Maximum sum of non-overlapping subarrays of length atmost K

Given an integer array ‘arr’ of length N and an integer ‘k’, select some non-overlapping subarrays such that each sub-array if of length at most ‘k’, no two sub-arrays are adjacent and sum of all the elements of the selected sub-arrays are maximum.
Examples:

Input : arr[] = {-1, 2, -3, 4, 5}, k = 2
Output : 11
Sub-arrays that maximizes sum will be {{2}, {4, 5}}.
Thus, the answer will be 2+4+5 = 11.

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

Naive Approach : A simple way is to generate all possible subsets of elements satisfying above conditions recursively and find the subset with maximum sum. 
Time Complexity : O(2^N) 
Efficient Approach: A better approach is to use 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 {i, n-1} satisfying above conditions.
We will have ‘K+1’ possible choices i.e.

1. Reject ‘i’ and solve for ‘i+1’.
2. Select sub-array {i} and solve for ‘i+2’
3. Select sub-array {i, i+1} and solve for ‘i+3’

Thus, recurrence relation will be: 

dp[i] = max(dp[i+1], arr[i]+dp[i+2], arr[i]+arr[i+1]+dp[i+3],
        ...arr[i]+arr[i+1]+arr[i+2]...+arr[i+k-1] + dp[i+k+1])

Below is the implementation of the above approach: 

C++

// C++ program to implement above approach
#include <bits/stdc++.h>
#define maxLen 10
using namespace std;

// Variable to store states of dp
int dp[maxLen];

// Variable to check if a given state has been solved
bool visit[maxLen];

// 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 >= n)
        return 0;

    // To check if a state has been solved
    if (visit[i])
        return dp[i];
    visit[i] = 1;

    // Variable to store
    // prefix sum for sub-array
    // {i, j}
    int tot = 0;
    dp[i] = maxSum(arr, i + 1, n, k);

    // Required recurrence relation
    for (int j = i; j < i + k and j < n; j++) {
        tot += arr[j];
        dp[i] = max(dp[i], tot +
                     maxSum(arr, j + 2, n, k));
    }

    // Returning the value
    return dp[i];
}

// Driver code
int main()
{
    // Input array
    int arr[] = { -1, 2, -3, 4, 5 };

    int k = 2;

    int n = sizeof(arr) / sizeof(int);

    cout << maxSum(arr, 0, n, k);

    return 0;
}

Java

// Java program to implement above approach
import java.io.*;

class GFG
{

    static int maxLen = 10;

    // Variable to store states of dp
    static int dp[] = new int[maxLen];

    // Variable to check
    // if a given state has been solved
    static boolean []visit = new boolean[maxLen];

    // 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 >= n)
            return 0;

        // To check if a state has been solved
        if (visit[i])
            return dp[i];
        visit[i] = true;

        // Variable to store
        // prefix sum for sub-array
        // {i, j}
        int tot = 0;
        dp[i] = maxSum(arr, i + 1, n, k);

        // Required recurrence relation
        for (int j = i; j < (i + k) &&
                            (j < n); j++)
        {
            tot += arr[j];
            dp[i] = Math.max(dp[i], tot +
                    maxSum(arr, j + 2, n, k));
        }

        // Returning the value
        return dp[i];
    }

    // Driver code
    public static void main (String[] args)
    {

        // Input array
        int arr[] = { -1, 2, -3, 4, 5 };

        int k = 2;

        int n = arr.length;

        System.out.println(maxSum(arr, 0, n, k));
    }
}

// This code is contributed by ajit.

Python3

# Python3 program to implement above approach
maxLen = 10

# Variable to store states of dp
dp = [0]*maxLen;

# Variable to check if a given state has been solved
visit = [0]*maxLen;

# Function to find the maximum sum subsequence
# such that no two elements are adjacent
def maxSum(arr, i, n, k) :

    # Base case
    if (i >= n) :
        return 0;

    # To check if a state has been solved
    if (visit[i]) :
        return dp[i];

    visit[i] = 1;

    # Variable to store
    # prefix sum for sub-array
    # {i, j}
    tot = 0;
    dp[i] = maxSum(arr, i + 1, n, k);

    # Required recurrence relation
    j = i
    while (j < i + k and j < n) :
        tot += arr[j];
        dp[i] = max(dp[i], tot +
                    maxSum(arr, j + 2, n, k));

        j += 1

    # Returning the value
    return dp[i];


# Driver code
if __name__ == "__main__" :

    # Input array
    arr = [ -1, 2, -3, 4, 5 ];

    k = 2;

    n = len(arr);

    print(maxSum(arr, 0, n, k));

# This code is contributed by AnkitRai01

C#

// C# program to implement above approach
using System;

class GFG
{
static int maxLen = 10;

// Variable to store states of dp
static int []dp = new int[maxLen];

// Variable to check
// if a given state has been solved
static bool []visit = new bool[maxLen];

// 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 >= n)
        return 0;

    // To check if a state has been solved
    if (visit[i])
        return dp[i];
    visit[i] = true;

    // Variable to store
    // prefix sum for sub-array
    // {i, j}
    int tot = 0;
    dp[i] = maxSum(arr, i + 1, n, k);

    // Required recurrence relation
    for (int j = i; j < (i + k) &&
                        (j < n); j++)
    {
        tot += arr[j];
        dp[i] = Math.Max(dp[i], tot +
                  maxSum(arr, j + 2, n, k));
    }

    // Returning the value
    return dp[i];
}

// Driver code
static public void Main ()
{

    // Input array
    int []arr = { -1, 2, -3, 4, 5 };

    int k = 2;

    int n = arr.Length;

    Console.WriteLine (maxSum(arr, 0, n, k));
}
}

// This code is contributed by ajit.

Javascript

<script>
    // Javascript program to implement above approach

    let maxLen = 10;

    // Variable to store states of dp
    let dp = new Array(maxLen);

    // Variable to check
    // if a given state has been solved
    let visit = new Array(maxLen);

    // Function to find the maximum sum subsequence
    // such that no two elements are adjacent
    function maxSum(arr, i, n, k)
    {
        // Base case
        if (i >= n)
            return 0;

        // To check if a state has been solved
        if (visit[i])
            return dp[i];
        visit[i] = true;

        // Variable to store
        // prefix sum for sub-array
        // {i, j}
        let tot = 0;
        dp[i] = maxSum(arr, i + 1, n, k);

        // Required recurrence relation
        for (let j = i; j < (i + k) &&
                            (j < n); j++)
        {
            tot += arr[j];
            dp[i] = Math.max(dp[i], tot +
                      maxSum(arr, j + 2, n, k));
        }

        // Returning the value
        return dp[i];
    }

    // Input array
    let arr = [ -1, 2, -3, 4, 5 ];

    let k = 2;

    let n = arr.length;

    document.write(maxSum(arr, 0, n, k));

</script>

Output:

11

Time Complexity: O(n*k)