Maximize sum of atmost K elements in array by taking only corner elements | Set 2

Given an array arr[] and an integer K, the task is to find and maximize the sum of at most K elements in the Array by taking only corner elements.

A corner element is an element from the start of the array or from the end of the array.

Examples:

Input: N = 8, arr[] = {6, -1, 14, -15, 2, 1, 2, -5}, K = 4 
Output: 19 
Explanation:
Here the optimal choice is to pick three cards from the beginning. After that if we want to pick the next card, our points will decrease. So maximum points is arr[0] + arr[1] + arr[2] = 19.

Input : N = 5, arr[] = {-2, -1, -6, -3, 1}, K = 2 
Output : 1 
Here optimal choice is to pick last card. So maximum possible points is arr[4] = 1. Any further selection will reduce the value. 

Naive Approach:
To solve the problem mentioned above we will use Recursion. As we can only take a start or end index value hence initialize two variables and take at most K steps and return the maximum sum among all the possible combinations. Update the maximum sum only if it is greater than the previous sum otherwise skip to the next possible combination. The recursive approach has exponential complexity due to its overlapping subproblem and optimal substructure property.

Below is the implementation of the above approach: 

C++

// C++ implementation to Maximize sum of atmost
// K elements in Array by taking only corner elements
#include <bits/stdc++.h>
using namespace std;

// Function to return maximum points
int maxPointCount(int arr[], int K, int start, int end,
                  int points, int max_points)
{
    if (K == 0) {
        return max_points;
    }
    // Pick the start index
    int points_start = points + arr[start];

    // Update maximum points if necessary
    max_points = max(max_points, points_start);

    // Pick the end index
    int points_end = points + arr[end];

    // Update maximum points if necessary
    max_points = max(max_points, points_end);

    // Recursive call to get max value
    return max(maxPointCount(arr, K - 1, start + 1, end,
                                points_start, max_points),
               maxPointCount(arr, K - 1, start, end - 1,
                                points_end, max_points));
}

// Driver code
int main()
{
    int arr[] = { -2, -1, -6, -3, 1 };
    int N = sizeof(arr) / sizeof(arr[0]);

    int K = 2;

    int points = 0;

    int max_points = 0;

    // beginning index
    int start = 0;

    // end index
    int end = N - 1;

    cout << maxPointCount(arr, K, start,
                end, points, max_points);

    return 0;
}

Java

// Java implementation to Maximize
// sum of atmost K elements in Array
// by taking only corner elements
import java.util.*;

class GFG{

// Function to return maximum points
static int maxPointCount(int arr[], int K,
                         int start, int end,
                         int points, int max_points)
{
    if (K == 0)
    {
        return max_points;
    }

    // Pick the start index
    int points_start = points + arr[start];

    // Update maximum points if necessary
    max_points = Math.max(max_points, points_start);

    // Pick the end index
    int points_end = points + arr[end];

    // Update maximum points if necessary
    max_points = Math.max(max_points, points_end);

    // Recursive call to get max value
    return Math.max(maxPointCount(arr, K - 1,
                                  start + 1, end,
                                  points_start, max_points),
                    maxPointCount(arr, K - 1,
                                  start, end - 1,
                                  points_end, max_points));
}

// Driver code
public static void main(String[] args)
{
    int arr[] = { -2, -1, -6, -3, 1 };
    int N = arr.length;
    int K = 2;
    int points = 0;
    int max_points = 0;

    // Beginning index
    int start = 0;

    // End index
    int end = N - 1;

    System.out.print(maxPointCount(arr, K, start,
                                   end, points,
                                   max_points));
}
}

// This code is contributed by Princi Singh

Python3

# Python3 implementation to maximize sum
# of atmost K elements in array by taking
# only corner elements

# Function to return maximum points
def maxPointCount(arr, K, start, end,
                  points, max_points):

    if (K == 0):
        return max_points

    # Pick the start index
    points_start = points + arr[start]

    # Update maximum points if necessary
    max_points = max(max_points, points_start)

    # Pick the end index
    points_end = points + arr[end]

    # Update maximum points if necessary
    max_points = max(max_points, points_end)

    # Recursive call to get max value
    return max(maxPointCount(arr, K - 1, start + 1, end,
                             points_start, max_points),
               maxPointCount(arr, K - 1, start, end - 1,
                             points_end, max_points))

# Driver code
if __name__ == "__main__":

    arr = [ -2, -1, -6, -3, 1 ]
    N = len(arr)

    K = 2
    points = 0
    max_points = 0

    # Beginning index
    start = 0

    # end index
    end = N - 1

    print(maxPointCount(arr, K, start,
                        end, points,
                        max_points))

