Produce a high-quality educational video that explains the Heavy-Light Decomposition (HLD) algorithm — a fundamental data structure technique used to optimize queries and updates on trees, especially for competitive programming and advanced algorithmic applications. The video should be designed to effectively teach both intuition and implementation, making it accessible to students and professionals preparing for contests such as IOI, Codeforces, or ICPC. ⸻ ✅ Video Requirements 1. Introduction (Motivation) • Define the problem HLD solves: efficiently handling queries on paths or subtrees of a tree. • Show real-world or contest-related examples (e.g., finding sum/max on a path in a tree). • Brief mention of other techniques (like Segment Tree, Binary Lifting) and why HLD is more efficient for path queries. 2. Conceptual Explanation • Explain decomposition of a tree into “heavy” and “light” edges. • Describe how the tree is divided into chains (or paths). • Visualize the HLD process step-by-step using diagrams or animations: • Identify heavy child (the subtree with largest size). • Mark light edges and explain their significance. • Number nodes using Euler tour or DFS order. 3. Implementation Walkthrough • Pseudocode first, then real code (preferably in C++ or Python). • Show: • DFS to compute subtree sizes. • Assigning heavy paths. • Building a Segment Tree on top of linearized nodes. • Explain how to query from one node to another using HLD. 4. Complexity Analysis • Time complexity for preprocessing and per-query. • Comparison with naive approaches and other methods. 5. Examples & Practice • Walk through at least one sample problem from platforms like Codeforces or AtCoder. • Implement a query like: “find max value between node u and node v”. • Bonus: Add subtree queries support if possible. 6. Visual Aids • Use animations, tree diagrams, and call stack tracing to visualize DFS and segment tree traversal. • Highlight heavy/light edges, color chains differently. 7. Advanced Tips (Optional Section) • How to extend HLD to support: • Lazy propagation • Dynamic trees (e.g., using Euler Tour Trees) • Link-Cut Trees comparison ⸻ 🎯 Target Audience • Advanced Data Structures students • Competitive programmers (rated 1600–2400+) • Interview prep learners (Google, Meta, etc.) ⸻ 📋 Deliverables • YouTube-optimized video with chapters and timestamps • GitHub repo (or Gist) with code samples and explanations • Markdown README with: • Summary of HLD • Code used • Problem links • Complexity table

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Heavy-Light Decomposition is a powerful technique for efficiently handling path queries and updates on trees. Consider this problem: given a tree with values on nodes, we want to find the maximum value on the path between any two nodes. A naive approach would traverse the entire path, taking O(N) time per query. HLD allows us to answer such queries in O(log squared N) time by decomposing the tree into heavy and light paths, making it ideal for competitive programming problems involving dynamic tree operations. The core concept of Heavy-Light Decomposition is classifying tree edges into heavy and light categories. A heavy edge connects a node to its child with the largest subtree size, while all other edges are light edges. In our example tree, we first compute subtree sizes for each node. Node 1 has subtree size 7, node 2 has size 3, and so on. The heavy edges, shown in red, connect nodes to their heaviest children. This classification has a crucial property: any path from root to leaf crosses at most logarithmic number of light edges, which is the foundation for HLD's efficiency. Heavy edges form heavy paths or chains in the tree. The key insight is linearization: we perform a DFS traversal that assigns position indices to nodes such that nodes on the same heavy path get contiguous positions. In our example, the heavy path from node 1 to 2 to 5 gets positions 0, 1, and 2. The heavy path from 3 to 6 gets positions 3 and 4. This linearization maps the tree structure to a linear array, enabling us to use segment trees for efficient range queries and updates on heavy paths. The position array becomes the foundation for all HLD operations. The HLD implementation consists of two main DFS passes. The first DFS computes subtree sizes and identifies the heavy child for each node - the child with the largest subtree. The second DFS assigns position indices and builds heavy path chains by traversing heavy edges first, ensuring nodes on the same heavy path get contiguous positions. We then build a segment tree on the linearized array for efficient range operations. The time complexity is O(N) for preprocessing and O(log squared N) for each path query or update, making HLD highly efficient for dynamic tree problems in competitive programming. Let's trace a path query from node 4 to node 7. We start at node 4 and move up its heavy chain, querying the segment tree for the range from position of node 4 to position of node 2. Then we jump to the parent of the chain head, which is node 1. We continue this process, querying the path from 1 to 3, then from 3 to 7. Each jump crosses a light edge, and since there are at most logarithmic light edges on any path, we perform at most logarithmic segment tree queries, achieving O(log squared N) total complexity. This makes HLD extremely efficient for competitive programming problems involving dynamic tree path operations.