Maximum Balanced Subsequence Sum

Problem Description

You are given a 0-indexed integer array nums. A subsequence of nums having length k and consisting of indices i_0 < i_1 < ... < i_{k-1} is balanced if the following holds:

• nums[i_j] - nums[i_{j-1}] >= i_j - i_{j-1}, for every j in the range [1, k - 1]. A subsequence of nums having length 1 is considered balanced. Return an integer denoting the maximum possible sum of elements in a balanced subsequence of nums.

Key Insights

• A balanced subsequence maintains a specific difference condition based on the indices of its elements.
• The problem can be approached using dynamic programming to maximize the sum of valid subsequences.
• We can utilize binary search to efficiently find valid indices for subsequences.

Space and Time Complexity

Time Complexity: O(n log n)
Space Complexity: O(n)

Solution

To solve this problem, we will use a dynamic programming approach combined with binary search. The idea is to maintain an array that represents the maximum sum of balanced subsequences up to each index.

1. Initialize a DP array of the same length as nums to store the maximum sums.
2. Iterate through the nums array. For each element, check all previous elements to see if they can form a balanced subsequence with the current element.
3. Use binary search to efficiently find valid indices that can form balanced subsequences based on the defined conditions.
4. Update the DP array to keep track of the maximum sums.
5. The answer will be the maximum value in the DP array.

Code Solutions