# This code is contributed by chitranayal

C#

// C# implementation to Maximize
// sum of atmost K elements in Array
// by taking only corner elements
using System;

class GFG{

// Function to return maximum points
static int maxPointCount(int []arr, int K,
                         int start, int end,
                         int points, int max_points)
{
    if (K == 0)
    {
        return max_points;
    }

    // Pick the start index
    int points_start = points + arr[start];

    // Update maximum points if necessary
    max_points = Math.Max(max_points, points_start);

    // Pick the end index
    int points_end = points + arr[end];

    // Update maximum points if necessary
    max_points = Math.Max(max_points, points_end);

    // Recursive call to get max value
    return Math.Max(maxPointCount(arr, K - 1,
                                  start + 1, end,
                                  points_start, max_points),
                    maxPointCount(arr, K - 1,
                                  start, end - 1,
                                  points_end, max_points));
}

// Driver code
public static void Main(String[] args)
{
    int []arr = { -2, -1, -6, -3, 1 };
    int N = arr.Length;
    int K = 2;
    int points = 0;
    int max_points = 0;

    // Beginning index
    int start = 0;

    // End index
    int end = N - 1;

    Console.Write(maxPointCount(arr, K, start,
                                end, points,
                                max_points));
}
}

// This code is contributed by sapnasingh4991

Javascript

<script>
// Javascript implementation to Maximize
// sum of atmost K elements in Array
// by taking only corner elements

// Function to return maximum points
function maxPointCount(arr,K,start,end,points,max_points)
{
    if (K == 0)
    {
        return max_points;
    }

    // Pick the start index
    let points_start = points + arr[start];

    // Update maximum points if necessary
    max_points = Math.max(max_points, points_start);

    // Pick the end index
    let points_end = points + arr[end];

    // Update maximum points if necessary
    max_points = Math.max(max_points, points_end);

    // Recursive call to get max value
    return Math.max(maxPointCount(arr, K - 1,
                                  start + 1, end,
                                  points_start, max_points),
                    maxPointCount(arr, K - 1,
                                  start, end - 1,
                                  points_end, max_points));
}

// Driver code
let arr=[-2, -1, -6, -3, 1];
let N = arr.length;
    let K = 2;
    let points = 0;
    let max_points = 0;

    // Beginning index
    let start = 0;

    // End index
    let end = N - 1;

    document.write(maxPointCount(arr, K, start,
                                   end, points,
                                   max_points));

// This code is contributed by avanitrachhadiya2155
</script>

Output:

1

Efficient Approach:
To optimize the above solution we will implement the sliding window concept. 

• Initially, the window size is 0 as we don’t pick any element from the array. We take two-variable curr_points and max_points to represents current points and maximum points.
• Consider K elements one by one from the beginning. So in each step we calculate current points and update maximum points if necessary and after including K elements from the array our sliding window size becomes K, which is the maximum possible.
• After that in each step, we pick elements from the end and remove the rightmost element from the previously selected window with first K elements. Update curr_points and max_points. In the end, the window contains K cards from the end of the array.
• Finally, in each step remove the leftmost card from the previously selected window with K elements from the end. Update the values for curr_points and max_points. In the end, the window size will be 0 again.

Let us look at this example to understand it better, arr[] = {-2, -1, -6, -3, 1}, K = 2

Below is the implementation of the above approach: 

C++

// C++ implementation to Maximize sum
// of atmost K elements in Array by taking
// only corner elements
#include <bits/stdc++.h>
using namespace std;

// Function to return maximum points
int maxPointCount(int arr[], int K, int size)
{
    // Initialization of current points
    // and max points so far
    int curr_points = 0;
    int max_points = 0;

    // Add elements from the beginning
    for (int i = 0; i < K; i++) {
        curr_points += arr[i];
        max_points = max(curr_points, max_points);
    }

    // Points to the end of array element
    int j = size - 1;

    // Add K elements from end of array
    for (int i = K - 1; i >= 0; i--) {
        curr_points = curr_points + arr[j] - arr[i];
        max_points = max(curr_points, max_points);

        // Decrement the value for j
        j--;
    }

    j = size - K;

    for (; j < size; j++) {
        curr_points = curr_points - arr[j];
        max_points = max(curr_points, max_points);
    }

    // Return the final result
    return max_points;
}

// Driver code
int main()
{
    int arr[] = { -2, -1, -6, -3, 1 };

    int N = sizeof(arr) / sizeof(arr[0]);

    int K = 2;

    cout << maxPointCount(arr, K, N);

    return 0;
}

Java

// Java implementation to maximize
// sum of atmost K elements in Array
// by taking only corner elements
import java.util.Scanner;
import java.util.Arrays;

class GFG{

// Function to return maximum points
public static int maxPointCount(int arr[],
                                int K,
                                int size)
{

    // Initialization of current points
    // and max points so far
    int curr_points = 0;
    int max_points = 0;

