Maximise array sum after taking non-overlapping sub-arrays of length K

Given an integer array arr[] of length N and an integer K, the task is to select some non-overlapping sub-arrays such that each sub-array is exactly of length K, no two sub-arrays are adjacent and sum of all the elements of the selected sub-arrays is maximum.

Examples:

Input: arr[] = {1, 2, 3, 4, 5}, K = 2
Output: 12
Sub-arrays that maximizes sum will be {{1, 2}, {4, 5}}.
Thus, the answer will be 12.



Input: arr[] = {1, 1, 1, 1, 1}, K = 1
Output: 3

Approach: This problem can be solved using dynamic programming. Let’s suppose we are at an index i. Let dp[i] be defined as the maximum sum of elements of all possible subsets of sub-array arr[i…n-1] satisfying above conditions.
We will have two possible choices i.e. select the sub-array arr[i…i+k-1] and solve for dp[i + k + 1] or reject it and solve for dp[i + 1].

Thus, recurrence relation will be

dp[i] = max(dp[i + 1], arr[i] + arr[i + 1] + arr[i + 2] + … + arr[i + k – 1] + dp[i + k + 1])

Since, the values of K can be large, we will use prefix-sum array to find the sum of all the elements of the sub-array arr[i…i + k – 1] in O(1).
Overall, time complexity of the algorithm will be O(N).

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
#define maxLen 10
using namespace std;
  
// To store the states of dp
int dp[maxLen];
  
// To check if a given state
// has been solved
bool v[maxLen];
  
// To store the prefix-sum
int prefix_sum[maxLen];
  
// Function to fill the prefix_sum[] with
// the prefix sum of the given array
void findPrefixSum(int arr[], int n)
{
    prefix_sum[0] = arr[0];
    for (int i = 1; i < n; i++)
        prefix_sum[i] = arr[i] + prefix_sum[i - 1];
}
  
// Function to find the maximum sum subsequence
// such that no two elements are adjacent
int maxSum(int arr[], int i, int n, int k)
{
    // Base case
    if (i + k > n)
        return 0;
  
    // To check if a state has
    // been solved
    if (v[i])
        return dp[i];
    v[i] = 1;
  
    int x;
  
    if (i == 0)
        x = prefix_sum[k - 1];
    else
        x = prefix_sum[i + k - 1] - prefix_sum[i - 1];
  
    // Required recurrence relation
    dp[i] = max(maxSum(arr, i + 1, n, k),
                x + maxSum(arr, i + k + 1, n, k));
  
    // Returning the value
    return dp[i];
}
  
// Driver code
int main()
{
    int arr[] = { 1, 3, 7, 6 };
    int n = sizeof(arr) / sizeof(int);
    int k = 1;
  
    // Finding prefix-sum
    findPrefixSum(arr, n);
  
    // Finding the maximum possible sum
    cout << maxSum(arr, 0, n, k);
  
    return 0;
}

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Java

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// Java implementation of the approach
import java.util.*;
  
class GFG 
{
  
    static int maxLen = 10;
  
    // To store the states of dp
    static int[] dp = new int[maxLen];
  
    // To check if a given state
    // has been solved
    static boolean[] v = new boolean[maxLen];
  
    // To store the prefix-sum
    static int[] prefix_sum = new int[maxLen];
  
    // Function to fill the prefix_sum[] with
    // the prefix sum of the given array
    static void findPrefixSum(int arr[], int n) 
    {
        prefix_sum[0] = arr[0];
        for (int i = 1; i < n; i++)
        {
            prefix_sum[i] = arr[i] + prefix_sum[i - 1];
        }
    }
  
    // Function to find the maximum sum subsequence
    // such that no two elements are adjacent
    static int maxSum(int arr[], int i, int n, int k)
    {
        // Base case
        if (i + k > n) 
        {
            return 0;
        }
  
        // To check if a state has
        // been solved
        if (v[i]) 
        {
            return dp[i];
        }
        v[i] = true;
  
        int x;
  
        if (i == 0)
        {
            x = prefix_sum[k - 1];
        
        else
        {
            x = prefix_sum[i + k - 1] - prefix_sum[i - 1];
        }
  
