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Find maximum Subsequence Sum according to given conditions
• Last Updated : 12 Jun, 2021

Given an integer array nums and an integer K, The task is to find the maximum sum of a non-empty subsequence of the array such that for every two consecutive integers in the subsequence, nums[i] and nums[j], where i < j, the condition j – i <= K is satisfied.

A subsequence of an array is obtained by deleting some number of elements (can be zero) from the array, leaving the remaining elements in their original order.

Examples:

```Input: nums = [10, 2, -10, 5, 20], K = 2
Output: 37
Explanation:
The subsequence is [10, 2, 5, 20].

Input: nums = [-1, -2, -3], K = 1
Output: -1

Input: nums = [10, -2, -10, -5, 20], K = 2
Output: 23```

Approach:

• The optimal solution of this problem can be achieved by using the Sliding Window Maximum and Dynamic Programming .
• At any index i after calculating the maximum sum for i, nums[i] will now store the maximum possible sum that can be obtained from a subsequence ending at i.
• To calculate the maximum possible sum for every index i , check what is the maximum value that can be obtained from a window of size K before it. If the maximum value is negative, use zero instead.
• To use the values of calculated sums optimally we use a deque to store the sums along with their indexes in an increasing order of the sums . We also pop the element from the back when its index goes out of the window k of current index i.

## CPP

 `// C++ program to find the maximum sum``// subsequence under given constraint``#include ``using` `namespace` `std;` `// Function return the maximum sum``int` `ConstrainedSubsetSum(vector<``int``>& nums,``                         ``int` `k)``{``    ``deque > d;``    ` `    ``// Iterating the given array``    ``for` `(``int` `i = 0; i < nums.size(); i++)``    ``{``        ``// Check if deque is empty``        ``nums[i] += d.size()``            ``? max(d.back().first, 0) : 0;``        ` `        ``while` `(d.size() &&``               ``d.front().first < nums[i])``            ``d.pop_front();``        ` `        ``d.push_front({ nums[i], i });``        ` `        ``if` `(d.back().second == i - k)``            ``d.pop_back();``    ``}` `    ``int` `ans = nums;``    ` `    ``for` `(``auto` `x : nums)``        ``ans = max(ans, x);` `    ``return` `ans;``}` `// Driver code``int` `main()``{``    ``vector<``int``> nums = { 10, -2, -10,``                        ``-5, 20 };``    ``int` `K = 2;``    ` `    ``// Function call``    ``cout << ConstrainedSubsetSum(nums, K)``        ``<< endl;``    ``return` `0;``}`
Output:
`23`

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

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