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 atmost 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++

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// 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;
}

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Java

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// 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

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C#

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// 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

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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. At 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:

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// 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;
}

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Output:

1

Time Complexity: O(n)

Auxiliary Space: O(1)

Similar article: Maximize sum of K elements in Array by taking only corner elements

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