Reduce given Array by replacing adjacent elements with their difference

Given an array arr[] consisting of N elements(such that N = 2^k for some k ≥ 0), the task is to reduce the array and find the last remaining element after merging all elements into one. The array is reduced by performing the following operation:

• Merge the adjacent elements i.e merge elements at indices 0 and 1 into one, 2 and 3 into one and so on. 
• Upon merging the newly formed element will become the absolute difference between the two elements merged.

Examples:

Input: N = 4, arr[] = [1, 2, 3, 4]
Output: 0
Explanation: First operation:
On merging 1st and 2nd elements we will have a element with value1. 
On merging 3rd and 4th elements, we will have a element with value1. 
Therefore, we are left with two elements where each of them having cost 1.
Second operation:
On merging the 1st and 2nd elements we will get a new element with value 0.
This is because both elements had the same value of 1.

Input: N = 1, arr[] = [20]
Output: 20
Explanation: We can’t perform any operation because performing an operation requires at least 2 elements. Hence, 20 is cost of the last remaining element

Approach: This problem can be solved using the Divide and Conquer approach.

• Create a recursive function.
  • The base condition for recursion will be if the size of the array is 1 then the answer will be the only array element in it.
  • Return the absolute difference between the first half of the array and the second half of the array by calling the recursive function for both halves.
  • Merge both halves and get the answer.

Below is the implementation of the above approach:

C++

// C++ code to implement the approach

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

// Function to get the last remaining element
// by using divide and conquer
int f(int l, int e, int a[])
{
    // Base condition
    if (l == e)
        return a[l];
    return abs(f(l, l + (e - l) / 2, a)
               - f(l + (e - l) / 2 + 1, e, a));
}

int find(int n, int a[])
{
    return f(0, n - 1, a);
}

// Driver code
int main()
{
    int arr[] = { 1, 2, 3, 4 };
    int N = sizeof(arr) / sizeof(arr[0]);

    // Function Call
    cout << find(N, arr);
    return 0;
}

Java

// Java code to implement the approach
import java.io.*;
class GFG {

  // Function to get the last remaining element
  // by using divide and conquer
  static int f(int l, int e, int[] a)
  {
    // Base condition
    if (l == e) {
      return a[l];
    }
    return Math.abs(f(l, l + (e - l) / 2, a)
                    - f(l + (e - l) / 2 + 1, e, a));
  }

  static int find(int n, int[] a)
  {
    return f(0, n - 1, a);
  }

  public static void main(String[] args)
  {
    int[] arr = { 1, 2, 3, 4 };
    int N = arr.length;

    // Function call
    System.out.print(find(N, arr));
  }
}

// This code is contributed by lokeshmvs21.

Python3

# Python code to implement the approach

# Function to get the last remaining element
# by using divide and conquer
import sys

sys.setrecursionlimit(1500)


def f(l, e, a):
    # Base condition
    if (l == e):
        return a[l]
    return abs(f(l, l + (e - l) // 2, a) - f(l + (e - l) // 2 + 1, e, a))


def find(n, a):
    return f(0, n - 1, a)


# Driver code
if __name__ == "__main__":
    arr = [1, 2, 3, 4]
    N = len(arr)
    # Function Call
    print(find(N, arr))

# This code is contributed by Rohit Pradhan

C#

// C# implementation
using System;
public class GFG{

  // Function to get the last remaining element
  // by using divide and conquer
  public static int f(int l, int e, int []a)
  {
    
    // Base condition
    if (l == e)
      return a[l];
    return Math.Abs(f(l, l + (e - l) / 2, a)
                    - f(l + (e - l) / 2 + 1, e, a));
  }

  public static int find(int n, int []a)
  {
    return f(0, n - 1, a);
  }

  static public void Main (){
    int []arr = { 1, 2, 3, 4 };
    int N = arr.Length;

    // Function Call
    Console.WriteLine(find(N, arr));
  }
}

// This code is contributed by ksam24000

Javascript

// Javascript code to implement the approach

// Function to get the last remaining element
// by using divide and conquer
function f(l, e, a) {
    // Base condition
    if (l == e) {
        return a[l];
    }
    return Math.abs(f(l, l + Math.floor((e - l) / 2), a)
        - f(l + Math.floor((e - l) / 2) + 1, e, a));
}

function find(n, a) {
    return f(0, n - 1, a);
}

let arr = [1, 2, 3, 4];
let N = arr.length;

// Function call
console.log(find(N, arr));

// This code is contributed by Saurabh Jaiswal

Output

0

Time Complexity: O(2^log₂N)
Auxiliary Space: O(N)

My Personal Notes arrow_drop_up

Save