What is Lowest Common Ancestor (LCA)?

The Lowest Common Ancestor (LCA) of two nodes in a tree is the deepest (i.e., lowest) node that is an ancestor of both nodes. In other words, it is the shared ancestor of the two nodes that is located farthest from the root. The concept is fundamental in computer science, especially in the context of tree data structures, such as binary trees, general trees, and state machines.

For example, given two nodes in a tree, their LCA is the node that appears on the path from the root to both nodes and is the last such node before the paths to the two nodes diverge. The LCA is useful in various applications, including finding relationships in hierarchical data, optimizing queries in databases, and analyzing state transitions in finite state machines.

Real-World Uses of LCA

The concept of Lowest Common Ancestor is widely used in real-world applications where hierarchical relationships exist. Here are a few examples:

1. Filesystem Navigation

In operating systems, files and directories are organized in a tree structure. To find the closest shared directory between two files, the LCA algorithm is used. For example, if you want to find the shortest path to move from one file to another, you first find their LCA (the common parent directory), then traverse up and down the tree accordingly.

Example:

Suppose you have the following directory structure:

/home/user/
├── documents/
│   ├── work/
│   │   └── report.docx
│   └── personal/
│       └── diary.txt

To find the shortest path from report.docx to diary.txt, the LCA is the documents directory. The path is:

report.docx → work → documents → personal → diary.txt

2. Organizational Hierarchies

In companies, employees are often organized in a management tree. To find the closest common manager between two employees, the LCA algorithm is used. This is useful for determining reporting structures or resolving permissions.

3. State Machines

In finite state machines, states can be organized hierarchically. When transitioning between two states, the LCA helps determine the minimal set of exit and entry actions required, optimizing the transition process.

4. Human Daily Activity FSM: LCA in Action

Let’s model a daily routine as a hierarchical finite state machine (FSM):

High-Level States

  • Sleeping
  • Working
  • Relaxing

Nested States (under Working)

  • Focused Work
  • Meeting
  • On Break

State Hierarchy (Tree View)

HumanDay
├── Sleeping
├── Working        ← LCA
│   ├── FocusedWork
│   ├── Meeting
│   └── OnBreak
└── Relaxing

LCA Example 1: Transition Within Working

Suppose you’re coding (FocusedWork) and a meeting starts. You transition from FocusedWork to Meeting:

Transition: FocusedWork → Meeting LCA: Working

Internal Actions:

  • Exit FocusedWork
  • Stay inside Working (LCA)
  • Enter Meeting

No need to exit Working since both states share it as their lowest common ancestor.

LCA Example 2: Another Transition Within Working

Transition from Meeting to OnBreak (e.g., lunch time):

Transition: Meeting → OnBreak LCA: Working

Internal Actions:

  • Exit Meeting
  • Enter OnBreak
  • Working remains active

LCA Example 3: Transition Across Top-Level States

Transition from FocusedWork (nested under Working) to Sleeping (top-level):

Transition: FocusedWork → Sleeping LCA: HumanDay (root)

Internal Actions:

  • Exit FocusedWork
  • Exit Working
  • Enter Sleeping

Here, you must exit all nested states up to the root before entering the new top-level state.

These examples show how LCA helps minimize unnecessary exits and entries, making transitions efficient in both human routines and software state machines.