22.1. Applications of BFS and DFS

We have seen how to use Breadth First and Depth First search to explore whether there is a path from a source to a target vertex. But one can do a lot more with these two procedures.

So, recall that a graph is a set of vertices and a set of edges which have connections between the vertices, and these maybe directed or undirected.

In this chunk, we observe the foundational concepts of graphs and the basic search techniques, Breadth First Search (BFS) and Depth First Search (DFS). A graph is made up of nodes (vertices) connected by edges. BFS explores the graph layer by layer, starting from a given node, visiting all its adjacent nodes before moving on to the next level. In contrast, DFS explores as far down a branch as possible before backing up to explore other branches. This initial understanding sets the stage for exploring more complex graph properties.

Think of a city map as a graph where intersections are vertices and the roads connecting them are the edges. When using BFS, it’s like checking all the streets at one intersection before moving to the next. With DFS, it’s akin to taking one road until you reach the end, and then deciding whether to explore a new street from where you started.

So, one fundamental property of undirected graph is whether or not disconnected, can we go from every vertex to every other vertex. ... We want to grouped together those vertices which are connected to each other into one unit and find out which of these units are there and how many such units are there.

This chunk introduces the idea of connected components in a graph. A connected component is a subset of a graph where there is a path between any two vertices. To identify these components using BFS or DFS, we can start at one unvisited vertex, marking all reachable vertices as part of the same component. Once all vertices reachable from a starting vertex are marked, we then look for another unvisited vertex to start a new search, continuing until all vertices are explored.

Imagine a group of friends at a party. Each friend represents a vertex, and a direct friendship represents an edge. Some friends might know each other, forming a clique while others don’t. By starting with a single friend (vertex) and finding out all who they know (connected friends), you identify one social circle (connected component). Repeating this process until you’ve accounted for all friends allows you to map out different social circles at the party.

And other interesting structural property of a graph is whether or not it has cycles. ... So, having a cycle is equivalent to finding a non-tree edge while doing BFS.

This chunk explores how both BFS and DFS can be used to detect cycles in graphs. A cycle occurs when you can start at a vertex and return back to it through a path without retracing any edges. During BFS or DFS, if you encounter an edge leading to a vertex that was already visited (and still reachable in the context of the search), it implies the presence of a cycle. Essentially, you can use these search strategies not just to explore connectivity, but also to reveal intricate structures like cycles.

Consider a circular track in an amusement park. If you start running and end up back where you began without taking a different path, you've formed a cycle. In a graph, identifying this kind of loop through BFS or DFS helps reveal similar loop structures in networks or relationships.

So, let us do the same thing, but DFS and let us compute the pre and post numbers. ... so these will again beet edges which are outside the tree.

In this chunk, we introduce pre and post numbering in the context of DFS to classify different types of edges in a graph. As DFS traverses a graph, it assigns a 'pre' number when it first encounters a vertex and a 'post' number once it finishes exploring all its neighbors. These numbers help in distinguishing between tree edges (part of the DFS tree), back edges (leading to ancestors in the DFS), forward edges, and cross edges in directed graphs. Understanding these edge classifications is crucial for analyzing graph properties.

Think of a journalist documenting a city’s neighborhoods. When they first enter a neighborhood, they note it down (pre number) and once they finish reporting on it, they make another note (post number). An edge connecting two neighborhoods that have been documented tells us which ones are directly connected (tree edge), whereas if a journalist encounters an already documented neighborhood, it can tell us about previously existing paths (back edge).

So, we have seen some concrete examples ... design more efficient procedures or to identify other things that need to be done.

This chunk discusses the practical applications of BFS and DFS in identifying critical components like articulation points and bridges in graphs. An articulation point is a vertex that, when removed, increases the number of connected components in the graph. Identifying these critical points allows us to understand vulnerabilities in networks (like communication, transportation, etc.), where the failure of one component could disrupt connectivity. BFS and DFS provide efficient methods to uncover these vital features within graph structures.

Consider a city's infrastructure: the intersections where major roads meet can be thought of as articulation points. If one of these intersections is blocked (vertex removed), traffic flow to certain areas might be severed. Understanding where these key points are enables urban planners to fortify and optimize the city's layout to avoid disruption.

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