Why is the time complexity of both DFS and BFS O(

Question

The basic algorithm for BFS:

set start vertex to visited

load it into queue

while queue not empty

   for each edge incident to vertex

        if its not visited

            load into queue

            mark vertex

So I would think the time complexity would be:

v1 + (incident edges) + v2 + (incident edges) + .... + vn + (incident edges) 

where v is vertex 1 to n

Firstly, is what I've said correct? Secondly, how is this O(N + E), and intuition as to why would be really nice. Thanks

Answer 1

Time complexity is O(E+V) instead of O(2E+V) because if the time complexity is n^2+2n+7 then it is written as O(n^2).

Hence, O(2E+V) is written as O(E+V)

because difference between n^2 and n matters but not between n and 2n.

Answer 2

I think every edge has been considered twice and every node has been visited once, so the total time complexity should be O(2E+V).

Answer 3

Very simplified without much formality: every edge is considered exactly twice, and every node is processed exactly once, so the complexity has to be a constant multiple of the number of edges as well as the number of vertices.

Answer 4

An intuitive explanation to this is by simply analysing a single loop:

1. visit a vertex -> O(1)
2. a for loop on all the incident edges -> O(e) where e is a number of edges incident on a given vertex v.

So the total time for a single loop is O(1)+O(e). Now sum it for each vertex as each vertex is visited once. This gives

Sigma_i

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<p>
    <span>&Sigma;</span>
    O(1) + O(e)
=> 
    <span>&Sigma;</span>
    O(1)
    +
   <span>&Sigma;</span>
    O(e)

=> O(V) + O(E)

</p>

[ O(1) + O(e)]

Answer 5

DFS(analysis):

• Setting/getting a vertex/edge label takes O(1) time
• Each vertex is labeled twice
  • once as UNEXPLORED
  • once as VISITED
• Each edge is labeled twice
  • once as UNEXPLORED
  • once as DISCOVERY or BACK
• Method incidentEdges is called once for each vertex
• DFS runs in O(n + m) time provided the graph is represented by the adjacency list structure
• Recall that Σv deg(v) = 2m

BFS(analysis):

• Setting/getting a vertex/edge label takes O(1) time
• Each vertex is labeled twice
  • once as UNEXPLORED
  • once as VISITED
• Each edge is labeled twice
  • once as UNEXPLORED
  • once as DISCOVERY or CROSS
• Each vertex is inserted once into a sequence Li
• Method incidentEdges is called once for each vertex
• BFS runs in O(n + m) time provided the graph is represented by the adjacency list structure
• Recall that Σv deg(v) = 2m

Answer 6

Short but simple explanation:

I the worst case you would need to visit all the vertex and edge hence the time complexity in the worst case is O(V+E)