Understanding Breadth First Traversal
Breadth First Traversal (BFS) is a fundamental algorithm used to search and traverse graphs or tree data structures. It is a recursive approach that categorizes each vertex into one of two groups: visited or not visited. The primary goal of BFS is to mark each vertex as visited while avoiding cycles.
How BFS Works
- Initialization: Choose any vertex from the graph and place it at the end of a queue.
- Visitation: Take the front item from the queue and add it to the visited list.
- Exploration: Create a list of adjacent nodes for the current vertex. Add the unvisited nodes to the end of the queue.
- Repeat: Continue steps 2 and 3 until the queue is empty.
Example Walkthrough
Consider an undirected graph with five vertices. We start from vertex 0.
- Vertex 0 is added to the visited list, and its adjacent vertices are placed in the queue.
- We visit the front element of the queue (vertex 1) and explore its adjacent nodes. Since vertex 0 has already been visited, we move on to vertex 2.
- Vertex 2 has an unvisited adjacent node (vertex 4), which is added to the queue. We then visit vertex 3, which is at the front of the queue.
- Only vertex 4 remains in the queue, as the only adjacent node of vertex 3 (vertex 0) has already been visited. We visit vertex 4.
- With the queue now empty, we have completed the Breadth First Traversal of the graph.
Pseudocode and Example Code
The code for the Breadth First Search Algorithm is provided below in Python, Java, and C/C++.
Time and Space Complexity
The time complexity of the BFS algorithm is O(V + E), where V represents the number of nodes and E represents the number of edges. The space complexity is O(V).
Applications of BFS
Breadth First Search has various applications:
- Building indexes by search index
- GPS navigation
- Pathfinding algorithms
- Ford-Fulkerson algorithm to find maximum flow in a network
- Cycle detection in an undirected graph
- Minimum spanning tree