Cycle Detection

Imagine sending a package that keeps getting forwarded between the same three cities forever. That’s a cycle. Detecting these infinite loops is critical for preventing system crashes, deadlocks, and infinite recursion.

Cycle Detection is an algorithmic technique used to determine if a data structure (like a linked list or graph) contains a path that eventually repeats a node. It prevents programs from getting stuck in infinite loops and is essential for validating dependencies.

Real-World Analogy

The Race Track: Imagine two runners on a race track.

• If the track is a straight line, the faster runner will simply finish the race before the slower one. They will never meet again after the start.
• If the track is circular (contains a cycle), the faster runner will eventually “lap” or catch up to the slower runner from behind.

In the Tortoise and Hare algorithm (Floyd’s Cycle Finding), we use this principle. We have two pointers moving at different speeds. If they meet, there is a cycle. If the fast pointer reaches the end, there is no cycle.

Visual Explanation

Here is how the fast and slow pointers move in a structure with a loop:

  graph LR
    subgraph Iteration 1
    A((1))-->B((2))
    B-->C((3))
    C-->D((4))
    D-->E((5))
    E-->C
    style A fill:#f9f,stroke:#333
    style A stroke-width:4px
    end

In a cycle detection algorithm like DFS, we track the state of each node to see if we visit a node that is currently in our recursion stack.

  stateDiagram-v2
    direction LR
    [*] --> Unvisited
    Unvisited --> Visiting: Start DFS
    Visiting --> Visited: Finish Neighbors
    Visiting --> Visiting: Cycle Detected!
    Visited --> [*]

When to Use This Pattern

Use Cycle Detection when your problem involves:

• Infinite Loops: You need to check if a process will terminate.
• Dependencies: You verify if a dependency graph (like a build system) is valid (DAG).
• Resource Allocation: You need to detect deadlocks in an operating system.
• Linked List Validation: You need to check if a linked list is corrupted with a loop.
• Path Checking: Determining if a path leads back to a previous state.

Complexity Analysis

• Time: Most cycle detection algorithms visit each node and edge at most once, resulting in linear time relative to the size of the graph or list.
• Space:
  • Floyd’s: Uses only two pointers, so it is constant space.
  • DFS/BFS: Requires storage for visited sets or recursion stacks, proportional to the number of vertices.

Common Mistakes

• Incorrect Base Cases: Failing to handle empty lists or single nodes correctly.
• Pointer Initialization: Starting slow and fast pointers at the wrong positions (e.g., both at null).
• Recursion Depth: For deep graphs, DFS might cause a stack overflow; iterative approaches or Morris Traversal might be needed.
• Confusing Cycles with Duplicates: A value appearing twice is not a cycle; the same node reference appearing twice in a path is a cycle.

Next Steps

• Code Templates: Get copy-paste ready code for Floyd’s Algorithm, DFS, and Union-Find in 6 languages.
• Practice Problems: Solve rated LeetCode problems to test your knowledge.