Clarify the locking semantics and constraints, then propose an efficient solution using a combination of ancestor tracking and descendant checks. Use a data structure like a segment tree or binary indexed tree to maintain lock status and support O(log N) operations. Discuss trade-offs between time and space, and consider edge cases like locking the root or a leaf.
Pro tip: Emphasize that the key challenge is efficiently checking both ancestors and descendants; using a single data structure like a Fenwick tree with Euler tour can handle both in O(log N). Also, mention that you'd confirm whether concurrent locks are needed, as that would require synchronization.
Ask about the tree structure (static or dynamic), locking semantics (exclusive locks, reentrancy), and performance constraints. Confirm that lock(x) fails if any ancestor or descendant is locked.
Propose maintaining a lock count for each node and using a data structure to quickly query if any ancestor or descendant is locked. For ancestors, store the nearest locked ancestor; for descendants, use a segment tree over Euler tour to check if any descendant is locked.
For lock(x): check if x is already locked, if any ancestor is locked (via parent pointers or a separate structure), and if any descendant is locked (via segment tree query). If all clear, mark x as locked and update structures. For unlock(x): reverse the updates. For isLocked(x): return the lock status.
Show that each operation runs in O(log N) time: ancestor checks can be O(log N) with binary lifting or O(1) with a maintained nearest locked ancestor; descendant checks via segment tree are O(log N). Space is O(N).
Mention alternative approaches like using a single Fenwick tree with Euler tour for both ancestor and descendant checks, or using a balanced BST. Discuss handling concurrent locks with mutexes or read-write locks if needed.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.