for the above approach
import java.io.*;
import java.lang.*;
import java.util.*;

class GFG {

  // Function to find the number of ways to
  // achieve the required array
  static void countWays(int arr[])
  {
    int last_pos = -1;
    int front_pos = -1;
    int N = arr.length;
    int maxi = Integer.MIN_VALUE;
    for (int i = 0; i < N; i++) {
      maxi = Math.max(maxi, arr[i]);
    }
    for (int i = 0; i < N; i++) {
      if (arr[i] == maxi) {
        front_pos = i;
        break;
      }
    }
    for (int i = N - 1; i >= 0; i--) {
      if (arr[i] == maxi) {
        last_pos = i;
        break;
      }
    }

    if (N / 2 >= (last_pos - front_pos))
      System.out.println(N / 2 - (last_pos - front_pos));
    else
      System.out.println("0");
  }

  // Driver Code
  public static void main (String[] args) {
    int arr[] = { 2, 2, 5, 2, 2, 2 };

    // Function Call
    countWays(arr);
  }
}

// This code is contributed by hrithikgarg03188.

# python3 program for above approach
INT_MIN = -2147483648

# Function to find the number of ways to
# achieve the required array
def countWays(arr):

    last_pos = -1
    front_pos = -1
    N = len(arr)
    maxi = INT_MIN
    for i in range(0, N):
        maxi = max(maxi, arr[i])

    for i in range(0, N):
        if (arr[i] == maxi):
            front_pos = i
            break

    for i in range(N - 1, -1, -1):
        if (arr[i] == maxi):
            last_pos = i
            break

    if (N // 2 >= (last_pos - front_pos)):
        print(N // 2 - (last_pos - front_pos))
    else:
        print("0")

# Driver Code
if __name__ == "__main__":

    arr = [2, 2, 5, 2, 2, 2]

    # Function Call
    countWays(arr)

    # This code is contributed by rakeshsahni

// C# program for the above approach
using System;
using System.Collections.Generic;
public class GFG {

  // Function to find the number of ways to
  // achieve the required array
  static void countWays(int []arr)
  {
    int last_pos = -1;
    int front_pos = -1;
    int N = arr.Length;
    int maxi = int.MinValue;
    for (int i = 0; i < N; i++) {
      maxi = Math.Max(maxi, arr[i]);
    }
    for (int i = 0; i < N; i++) {
      if (arr[i] == maxi) {
        front_pos = i;
        break;
      }
    }
    for (int i = N - 1; i >= 0; i--) {
      if (arr[i] == maxi) {
        last_pos = i;
        break;
      }
    }

    if (N / 2 >= (last_pos - front_pos))
      Console.WriteLine(N / 2 - (last_pos - front_pos));
    else
      Console.WriteLine("0");
  }

  // Driver Code
  public static void Main(String[] args)
  {
    int []arr = { 2, 2, 5, 2, 2, 2 };

    // Function Call
    countWays(arr);
  }
}

// This code is contributed by shikhasingrajput 

  <script>
        // JavaScript code for the above approach 

        // Function to find the number of ways to
        // achieve the required array
        function countWays(arr)
        {
            let last_pos = -1;
            let front_pos = -1;
            let N = arr.length;
            let maxi = Number.MIN_VALUE;
            for (let i = 0; i < N; i++) {
                maxi = Math.max(maxi, arr[i]);
            }
            for (let i = 0; i < N; i++) {
                if (arr[i] == maxi) {
                    front_pos = i;
                    break;
                }
            }
            for (let i = N - 1; i >= 0; i--) {
                if (arr[i] == maxi) {
                    last_pos = i;
                    break;
                }
            }

            if (Math.floor(N / 2) >= (last_pos - front_pos))
                document.write(Math.floor(N / 2) - (last_pos - front_pos));
            else
                document.write("0");
        }

        // Driver Code
        let arr = [2, 2, 5, 2, 2, 2];

        // Function Call
        countWays(arr);

       // This code is contributed by Potta Lokesh
    </script>

3

Time Complexity: O(N) since one traversal of the array is required to complete all operations hence the overall time required by the algorithm is linear
Auxiliary Space: O(1) since no extra array is used so the space taken by the algorithm is constant