Breadth First Search | Graph Traversals | DAA

Graph Traversals

There are a number of approaches used for solving problems on graphs. One of the most important approaches is based on the notion of systematically visiting all the vertices and edges of a graph. The reason for this is that these traversals impose a type of tree structure (or generally a forest) on the graph, and trees are usually much easier to reason about than general graphs.

Breadth-first search

This is one of the simplest methods of graph searching. Choose some vertex arbitrarily as a root. Add all the vertices and edges that are incident in the root. The new vertices added will become the vertices at level 1 of the BFS tree. Form the set of the added vertices of level 1, and find other vertices, such that they are connected by edges at level 1 vertices. Follow the above step until all the vertices are added.

Algorithm:

BFS(G, s) // s is start vertex
{
    T = {s};
    L = Φ; // an empty queue
    Enqueue(L, s);
    while (L != Φ)
    {
        v = dequeue(L);
for each
    neighbor w to v
        if (w Ï L and w Ï T)
    {
        enqueue(L, w);
        T = T U{w}; // put edge {v,w} also
    }
    }
}

Example: Use breadth-first search to find a BFS tree of the following graph.

Analysis

From the algorithm above all the vertices are put once in the queue and they are accessed. For each accessed vertex from the queue, their adjacent vertices are looked for and this can be done in O(n) time(for the worst case the graph is complete). This computation for all the possible vertices that may be in the queue i.e. n, produce the complexity of an algorithm as O(n2 ). Also from the aggregate analysis, we can write the complexity as O(E+V) because the inner loop executes E times in total

Depth-first search