Graphs and Breadth-First Search

Graphs are an essential data structure in computer science that can be used to model real-world problems. They consist of a set of vertices (or nodes) and edges that connect these vertices. Graphs can be used to model complex relationships between objects, such as roads between cities, social connections between people, or relationships between web pages.

In this post, we'll be discussing a specific type of graph called a bipartite graph, and how the Breadth-First Search (BFS) algorithm can be used to determine whether a given graph is bipartite or not.

What is a Bipartite Graph?

A bipartite graph is a type of graph where vertices can be split into two disjoint sets, such that all edges connect vertices from different sets. In other words, vertices in a bipartite graph cannot have edges to other vertices in the same set.

A real-world example of a bipartite graph could be a job matching system, where one set of vertices represents job seekers and the other set represents job openings. An edge would represent a match between a job seeker and a job opening. In this example, it would not make sense for a job seeker to have an edge connecting to another job seeker, or for a job opening to have an edge connecting to another job opening.

Breadth-First Search (BFS)

The Breadth-First Search algorithm is a graph traversal algorithm that visits all vertices in a graph by exploring the edges in a breadth-first manner. In other words, the algorithm starts at an arbitrary vertex and visits its neighbors before visiting its neighbor's neighbors, and so on.

BFS is a useful algorithm for a variety of tasks, such as finding the shortest path between two vertices, checking for connectivity in a graph, and determining whether a graph is bipartite or not.

Using BFS to Determine if a Graph is Bipartite

The idea behind using BFS to determine whether a graph is bipartite is to assign a color to each vertex and check that no two adjacent vertices have the same color. In a bipartite graph, vertices in one set should have one color, and vertices in the other set should have another color.

Here is an example implementation of the BFS algorithm to determine if a given graph is bipartite in Python:

collections import defaultdict

def is_bipartite(graph):
    color = defaultdict(int)
    queue = []

    for node in graph:
        if color[node] == 0:
            color[node] = 1
            queue.append(node)

            while queue:
                current = queue.pop(0)
                for neighbor in graph[current]:
                    if color[neighbor] == color[current]:
                        return False
                    if color[neighbor] == 0:
                        color[neighbor] = 1 - color[current]
                        queue.append(neighbor)
    return True

graph = {
  0: [1, 3],
  1: [0, 2],
  2: [1, 3],
  3: [0, 2]
}

print(is_bipartite(graph)) # True

In this example, the is_bipartite function takes a graph represented as an adjacency list as an input and returns a boolean indicating whether the graph is bipartite or not. The color dictionary is used to store the color assigned to each vertex, with 0 representing uncolored vertices. The queue is used to store the vertices that need to be colored.

The algorithm starts by choosing an arbitrary node from the graph and assigning it a color of 1. It then adds the node to the queue and enters a loop that continues until the queue is empty. In each iteration of the loop, the first node in the queue is popped and its neighbors are colored with a color that is different from its own. If any adjacent vertices have the same color, the algorithm immediately returns False, indicating that the graph is not bipartite.

In the example above, the graph is a bipartite graph and the is_bipartite function returns True.

Conclusion

Bipartite graphs are a specific type of graph where vertices can be split into two disjoint sets, and edges only connect vertices from different sets. The Breadth-First Search algorithm can be used to determine whether a given graph is bipartite by assigning colors to vertices and checking that no two adjacent vertices have the same color. This can be useful in various real-world problems, such as job matching systems, where a bipartite graph can model the relationships between job seekers and job openings.

I hope you found this overview of bipartite graphs and the BFS algorithm useful. If you have any questions or comments, please leave them in the comments section below. Happy coding! 💻🚀