What is time complexity of BFS depending on the representation of the graph?

Question

I was wondering what is the time complexity of BFS, if I use:

• an adjacency matrix
• adjacency list
• edge list

Is it same as their spa

Answer 1

Time complexities for different representations of Graph:

1. Edge List:

Edge list consists of all the edges in a list. In order to do the BFS time complexity is O(E^2).Because for every edge u->v, you have to traverse through entire edge list and find the edges whose source vertex is u and explore them, then explore the vertices 'v' which are in u->v to do the BFS.

Where E is the number of edges.

If you sort the edges based on source index and destination index, then the sorted list will be in BFS order. Just traverse the list, you will get the BFS.

Time complexity is O(E*log(E)) for sorting the edge list.

2. Adjacency List

Adjacency is a map of keys, where every vertex is a key and points to a list of vertices which are incident from or adjacent to that key vertex.

In order to perform BFS, put any vertex in the queue and make it as visited, pop the queue[0], pick the starting vertex, explore all its adjacent vertexes, make them as visited and put them in the queue and similarly pop the queue[0] and explore all the non-visited vertices until the queue becomes empty. For every vertex, you are traversing through only its adjacent non-visited vertexes(nothing but edges).

So, the time complexity is O(V+E)

3. Matrix

In matrix representation, for every vertex, you have to traverse through all the vertices and check whether there is a non-visited vertex. Since, for every vertex, we are traversing through all the vertices,

The time complexity is O(V^2)