Reverse a subarray to maximize sum of even-indexed elements of given array

Given an array arr[], the task is to maximize the sum of even-indexed elements by reversing a subarray and print the maximum sum obtained.

Examples:

Input: arr[] = {1, 2, 1, 2, 1}
Output: 5
Explanation:
Sum of intial even-indexed elements = a[0] + a[2] + a[4] = 1 + 1 + 1 = 3
Reversing subarray {1, 2, 1, 2} modifies the array to {2, 1, 2, 1, 1}.
Hence, the maximized sum = 2 + 2 + 1 = 5

Input: arr[] = {7, 8, 4, 5, 7, 6, 8, 9, 7, 3}
Output: 37

Naive Approach:
The simplest approach to solve the problem is to generate all the possible permutations by reversal of elements one by one and calculate the sum at even indices for each permutation. Print the maximum possible sum among all the permutations.
Time Complexity: O(N3)
Auxiliary Space: O(N)



Efficient Approach:
The above approach can be further optimized to O(N) computational complexity by using Dynamic Programming to check the maximum difference by rotation of arrays.

Follow the steps below to solve the problem:

  • Compare the elements at odd index with even index and also keep track of them.
  • Initialize two arrays leftDP[] and rightDP[].
  • For every odd index, leftDP[] stores the difference of the element at current index with the element on its left and rightDP[] stores that of the right.
  • If the difference calculated for the previous index is positive, add it to the current difference:

    if(dp[i – 1] > 0)
    dp[i] = dp[i-1] + curr_diff

  • Otherwise, store the current difference:

    dp[i] = curr_diff;

Below is the implementation of the above approach:

Java

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// Java Program to implement
// the above approach
import java.io.*;
  
class GFG {
  
    // Function to return maximized sum
    // at even indices
    public static int maximizeSum(int[] arr)
    {
  
        int n = arr.length;
        int sum = 0;
        for (int i = 0; i < n; i = i + 2)
            sum += arr[i];
  
        // Stores difference with
        // element on the left
        int leftDP[] = new int[n / 2];
  
        // Stores difference with
        // element on the right
        int rightDP[] = new int[n / 2];
  
        int c = 0;
  
        for (int i = 1; i < n; i = i + 2) {
  
            // Compute and store
            // left difference
            int leftDiff = arr[i]
                           - arr[i - 1];
  
            // For first index
            if (c - 1 < 0)
                leftDP = leftDiff;
  
            else {
  
                // If previous difference
                // is positive
                if (leftDP > 0)
                    leftDP = leftDiff
                                + leftDP;
  
                // Otherwise
                else
                    leftDP[i] = leftDiff;
            }
  
            int rightDiff;
  
            // For the last index
            if (i + 1 >= arr.length)
                rightDiff = 0;
  
            // Otherwise
            else
                rightDiff = arr[i]
                            - arr[i + 1];
  
            // For first index
            if (c - 1 < 0)
                rightDP = rightDiff;
            else {
  
                // If the previous difference
                // is positive
                if (rightDP > 0)
                    rightDP = rightDiff
                                 + rightDP;
                else
                    rightDP = rightDiff;
            }
            c++;
        }
        int max = 0;
        for (int i = 0; i < n / 2; i++) {
            max = Math.max(max,
                           Math.max(
                               leftDP[i],
                               rightDP[i]));
        }
  
        return max + sum;
    }
  
    // Driver Code
    public static void main(String[] args)
    {
        int arr[] = { 7, 8, 4, 5, 7, 6,
                      8, 9, 7, 3 };
        int ans = maximizeSum(arr);
        System.out.println(ans);
    }
}

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Python3

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# Python3 program to implement
# the above approach
  
# Function to return maximized sum
# at even indices
def maximizeSum(arr):
  
    n = len(arr)
    sum = 0
  
    for i in range(0, n, 2):
        sum += arr[i]
  
    # Stores difference with
    # element on the left
    leftDP = [0] * (n)
  
    # Stores difference with
    # element on the right
    rightDP = [0] * (n)
  
    c = 0
    for i in range(1, n, 2):
  
        # Compute and store
        # left difference
        leftDiff = arr[i] - arr[i - 1]
  
        # For first index
        if (c - 1 < 0):
            leftDP[i] = leftDiff
        else:
  
            # If previous difference
            # is positive
            if (leftDP[i] > 0):
                leftDP[i] = (leftDiff + 
                             leftDP[i - 1])
  
            # Otherwise
            else:
                leftDP[i] = leftDiff
  
        rightDiff = 0
  
        # For the last index
        if (i + 1 >= len(arr)):
            rightDiff = 0
              
        # Otherwise
        else:
            rightDiff = arr[i] - arr[i + 1]
  
        # For first index
        if (c - 1 < 0):
            rightDP[i] = rightDiff
        else:
  
            # If the previous difference
            # is positive
            if (rightDP[i] > 0):
                rightDP[i] = (rightDiff +
                              rightDP[i - 1])
            else:
                rightDP[i] = rightDiff
                  
        c += 1
  
    maxm = 0
  
    for i in range(n // 2):
        maxm = max(maxm, max(leftDP[i],
                            rightDP[i]))
  
    return maxm + sum
  
# Driver Code
if __name__ == '__main__':
  
    arr = [ 7, 8, 4, 5, 7,
            6, 8, 9, 7, 3 ]
    ans = maximizeSum(arr)
  
    print(ans)
  
# This code is contributed by mohit kumar 29

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C#

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// C# program to implement 
// the above approach 
using System;
  
class GFG{ 
  
// Function to return maximized sum 
// at even indices 
public static int maximizeSum(int[] arr) 
    int n = arr.Length; 
    int sum = 0; 
      
    for(int i = 0; i < n; i = i + 2) 
        sum += arr[i]; 
  
    // Stores difference with 
    // element on the left 
    int []leftDP = new int[n / 2]; 
  
    // Stores difference with 
    // element on the right 
    int []rightDP = new int[n / 2]; 
  
    int c = 0; 
  
    for(int i = 1; i < n; i = i + 2)
    
          
        // Compute and store 
        // left difference 
        int leftDiff = arr[i] - arr[i - 1]; 
  
        // For first index 
        if (c - 1 < 0) 
            leftDP = leftDiff; 
              
        else 
        
              
            // If previous difference 
            // is positive 
            if (leftDP > 0) 
                leftDP = leftDiff + 
                            leftDP; 
                              
            // Otherwise 
            else
                leftDP = leftDiff; 
        
  
        int rightDiff; 
  
        // For the last index 
        if (i + 1 >= arr.Length) 
            rightDiff = 0; 
  
        // Otherwise 
        else
            rightDiff = arr[i] - arr[i + 1]; 
  
        // For first index 
        if (c - 1 < 0) 
            rightDP = rightDiff; 
              
        else 
        
              
            // If the previous difference 
            // is positive 
            if (rightDP > 0) 
                rightDP = rightDiff + 
                             rightDP; 
            else
                rightDP = rightDiff; 
        
        c++; 
    
      
    int max = 0; 
      
    for(int i = 0; i < n / 2; i++)
    
        max = Math.Max(max, 
                       Math.Max(leftDP[i], 
                               rightDP[i])); 
    
    return max + sum; 
  
// Driver Code 
public static void Main(String[] args) 
    int []arr = { 7, 8, 4, 5, 7, 6, 
                  8, 9, 7, 3 }; 
    int ans = maximizeSum(arr); 
      
    Console.WriteLine(ans); 
  
// This code is contributed by 29AjayKumar

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Output:

37

Time Complexity: O(N)
Auxiliary Space: O(N)

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