The setup clicked pretty fast but the greedy part took me a minute.
Model the problem as a tree where each node has a depth from the CEO. Use a greedy bottom-up approach: for each node, compute the maximum depth in its subtree; if it exceeds h, promote the node itself (or an ancestor) to cut the depth. The goal is to minimize promotions while ensuring all leaves are within depth h.
Pro tip: Clarify whether promotions can be applied to any employee or only to managers, and whether the tree structure changes after promotion. Also, discuss the trade-off between greedy and dynamic programming approaches, showing awareness of optimality.
Ask about constraints: tree size, h value, whether promotions can be repeated, and if the CEO's direct reports count as depth 1. Confirm that promoting an employee means they and their subtree move up.
Define depth as number of edges from CEO. Promoting a node reduces the depth of its entire subtree by the difference between its current depth and 1 (or 0 if CEO is depth 0).
Traverse the tree post-order. For each node, compute the maximum depth of any leaf in its subtree. If that depth exceeds h, promote the node (or an ancestor) to cut the subtree. Count promotions.
Argue that promoting the highest possible node that still keeps the subtree within h is optimal, as it covers the most nodes with one promotion. Use exchange argument.
The algorithm runs in O(n) time and O(n) space for a tree with n nodes, using a single DFS. Mention that if h is small, we might need multiple promotions per path.
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