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Problem solution · Python

Maximum Balanced Subsequence Sum

Maximum Balanced Subsequence Sum: a Python solution using sorting and greedy selection. Learn the idea, check the complexity, and read the full code, with credit to walkccc LeetCode Solutions.

Technique
Sorting and greedy selection
Source
walkccc LeetCode Solutions
Length
41 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Sorting and greedy selection

For Maximum Balanced Subsequence Sum, the implementation first exposes a useful order, then scans that order while making locally justified choices.

  1. Choose the key that reveals the greedy or grouping structure.
  2. Sort the relevant records by that key.
  3. Scan in order, maintaining the invariant that makes each local choice safe.

Code notes

  • 41 lines of Python from the credited upstream file 2926.py.
  • The implementation visibly relies on sequence storage.
  • No explicit loop blocks detected.

Complexity

Sorting is typically the dominant term unless the subsequent scan uses a more expensive nested operation.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from walkccc LeetCode Solutions by P.-Y. Chen (walkccc) and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeMaximum Balanced Subsequence Sum · PythonPython
Use this to learn the idea, then write your own version.
class FenwickTree:  def __init__(self, n: int):    self.vals = [0] * (n + 1)   def maximize(self, i: int, val: int) -> None:    """Updates the maximum sum of subsequence ending in (i - 1) with `val`."""    while i < len(self.vals):      self.vals[i] = max(self.vals[i], val)      i += FenwickTree.lowbit(i)   def get(self, i: int) -> int:    """Returns the maximum sum of subsequence ending in (i - 1)."""    res = 0    while i > 0:      res = max(res, self.vals[i])      i -= FenwickTree.lowbit(i)    return res   @staticmethod  def lowbit(i: int) -> int:    return i & -i  class Solution:  def maxBalancedSubsequenceSum(self, nums: list[int]) -> int:    # Let's define maxSum[i] := subsequence with the maximum sum ending in i    # By observation:    #    nums[i] - nums[j] >= i - j    # => nums[i] - i >= nums[j] - j    # So, if nums[i] - i >= nums[j] - j, where i > j,    # maxSum[i] = max(maxSum[i], maxSum[j] + nums[i])    ans = -math.inf    tree = FenwickTree(len(nums))     for _, i in sorted([(num - i, i) for i, num in enumerate(nums)]):      subseqSum = tree.get(i) + nums[i]      tree.maximize(i + 1, subseqSum)      ans = max(ans, subseqSum)     return ans

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