        // Required recurrence relation
        dp[i] = Math.max(maxSum(arr, i + 1, n, k),
                x + maxSum(arr, i + k + 1, n, k));
  
        // Returning the value
        return dp[i];
    }
  
    // Driver code
    public static void main(String[] args)
    {
        int arr[] = {1, 3, 7, 6};
        int n = arr.length;
        int k = 1;
  
        // Finding prefix-sum
        findPrefixSum(arr, n);
  
        // Finding the maximum possible sum
        System.out.println(maxSum(arr, 0, n, k));
    }
}
  
// This code contributed by Rajput-Ji

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Python3

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# Python3 implementation of the approach 
  
maxLen = 10
  
# To store the states of dp 
dp = [0]*maxLen; 
  
# To check if a given state 
# has been solved 
v = [0]*maxLen; 
  
# To store the prefix-sum 
prefix_sum = [0]*maxLen; 
  
# Function to fill the prefix_sum[] with 
# the prefix sum of the given array 
def findPrefixSum(arr, n) : 
  
    prefix_sum[0] = arr[0]; 
    for i in range(n) :
        prefix_sum[i] = arr[i] + prefix_sum[i - 1]; 
  
  
# Function to find the maximum sum subsequence 
# such that no two elements are adjacent 
def maxSum(arr, i, n, k) : 
  
    # Base case 
    if (i + k > n) :
        return 0
  
    # To check if a state has 
    # been solved 
    if (v[i]) :
        return dp[i]; 
          
    v[i] = 1
  
    if (i == 0) :
        x = prefix_sum[k - 1]; 
    else :
        x = prefix_sum[i + k - 1] - prefix_sum[i - 1]; 
  
    # Required recurrence relation 
    dp[i] = max(maxSum(arr, i + 1, n, k), 
                x + maxSum(arr, i + k + 1, n, k)); 
  
    # Returning the value 
    return dp[i]; 
  
  
# Driver code 
if __name__ == "__main__"
  
    arr = [ 1, 3, 7, 6 ];
      
    n = len(arr); 
    k = 1
  
    # Finding prefix-sum 
    findPrefixSum(arr, n); 
  
    # Finding the maximum possible sum 
    print(maxSum(arr, 0, n, k)); 
      
# This code is contributed by AnkitRai01

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C#

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// C# implementation of the approach
using System;
  
class GFG 
  
    static int maxLen = 10; 
  
    // To store the states of dp 
    static int[] dp = new int[maxLen]; 
  
    // To check if a given state 
    // has been solved 
    static bool[] v = new bool[maxLen]; 
  
    // To store the prefix-sum 
    static int[] prefix_sum = new int[maxLen]; 
  
    // Function to fill the prefix_sum[] with 
    // the prefix sum of the given array 
    static void findPrefixSum(int []arr, int n) 
    
        prefix_sum[0] = arr[0]; 
        for (int i = 1; i < n; i++) 
        
            prefix_sum[i] = arr[i] + prefix_sum[i - 1]; 
        
    
  
    // Function to find the maximum sum subsequence 
    // such that no two elements are adjacent 
    static int maxSum(int []arr, int i, int n, int k) 
    
        // Base case 
        if (i + k > n) 
        
            return 0; 
        
  
        // To check if a state has 
        // been solved 
        if (v[i]) 
        
            return dp[i]; 
        
        v[i] = true
  
        int x; 
  
        if (i == 0) 
        
            x = prefix_sum[k - 1]; 
        
        else
        
            x = prefix_sum[i + k - 1] - prefix_sum[i - 1]; 
        
  
        // Required recurrence relation 
        dp[i] = Math.Max(maxSum(arr, i + 1, n, k), 
                x + maxSum(arr, i + k + 1, n, k)); 
  
        // Returning the value 
        return dp[i]; 
    
  
    // Driver code 
    public static void Main(String[] args) 
    
        int []arr = {1, 3, 7, 6}; 
        int n = arr.Length; 
        int k = 1; 
  
        // Finding prefix-sum 
        findPrefixSum(arr, n); 
  
        // Finding the maximum possible sum 
        Console.Write(maxSum(arr, 0, n, k)); 
    
  
// This code is contributed by Princi Singh

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Output:

9


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