Maximum sum subsequence made up of at most K distant elements including the first and last array elements

Last Updated : 24 Feb, 2021

Given an array arr[] consisting of N integers and an integer K, the task is to print the maximum sum possible in a subsequence satisfying the following conditions:

The elements arr[N – 1] and arr[0] are included in the subsequence.
Adjacent elements in the subsequence can be at a distance of at most K indices.

Examples:

Input: arr[] = {10, -5, -2, 4, 0, 3}, K = 3
Output: 17
Explanation:
One of possible way is as follows:
Include arr[0] into the subsequence. Sum = 10.
Include arr[3] in the subsequence. Therefore, sum = 10 + 4 = 14.
Include arr[5] in the subsequence. Therefore, total sum = 14 + 3 = 17.
Therefore, the maximum sum possible is 17.
Input: arr[] = {1, -5, -20, 4, -1, 3, -6, -3}, K = 2
Output: 0

Naive Approach: The simplest approach is to find all subsequences possible from arr[] with at most K difference between indices of adjacent elements, starting from index 0 and ending at index (N – 1). Calculate sum of all such subsequences. Finally, print the maximum of all the sums obtained.
Time Complexity: O(N*2^N)
Auxiliary Space: O(N)

Efficient Approach: The above approach can be optimized by using a Greedy Algorithm and deque. Follow the steps below to solve the problem:

Initialize an array, say dp[], to store the maximum value obtained till the current index.
Initialize a deque of pairs, say Q, to store the pair {dp[i], i}.
Assign the value of arr[0] to dp[0] and push the pair {dp[0], 0} into the deque.
Traverse the given array arr[] using the variable i and perform the following steps:
- Increment dp[i] by the sum of arr[i] and maximum value in the deque, i.e. dp[i] = arr[i] + Q[0][0].
- Traverse over the deque Q from the end and pop the last element if Q[-1][0] is less than dp[i].
- Append the pair {dp[i], i} in the deque.
- Check if the index of the first element of the deque q is equal to (i – K) or not and then, pop the first element from the deque Q.
After completing the above steps, print the value stored at the last index of dp[], i.e. dp[N – 1] as the result.

Below is the implementation of the above approach:

C++

// CPP program for the above approach
#include<bits/stdc++.h>
using namespace std;
 
// Function to find maximum sum
// of a subsequence satisfying
// the given conditions
int maxResult(int arr[], int k, int n){
 
  // Stores the maximum sum
  int dp[n] = {0};
 
  // Starting index of
  // the subsequence
  dp[0] = arr[0];
 
  // Stores the pair of maximum value
  // and the index of that value
  deque<pair<int,int>> q;
  q.push_back({arr[0], 0});
 
  // Traverse the array
  for (int i = 1; i < n; i++)
  {
 
    // Increment the first value
    // of deque by arr[i] and
    // store it in dp[i]
    dp[i] = arr[i] + q.front().first;
 
    // Delete all the values which
    // are less than dp[i] in deque
    while (q.size() > 0 and q.back().first < dp[i])
      q.pop_back();
 
    // Append the current pair of
    // value and index in deque
    q.push_back({dp[i], i});
 
    // If first value of the
    // queue is at a distance > K
    if (i - k == q.front().second)
      q.pop_front();
  }
 
  // Return the value at the last index
  return dp[n - 1];
}
 
// Driver Code
int main()
{
  int arr[] = {10, -5, -2, 4, 0, 3};
  int K = 3;
  int n = sizeof(arr)/sizeof(arr[0]);
  cout<<maxResult(arr, K,n);
 
}
 
// This code is contributed by ipg2016107.

Python3

# Python program for the above approach
from collections import deque
 
# Function to find maximum sum
# of a subsequence satisfying
# the given conditions
def maxResult(arr, k):
 
    # Stores the maximum sum
    dp = [0]*len(arr)
 
    # Starting index of
    # the subsequence
    dp[0] = arr[0]
 
    # Stores the pair of maximum value
    # and the index of that value
    q = deque([(arr[0], 0)])
 
    # Traverse the array
    for i in range(1, len(arr)):
       
        # Increment the first value
        # of deque by arr[i] and
        # store it in dp[i]
        dp[i] = arr[i] + q[0][0]
 
        # Delete all the values which
        # are less than dp[i] in deque
        while q and q[-1][0] < dp[i]:
            q.pop()
 
        # Append the current pair of
        # value and index in deque
        q.append((dp[i], i))
         
        # If first value of the
        # queue is at a distance > K
        if i - k == q[0][1]:
            q.popleft()
 
    # Return the value at the last index
    return dp[-1]
 
# Driver Code
 
arr = [10, -5, -2, 4, 0, 3]
K = 3
print(maxResult(arr, K))

Output:

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

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