Maximize Kth largest element after splitting the given Array at most C times

Given an array arr[] and two positive integers K and C, the task is to maximize the K^th maximum element obtained after splitting an array element arr[] into two parts(not necessarily an integer) C number of times. Print -1 if there doesn’t exist K^th maximum element.

Note: It is compulsory to do splitting operation until the size of the array arr[] is ≥ K.

Examples:

Input: arr[] = {5, 8}, K = 1, C = 1
Output: 8.0
Explanation: There is no need to perform any operations. The finally array will be {8, 5} Hence 8.0 is the maximum achievable value.

Input: arr[] = {5, 9}, K = 3, C = 1
Output: 4.5
Explanation: The value 9 can be splitted as 4.5 and 4.5. The final array will be {5, 4.5, 4.5} where the 3rd value is 4.5 which is maximum achievable.

Approach: The given problem can be solved by using Binary Search on the answer. Follow the steps below to solve the given problem.

• Initialize two variables, say low and high as 0 and 10⁹ respectively that represents the range where Binary Search can be performed.
• Perform the Binary Search using the following steps:
  • Find the value of mid as (low + high)*0.5.
  • Traverse the given array arr[] and store the count of elements which is at least the value of mid in the variable, say A and also find the number of operations performed in the variable, say B.
  • If the value of (A ≥ K) and (B + C ≥ K) then update the value of low as mid. Otherwise, update the value of high as mid.
• After completing the above steps, the value stored in the variable low is the resultant maximized K^th maximum element.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

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

// Function to find the K-th maximum
// element after upto C operations
double maxKth(int arr[], int N,
              int C, int K)
{
    // Check for the base case
    if (N + C < K) {
        return -1;
    }
    // Stores the count iterations of BS
    int iter = 300;

    // Create the left and right bounds
    // of binary search
    double l = 0, r = 1000000000.0;

    // Perform binary search
    while (iter--) {

        // Find the value of mid
        double mid = (l + r) * 0.5;
        double a = 0;
        double b = 0;

        // Traverse the array
        for (int i = 0; i < N; i++) {
            a += int((double)arr[i] / mid);
            if ((double)arr[i] >= mid) {
                b++;
            }
        }

        // Update the ranges
        if (a >= K && b + C >= K) {
            l = mid;
        }
        else {
            r = mid;
        }
    }

    // Return the maximum value obtained
    return l;
}

// Driver Code
int main()
{
    int arr[] = { 5, 8 };
    int K = 1, C = 1;
    int N = sizeof(arr) / sizeof(arr[0]);

    cout << maxKth(arr, N, C, K);

    return 0;
}

Java

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

class GFG
{

    // Function to find the K-th maximum
    // element after upto C operations
    static double maxKth(int arr[], int N, int C, int K)
    {

        // Check for the base case
        if (N + C < K) {
            return -1;
        }

        // Stores the count iterations of BS
        int iter = 300;

        // Create the left and right bounds
        // of binary search
        double l = 0, r = 1000000000.0;

        // Perform binary search
        while (iter-- > 0) {

            // Find the value of mid
            double mid = (l + r) * 0.5;
            double a = 0;
            double b = 0;

            // Traverse the array
            for (int i = 0; i < N; i++) {
                a += (int)((double)arr[i] / mid);
                if ((double)arr[i] >= mid) {
                    b++;
                }
            }

            // Update the ranges
            if (a >= K && b + C >= K) {
                l = mid;
            }
            else {
                r = mid;
            }
        }

        // Return the maximum value obtained
        return l;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int arr[] = { 5, 8 };
        int K = 1, C = 1;
        int N = arr.length;

        System.out.println(maxKth(arr, N, C, K));
    }
}

// This code is contributed by Dharanendra L V.

Python3

# Python Program to implement
# the above approach

# Function to find the K-th maximum
# element after upto C operations
def maxKth(arr, N, C, K):

    # Check for the base case
    if (N + C < K):
        return -1

    # Stores the count iterations of BS
    iter = 300

    # Create the left and right bounds
    # of binary search
    l = 0
    r = 1000000000.0

    # Perform binary search
    while (iter):
        iter = iter - 1
        # Find the value of mid
        mid = (l + r) * 0.5
        a = 0
        b = 0

        # Traverse the array
        for i in range(N) :
            a += arr[i] // mid
            if (arr[i] >= mid) :
                b += 1



        # Update the ranges
        if (a >= K and b + C >= K) :
                l = mid

        else :
                r = mid


    # Return the maximum value obtained
    return int(l)


# Driver Code
arr = [5, 8]
K = 1
C = 1
N = len(arr)

print(maxKth(arr, N, C, K))

# This code is contributed by Saurabh Jaiswal

C#

// C# program for the above approach
using System;

class GFG
{

    // Function to find the K-th maximum
    // element after upto C operations
    static double maxKth(int []arr, int N, int C, int K)
    {

        // Check for the base case
        if (N + C < K) {
            return -1;
        }

        // Stores the count iterations of BS
        int iter = 300;

        // Create the left and right bounds
        // of binary search
        double l = 0, r = 1000000000.0;

        // Perform binary search
        while (iter-- > 0) {

            // Find the value of mid
            double mid = (l + r) * 0.5;
            double a = 0;
            double b = 0;

            // Traverse the array
            for (int i = 0; i < N; i++) {
                a += (int)((double)arr[i] / mid);
                if ((double)arr[i] >= mid) {
                    b++;
                }
            }

            // Update the ranges
            if (a >= K && b + C >= K) {
                l = mid;
            }
            else {
                r = mid;
            }
        }

        // Return the maximum value obtained
        return l;
    }

    // Driver Code
    public static void Main(String[] args)
    {
        int []arr = { 5, 8 };
        int K = 1, C = 1;
        int N = arr.Length;

        Console.Write(maxKth(arr, N, C, K));
    }
}

// This code is contributed by shivanisinghss2110

Javascript

<script>
       // JavaScript Program to implement
       // the above approach

       // Function to find the K-th maximum
       // element after upto C operations
       function maxKth(arr, N,
           C, K)
       {

           // Check for the base case
           if (N + C < K) {
               return -1;
           }
           // Stores the count iterations of BS
           let iter = 300;

           // Create the left and right bounds
           // of binary search
           let l = 0, r = 1000000000.0;

           // Perform binary search
           while (iter--) {

               // Find the value of mid
               let mid = (l + r) * 0.5;
               let a = 0;
               let b = 0;

               // Traverse the array
               for (let i = 0; i < N; i++) {
                   a += Math.floor(arr[i] / mid);
                   if (arr[i] >= mid) {
                       b++;
                   }
               }

               // Update the ranges
               if (a >= K && b + C >= K) {
                   l = mid;
               }
               else {
                   r = mid;
               }
           }

           // Return the maximum value obtained
           return l;
       }

       // Driver Code
       let arr = [5, 8];
       let K = 1, C = 1;
       let N = arr.length;

       document.write(maxKth(arr, N, C, K));

    // This code is contributed by Potta Lokesh

   </script>

Output:

8

Time Complexity: O(N*log M)
Auxiliary Space: O(1)