Find nearest common manager for multiple employees

Given an organizational hierarchy modeled as a rooted tree, build the tree from (employee, manager) pairs and implement a function that, for k ≥ 2 employees, returns their nearest common organization node (i.e., the lowest common ancestor for multiple nodes). Specify and implement APIs to construct the tree, validate input (detect cycles/disconnected components), and compute LCA for a set of nodes efficiently. Discuss and implement approaches for preprocessing (e.g., parent pointers with depths, binary lifting, or Euler tour + RMQ) and analyze time/space complexity. Include clear unit tests for typical and edge cases (employees in different subtrees, missing employees, and the root case).