CLRS pseudocode:

   for each vertex u in G.V - {s}, do
      u.color := white
      u.d := infinity
      u.p := NIL
   s.color := green
   s.d := 0
   s.p := NIL
   Q := NULL
   insert s into Q
   while Q is not null, do
      u = delete from Q
      for each v in adjacent to u, do
         if v.color = white
            v.color := green
            v.d := u.d + 1
            v.p := u
            insert v into Q
         end if
      u.color = dark_green

In the clrs it says that the insertion and deletion operations with the queue require an O (1) time, so the total time of operations with the queue is O (V). (V because each vertex is visited at least once). instead the time to inspect the adjacency list of each node is O (E). Then we have another O (V) for initialization. So the total time is O (V + E) which in the worst case becomes O (V ^ 2), this depends on whether the graph is dense or not.

Now my question is: Since we have two cycles nested the complexity shouldn't be O (V * E), I can't understand why a sum operation is performed instead of multiplying.

  • $\begingroup$ Imaging node $i$ has out-degree $d_i$ (the number of neighbors). Now you'll agree that since every node is visited only once we have for both loops combined: $\sum_{i=1}^{n} (1 + d_i)$ ($1$ is just the constant cost, $d_i$ the cost of the inner loop). So what does $\sum_{i = 1}^n d_i$ evaluate to? $\endgroup$
    – plshelp
    Oct 9, 2022 at 18:12
  • $\begingroup$ di should evaluate all edges so it should cost O (E) with E varying between 1 and V ^ 2 depending on the density of the graph. (perhaps on the summation you should use V instead of n.) $\endgroup$
    – mimmolg99
    Oct 9, 2022 at 18:56

1 Answer 1


I will suppose here that the graph is undirected, but the same reasonning can be done with a digraph.

Using an implemention of a graph as an array of adjacency lists, the adjacency list of any vertex $v$ is its number of neighbors, $\deg(v)$.

The BFS inspects each adjacency list at most once during the graph traversal (and exactly once if the graph is connected). That means that the total complexity is: $$\mathcal{O}\left(\sum\limits_{v\in V}(\deg(v) + 1)\right)$$ Here, the $+1$ is necessary, as inspecting an adjacency list costs $\mathcal{O}(1)$ time, even if the list is empty. Now, a very useful formula for complexity on graphs is: $$\sum\limits_{v\in V}\deg(v) = 2|E|$$ It can be proven quite easily when noting that each edge increases the sum of degrees by two (one for each extremity).

Finaly, we conclude that the complexity is: $$\mathcal{O}\left(\sum\limits_{v\in V}(\deg(v) + 1)\right) = \mathcal{O}(2|E| + |V|) = \mathcal{O}(|E| + |V|)$$

  • $\begingroup$ all very clear thanks for the explanation $\endgroup$
    – mimmolg99
    Oct 9, 2022 at 19:41
  • $\begingroup$ @mimmolg99 If you want to be more precise, the complexity is $\mathcal{O}(|E_C| + |V|)$, where $E_C$ is the set of edges of the connected component $C$ of the vertex $s$. $\endgroup$
    – Nathaniel
    Oct 9, 2022 at 19:46
  • $\begingroup$ it is correct to say that in the best case then the complexity will be O (V) in the worst case it will be O (V ^ 2) and in the average case O (E + V). In the best case I think it is O (V) because E should be 1 while in the worst case it should be V ^ 2. what do you think? $\endgroup$
    – mimmolg99
    Oct 9, 2022 at 19:52
  • $\begingroup$ While it is true that the best case is $\mathcal{O}(|V|)$ and the worst case is $\Theta(|V|^2)$, you can't really say that the average case is $\mathcal{O}(|E| + |V|)$ because 1) it is always $\mathcal{O}(|E| + |V|)$. 2) average complexity implies that you consider a certain random distribution of graphs (and with the uniform distribution over all graphs with $|V|$ vertices, the average number of edges is $\Theta(|V|^2)$). $\endgroup$
    – Nathaniel
    Oct 9, 2022 at 20:01
  • $\begingroup$ I understand.. thanks 🙏 $\endgroup$
    – mimmolg99
    Oct 9, 2022 at 20:21

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