What are some examples of algorithms that use breadth-first search?

The BFS algorithm systematically explores the edges of a graph G to "discover" each vertex that can be reached from a vertex "s". The algorithm calculates the distance (the shortest number of edges) from s to each reachable vertex. It also produces a "breadth-first search tree" with source "s" that contain all reachable vertices. For any vertex "v" that can be reached from "s", the simple path in the breadth-first search tree from the origin "s" to "v" corresponds to a "shortest path" from "s" to "v" in the graph "G", i.e., a path that contains the smallest number of edges.

Breadth-first search (BFS) is a versatile algorithm used in various applications: 1. Shortest path finding: BFS methodically explores neighbors to find the closest path between two points. 2. Graph traversal: BFS visits all nodes level-by-level, providing a structured exploration of the graph. 3. Minimum spanning tree: BFS helps find the cheapest way to connect all nodes in a graph. 4. Network routing: BFS guides data packets by finding the closest next hop in the network. 5. Web crawling: BFS explores the web breadth-first, discovering new pages from existing ones.

Embarking on the quest for the shortest path in graphs, we encounter two powerful algorithms: Breadth-First Search (BFS) and Dijkstra's. BFS, a beginner-friendly tool, efficiently handles unweighted graphs. Starting at a source node, we mark each visited node and explore neighbors until reaching the destination. In contrast, Dijkstra's algorithm takes center stage in weighted graphs. Dealing with varying edge weights, it meticulously computes optimal paths from a single source to all nodes. While BFS lays the foundation, Dijkstra's equips us for more complex terrains.

The breadth-first search (BFS) procedure assumes that the input graph G = (V, E) is represented using adjacency lists. It attaches several additional attributes to each vertex in the graph. We store whether each vertex has been visited or not for the vertices contained in the set of vertices of the graph "G" as well as its predecessor. If a vertex "u" has no predecessor or is equal to the source itself (for example, if u = s or if u has not been discovered), then the vertex's predecessor is set to NULL. An attribute of the currently explored vertex maintains the distance from the source "s" to the vertex "u" calculated by the algorithm. The algorithm also uses a first-in, first-out (FIFO) queue to manage the set of vertices already visited.

After initialization, breadth-first search never marks vertices in advance, ensuring that each vertex is placed in the queue at most once and therefore removed from the queue at most once. Hence, if any vertex "u" can be reached from the source vertex, the same holds for another vertex "v". In this case, the shortest path from "s" to "v" cannot be longer than the shortest path from "s" to "u" followed by the edge (u, v), and thus the inequality holds. If "u" cannot be reached from "s", then the distance from vertex "s" to vertex "u" is initially infinite, and the inequality is valid.

The enqueue and dequeue operations take O(1) time, and thus, the total time dedicated to queue operations is O(V). Since the procedure traverses the adjacency list of each vertex only when the vertex is dequeued, it traverses each adjacency list at most once. Given that the sum of the lengths of all adjacency lists is Θ(E), the total time spent scanning the adjacency lists is O(E). The initialization overhead is O(V), and therefore, the total runtime of the BFS procedure is O(V + E). Thus, breadth-first search is executed in linear time concerning the size of the adjacency list representation of G.