Maximize the sum of maximum elements of at least K-sized subarrays

Given an integer array arr[] of length N and an integer K, partition the array in some non-overlapping subarrays such that each subarray has size at least K and each element of the array should be part of a subarray. The task is to maximize the sum of maximum elements across all the subarrays.

Examples:

Input: N = 4 , K = 1, arr[] = {1, 2, 3, 4}
Output: 10
Explanation: The most optimal way of partition is {1}, {2}, {3}, {4} and total score is 1 + 2 + 3 + 4 = 10

Input: N = 6, K = 2, arr[] = {1, 2, 9, 8, 3, 4}
Output: 17
Explanation: The most optimal way of partition is {1, 2, 9}, {8, 3, 4} and total score is max({1, 2, 9}) + max({8, 3, 4}) = 9 + 8 = 17

Approach: To solve the problem, follow the below idea:

The idea in this is to explore all the combinations of subarrays possible which have a size of at least K. We can do this using recursion and explore every possibility. For every index, we have two choices: Either add the current element to the partition or start a new partition from this index. Explore both the choices on each index to maximize the sum of maximum elements of all subarrays.

Step-by-step algorithm:

• Declare a recursive function solve() that takes the current index idx, the minimum subarray length K, and the input array arr.
• Base case is if the idx is greater than equal to the size of the array then we return 0.
• If arr.size()-idx is less than K then we return -1e5.
• Initialize a variable maxi equal to -1 and cnt equal to 0 and ans equal to -1.
• Run a loop j from idx to arr.size().
• Increase cnt by 1 and check if arr[j] > maxi then update maxi to arr[j] and maxi.
• If the cnt is greater than or equal to K then call the solve() function for j+1 and add maxi to it and update the ans to the maximum of ans and the sum of maxi and value returned by recursive function.
• Return ans.

Below is the implementation of the algorithm:

C++

#include <bits/stdc++.h>
using namespace std;

// Function to iterate in the given array
long long dpf(int i, int K, vector<int>& arr)
{

    // If length of array is reached
    if (i >= arr.size())
        return 0;

    // If left array size is less than K
    if (arr.size() - i < K)
        return -1e15;

    int maxi = -1;
    int cnt = 0;
    long long ans = -1;

    // Iterate in array
    for (int j = i; j < arr.size(); j++) {

        // Update the array
        maxi = max(arr[j], maxi);
        cnt++;
        if (cnt >= K)
            ans = max(ans, maxi + dpf(j + 1, K, arr));
    }

    // Return the maximum score
    return ans;
}

// Function to find maximum score
long long MaximumScore(int N, int K, vector<int>& arr)
{
    // Function call to iterate
    return dpf(0, K, arr);
}

// Driver code
int main()
{

    int N = 4;
    int K = 1;
    vector<int> arr = { 1, 2, 3, 4 };

    // Function call
    cout << MaximumScore(N, K, arr);

    return 0;
}

Java

import java.util.ArrayList;

public class Main {

    // Function to iterate in the given array
    static long dpf(int i, int K, ArrayList<Integer> arr) {

        // If length of array is reached
        if (i >= arr.size())
            return 0;

        // If left array size is less than K
        if (arr.size() - i < K)
            return Long.MIN_VALUE;

        int maxi = -1;
        int cnt = 0;
        long ans = Long.MIN_VALUE;

        // Iterate in array
        for (int j = i; j < arr.size(); j++) {

            // Update the array
            maxi = Math.max(arr.get(j), maxi);
            cnt++;
            if (cnt >= K)
                ans = Math.max(ans, maxi + dpf(j + 1, K, arr));
        }

        // Return the maximum score
        return ans;
    }

    // Function to find maximum score
    static long MaximumScore(int N, int K, ArrayList<Integer> arr) {
        // Function call to iterate
        return dpf(0, K, arr);
    }

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

        int N = 4;
        int K = 1;
        ArrayList<Integer> arr = new ArrayList<Integer>() {{
            add(1);
            add(2);
            add(3);
            add(4);
        }};

        // Function call
        System.out.println(MaximumScore(N, K, arr));
    }
}

// This code is contributed by akshitaguprzj3

C#

// C# program for the above approach
using System;
using System.Collections.Generic;

public class MainClass {
    // Function to iterate in the given array
    public static long Dpf(int i, int K, List<int> arr)
    {
        // If length of array is reached
        if (i >= arr.Count)
            return 0;

        // If left array size is less than K
        if (arr.Count - i < K)
            return int.MinValue;

        int maxi = int.MinValue;
        int cnt = 0;
        long ans = int.MinValue;

        // Iterate in array
        for (int j = i; j < arr.Count; j++) {
            // Update the array
            maxi = Math.Max(arr[j], maxi);
            cnt++;
            if (cnt >= K)
                ans = Math.Max(ans,
                               maxi + Dpf(j + 1, K, arr));
        }

        // Return the maximum score
        return ans;
    }

    // Function to find maximum score
    public static long MaximumScore(int N, int K,
                                    List<int> arr)
    {
        // Function call to iterate
        return Dpf(0, K, arr);
    }

    // Driver code
    public static void Main(string[] args)
    {
        int N = 4;
        int K = 1;
        List<int> arr = new List<int>{ 1, 2, 3, 4 };

        // Function call
        Console.WriteLine(MaximumScore(N, K, arr));
    }
}

// This code is contributed by Susobhan Akhuli

Javascript

// Function to find maximum score
function maximumScore(N, K, arr) {
    // Function to iterate in the given array
    function dpf(i, K, arr) {
        // If length of array is reached
        if (i >= arr.length)
            return 0;

        // If left array size is less than K
        if (arr.length - i < K)
            return Number.MIN_SAFE_INTEGER;

        let maxi = -1;
        let cnt = 0;
        let ans = Number.MIN_SAFE_INTEGER;

        // Iterate in array
        for (let j = i; j < arr.length; j++) {
            // Update the array
            maxi = Math.max(arr[j], maxi);
            cnt++;
            if (cnt >= K)
                ans = Math.max(ans, maxi + dpf(j + 1, K, arr));
        }

        // Return the maximum score
        return ans;
    }

    // Function call to iterate
    return dpf(0, K, arr);
}

// Driver code
function main() {
    const N = 4;
    const K = 1;
    const arr = [1, 2, 3, 4];

    // Function call
    console.log(maximumScore(N, K, arr));
}

// Calling the main function to execute the code
main();

Python3

# Python program for the above approach

# Function to iterate in the given array
def dpf(i, K, arr):
    # If length of array is reached
    if i >= len(arr):
        return 0

    # If left array size is less than K
    if len(arr) - i < K:
        return -1e15

    maxi = -1
    cnt = 0
    ans = -1

    # Iterate in array
    for j in range(i, len(arr)):
        # Update the array
        maxi = max(arr[j], maxi)
        cnt += 1
        if cnt >= K:
            ans = max(ans, maxi + dpf(j + 1, K, arr))

    # Return the maximum score
    return ans

# Function to find maximum score
def MaximumScore(N, K, arr):
    # Function call to iterate
    return dpf(0, K, arr)

# Driver code
if __name__ == "__main__":
    N = 4
    K = 1
    arr = [1, 2, 3, 4]

    # Function call
    print(MaximumScore(N, K, arr))

# This code is contributed by Susobhan Akhuli

Output

10

Time Complexity: O(N * K), where N is the size of input array arr[] and K is the minimum size of each subarray.
Auxiliary Space: O(1)