[FEATURE REQUEST] Implement Heavy-Light Decomposition (HLD) for Efficient Tree Queries

[NithinU2802]

What would you like to Propose?

Description:
Implementation of Heavy-Light Decomposition (HLD), which is a powerful technique for efficiently processing path queries on trees. HLD allows operations like finding the maximum, sum, or updating values on a tree path in O(log N) time using segment trees.

Why is this needed?

• Efficiently handles path queries (e.g., max/sum in path, LCA-based queries).
• Optimizes tree traversal by decomposing it into heavy and light chains.
• Reduces query complexity from O(N) to O(log N).

Proposed Solution:

• Implement Heavy-Light Decomposition (HLD) in Java.
• Use segment trees for efficient range queries.
• Provide test cases to validate correctness.

Expected Outcome:
A fully functional HLD implementation in Java that can be used for tree-based queries in competitive programming and real-world applications.

Would love to contribute this feature! 🚀
Please refer to PR #6169 for the implementation. 😊

Issue details

Heavy-Light Decomposition (HLD) is a tree decomposition technique that enables efficient path queries (such as maximum, sum, and updates) and Lowest Common Ancestor (LCA) queries in O(log N) time.

Approach:

1. DFS (Depth First Search) for Subtree Sizes - Calculate subtree sizes to determine the heavy child of each node.
2. HLD Decomposition - Assign chain numbers to nodes. Keep heavy children in the same chain, and start a new chain for light children.
3. Segment Tree - Store node values and support efficient path queries (max, sum, etc.).
4. Query Execution - Move up chains and use the segment tree to get answers in O(log N).

Additional Information

Please let me know if any improvements are needed. 😊