This is basically tree diameter and I'd seen it before, so the DFS approach came pretty naturally.
Use a post-order DFS that returns the height of each subtree and tracks the maximum diameter (number of edges) among leaf nodes. At each node, combine the heights of its left and right subtrees to update the global maximum, but only consider paths that start and end at leaves.
Pro tip: Clarify that 'alive nodes' are leaves and that the path must be between two leaves, not any nodes. Mention that the algorithm should handle edge cases like a single leaf or a tree with only one leaf, where the answer is 0.
Confirm that alive nodes are leaves, and the path must be between two distinct leaves. Discuss edge cases: empty tree, single leaf, and tree with only one leaf.
Define a function that returns the height (max edges to a leaf) of the subtree rooted at the current node. For a leaf, return 0; for null, return -1 or a sentinel.
At each internal node, if both left and right subtrees contain at least one leaf, update the global maximum with left_height + right_height + 2. If only one side has leaves, propagate the height from that side.
If a node has only one child that contains leaves, return that child's height + 1. This ensures the height correctly represents the distance to the nearest leaf in that subtree.
Explain that the DFS visits each node once, giving O(n) time, and the recursion stack uses O(h) space. Return the global maximum as the answer.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Use a post-order traversal to compute, for each node, the maximum distance from that node to any alive node in its subtree. At each node, combine the two largest such distances from different children to update the global maximum, and return the maximum distance (or 0 if the node itself is alive) to the parent.
Pro tip: Clarify edge cases upfront: if there are fewer than two alive nodes, the answer is 0; also confirm whether the distance is measured in edges or nodes, as this affects the base case and final result.
Ask about the definition of distance (edges vs. nodes), whether the tree is binary or general, and what to return if there are 0 or 1 alive nodes.
For each node, compute the maximum distance from that node to any alive node in its subtree. If no alive node exists, return a sentinel like -1.
Recursively process children, collect their returned distances, and at the current node, consider the two largest distances from different children to form a path through the node.
Maintain a global variable for the maximum distance found so far. At each node, update it with the sum of the two largest child distances (if both exist) and also consider the node itself if alive.
Return the maximum distance from the current node to an alive node in its subtree (including itself if alive) to the parent. After traversal, the global maximum is the answer.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Yeah this is where things fell apart a bit.
Clarify the problem constraints (tree vs graph, number of nodes, update frequency) and discuss trade-offs between recomputation and dynamic data structures. Propose a solution using tree diameter properties (e.g., maintaining farthest nodes via LCA and segment trees) or a dynamic programming approach with lazy updates. Outline the algorithm, complexity, and potential optimizations for online updates.
Pro tip: Emphasize that in a tree, the diameter endpoints are always among the farthest nodes from any node, and you can maintain them using two BFS/DFS passes; for dynamic updates, consider using a segment tree over Euler tour to query farthest alive node quickly.
Ask about the graph type (tree vs general graph), number of nodes, frequency of updates, and whether updates are online. Confirm if the distance metric is edge count or weighted.
Mention that recomputing diameter after each update via BFS/DFS is O(N) per update, which may be too slow. Propose maintaining the diameter endpoints dynamically using data structures like segment trees or balanced BSTs.
For a tree, maintain the set of alive nodes and the current diameter endpoints. When a node toggles, update the set and recompute the diameter by checking distances from the new node to existing endpoints and between endpoints. Use LCA for O(log N) distance queries.
State that each update takes O(log N) time with O(N log N) preprocessing, or O(1) if using a different structure. Discuss memory vs time trade-offs and scalability.
Consider cases with 0 or 1 alive node, all nodes dead, or multiple components. If the graph is not a tree, mention that the problem becomes harder (e.g., dynamic graph diameter) and may require approximation or different techniques.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.