By Christopher Griffin

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We can assume the steps in Line 7 1Algorithm 6 is not optimal. 31 easily and we do not have to appeal to special data structures. 32 for more on this. 42 are free2. This means that for any iteration of the while loop, we will perform O(k(n − k)) operations. Thus, for the whole algorithm we will perform: n−1 O k=1 k(n − k) =O 1 3 1 n − n 3 6 Thus, the running time for Algorithm 6 is O(n3 ) = O(|V |3 ). 32. As it turns out, the implementation of Prim’s algorithm can have a substantial impact on the running time.

Investigating its neighbor set, we identify three vertices 2, 3 and 4 and the path length from Vertex 1 to each of these vertices is smaller than the initialized distance of ∞ and so these vertices are assigned a parent (p(v)) as Vertex 1 and the new distances are recorded. Vertex 1 is then removed from the set Q. In the second iteration, we see that Vertex 3 is closest to v0 (Vertex 1) and investigating its neighborhood, we see that the distance from Vertex 1 to 3 and then from 3 to 4 is 9 and smaller than the currently recorded distance of Vertex 1 to Vertex 4.

We can also build a spanning tree using a breadth first search on a graph. These algorithms are shown in Algorithms 3 and 4. Notice that instead of just appending vertices to w we also grow a tree that will eventually span the input graphs G (just in case G is connected). 11. 3. 4 Exercise 28: Show that a breadth first spanning tree returns a tree with the property that the walk from v0 to any other vertex has the smallest length. } Algorithm 3. } Recurse Input: G = (V, E) a graph, T = (V, E ) a tree, vnow current vertex, w the sequence (1) for each v ∈ N (vnow ) do (2) if v ∈ w then (3) Append v to w (4) Add {vnow , v} to E (5) Recurse(T, v, w) (6) end if (7) end for Algorithm 4.

### Graph and Network Algorithms [Lecture notes] by Christopher Griffin

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