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Quantum Network Stream Validator Solution

Problem Statement

Given a high-dimensional input dataset or state graph of length $N$, calculate the optimal result using the **Heavy-Light Decomposition** algorithm. Formally, implement an optimal sub-linear or $O(N \log N)$ solution capable of satisfying strict time and space complexity limits under maximum competitive edge cases.

Example 1
Input
arr = [7, 15, 23, 11]
Output
36

Explanation: By applying Heavy-Light Decomposition, the optimal hard constraint value evaluates to 36.

Example 2
Input
arr = [10, 8]
Output
9

Explanation: Minimal edge input evaluates to 9.

Constraints

  • 1 <= N <= 2 * 10^5
  • -10^9 <= arr[i] <= 10^9
  • Time Complexity: O(N log N) or O(N log^2 N)
  • Space Complexity: O(N)

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