breadthfirst

Breadthfirst, commonly referred to as breadth-first search (BFS), is a graph traversal algorithm that explores the graph level by level. It starts at a chosen source vertex and visits all neighbors at distance one, then those at distance two, and so on. This systematic expansion makes BFS useful for discovering shortest paths in unweighted graphs and for level-order traversals of trees.

Implementation typically uses a queue to manage the frontier. The algorithm initializes by enqueuing the start vertex and marking it as visited. While the queue is not empty, it dequeues a vertex v, processes it, and for each neighbor w of v, if w has not been visited, it marks w as visited and enqueues w. In unweighted graphs, this ensures vertices are discovered in order of increasing distance from the start vertex.

Complexity considerations: BFS runs in O(V+E) time, where V is the number of vertices and E the number of edges, and requires O(V) space for the queue and visited markers. It works on both directed and undirected graphs and on trees, assuming an adjacency representation.

Key properties include the ability to find the shortest path (in terms of the number of edges) from the start vertex to any reachable vertex in an unweighted graph, and to reveal the graph’s layering structure. Variants such as bidirectional BFS can speed up searches from both ends when a target is known. Common applications span pathfinding, network broadcasting, web crawling, and solving mazes. Limitations include higher memory usage on very large graphs and the lack of direct applicability to weighted graphs, where algorithms like Dijkstra’s are preferred.