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we consider a graph G= (V, E), with n=|V| and m=|E|. Describe an O(n+m) time algorithm...

we consider a graph G= (V, E), with n=|V| and m=|E|.

Describe an O(n+m) time algorithm to find such a vertex w. Hint: a depth-first search from u might be helpful.

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Answer #1

A graph G=(V,E), with n=|v| and m=|E|

If the time complexity is O(n+m) then this is depth- first-search because in depth-first-search (DFS)

Recursively explore graph, backtracking as necessary. In DFS stack data structure is used.

  • Visit each vertex once in DFS alone that's why its O(v)
  • DFS (V1, V2...., Vn) called at most once per vertex . That's why we have to pay | adj[V] |

therefore, O( summation of |adj [V]|)= O(E)

That's why we can say the time complexity  of DFS is O(V+E) or O(m+n)

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