    // Add elements from the beginning
    for(int i = 0; i < K; i++)
    {
        curr_points += arr[i];
        max_points = Math.max(curr_points,
                              max_points);
    }

    // Points to the end of array element
    int j = size - 1;

    // Add K elements from end of array
    for(int i = K - 1; i >= 0; i--)
    {
        curr_points = curr_points +
                      arr[j] - arr[i];
        max_points = Math.max(curr_points,
                              max_points);

        // Decrement the value for j
        j--;
    }

    j = size - K;
    for(; j < size; j++)
    {
        curr_points = curr_points - arr[j];
        max_points = Math.max(curr_points,
                              max_points);
    }

    // Return the final result
    return max_points;
}

// Driver code
public static void main(String args[])
{
    int []arr = { -2, -1, -6, -3, 1 };
    int N = arr.length;
    int K = 2;

    System.out.print(maxPointCount(arr, K, N));
}
}

// This code is contributed by SoumikMondal

Python3

# Python3 implementation to
# Maximize sum of atmost K
# elements in Array by taking
# only corner elements

# Function to return maximum
# points
def maxPointCount(arr, K, size):

    # Initialization of current
    # points and max points so far
    curr_points = 0;
    max_points = 0;

    # Add elements from
    # the beginning
    for i in range(K):

        curr_points += arr[i];
        max_points = max(curr_points,
                         max_points)

    # Points to the end
    # of array element
    j = size - 1;

    # Add K elements from
    # end of array
    for i in range(K - 1, -1, -1):

        curr_points = (curr_points +
                       arr[j] - arr[i]);
        max_points = max(curr_points,
                         max_points);

        # Decrement the
        # value for j
        j -= 1;

    for j in range(size - K, size):
        curr_points = (curr_points -
                       arr[j]);
        max_points = max(curr_points,
                         max_points);

    # Return the final result
    return max_points;

# Driver code
if __name__ == "__main__":

    arr = [-2, -1, -6, -3, 1]
    N = len(arr)
    K = 2;
    print(maxPointCount(arr,K,N))

# This code is contributed by rutvik_56

C#

// C# implementation to maximize
// sum of atmost K elements in Array
// by taking only corner elements
using System;

class GFG{

// Function to return maximum points
static int maxPointCount(int[] arr, int K,
                         int size)
{

    // Initialization of current points
    // and max points so far
    int curr_points = 0;
    int max_points = 0;

    // Add elements from the beginning
    for(int i = 0; i < K; i++)
    {
        curr_points += arr[i];
        max_points = Math.Max(curr_points,
                               max_points);
    }

    // Points to the end of array element
    int j = size - 1;

    // Add K elements from end of array
    for(int i = K - 1; i >= 0; i--)
    {
        curr_points = curr_points + arr[j] -
                                    arr[i];
        max_points = Math.Max(curr_points,
                               max_points);

        // Decrement the value for j
        j--;
    }

    j = size - K;
    for(; j < size; j++)
    {
        curr_points = curr_points - arr[j];
        max_points = Math.Max(curr_points,
                               max_points);
    }

    // Return the final result
    return max_points;
}

// Driver code
static void Main()
{
    int[] arr = { -2, -1, -6, -3, 1 };
    int N = arr.Length;
    int K = 2;

    Console.WriteLine(maxPointCount(arr, K, N));
}
}

// This code is contributed by divyeshrabadiya07

Javascript

<script>

// JavaScript implementation to maximize
// sum of atmost K elements in Array
// by taking only corner elements

// Function to return maximum points
function maxPointCount(arr,K,size)
{
    // Initialization of current points
    // and max points so far
    let curr_points = 0;
    let max_points = 0;

    // Add elements from the beginning
    for(let i = 0; i < K; i++)
    {
        curr_points += arr[i];
        max_points = Math.max(curr_points,
                              max_points);
    }

    // Points to the end of array element
    let j = size - 1;

    // Add K elements from end of array
    for(let i = K - 1; i >= 0; i--)
    {
        curr_points = curr_points +
                      arr[j] - arr[i];
        max_points = Math.max(curr_points,
                              max_points);

        // Decrement the value for j
        j--;
    }

    j = size - K;
    for(; j < size; j++)
    {
        curr_points = curr_points - arr[j];
        max_points = Math.max(curr_points,
                              max_points);
    }

    // Return the final result
    return max_points;
}

// Driver code

let arr=[ -2, -1, -6, -3, 1 ];
let N = arr.length;
let K = 2;
document.write(maxPointCount(arr, K, N));


// This code is contributed by rag2127

</script>

Output:

1

Time Complexity: O(n)
Auxiliary Space: O(1)
Similar article: Maximize sum of K elements in Array by taking only corner elements