Cycle Detection in a Directed Graph Using DFS

Introduction

• Detecting cycles in a directed graph using DFS algorithm.
• Explanation of directed graphs and cycles.

Understanding Directed Graphs

• Directed graph: Nodes connected by directed edges.
• Cycle: A path starting and ending at the same node.

Why Usual DFS Algorithm Fails

• The example of starting DFS from node 1 and visiting nodes sequentially.
• In directed graphs, usual DFS might falsely detect cycles.
  • Visiting nodes in a path is ambiguous in directed graphs.
  • Need to differentiate paths and cross-path visits.

Improved DFS Algorithm for Directed Graphs

• Using two arrays:
  1. Visited Array: Tracks all visited nodes.
  2. Path Visited Array: Tracks nodes in the current path.
• Adjacency list to represent the graph.

Steps to Implement the Algorithm

1. Initialization:
   • Create visited and path visited arrays of size equal to the number of graph nodes.
2. Component-wise Traversal:
   • Traverse the entire graph starting from each node.
3. DFS Call:
   • On visiting a node, mark it as visited and path-visited.
   • For each adjacent node:
     • If not visited, call DFS recursively.
     • If visited and path-visited, a cycle is detected.
     • Backtrack and unmark path-visited when returning.

Detailed DFS Implementation

• Code walkthrough of a component-wise DFS traversal.
• Checking conditions for visited and path-visited status.
• Detecting and handling cycles during traversal.
• Efficiently stopping traversal once a cycle is detected.

Loop through nodes

• Ensure no repeated work for already visited nodes.
• Focus on unvisited nodes for new DFS calls.

Example: Detailed Walkthrough

• Practical example with nodes 1 to 10.
• Explanation of path-tracing and detection logic.
• Handling adjacent nodes and backtracking.

Time and Space Complexity

• Time Complexity: O(V + E), where V and E are vertices and edges, respectively.
• Space Complexity: O(2N)
  • Note: Space can be optimized using a single array with different markers. (Homework)

Homework Challenge

• Optimize space by using a single visited array with distinct markers for visited and path-visited states.
• Think about how to implement this and share the solution.

Conclusion

• DFS algorithm successfully detects cycles in directed graphs.
• Component-wise traversal and separate path tracking ensure accuracy.
• Encourage viewers to implement and practice further.

Final Note

• Subscribing, liking, and exploring channel content (like DP and HDC series).
• Links to additional resources in the video description.

Sign-off

• Encouraging message to viewers.