CPSC 220
Graph Exercises

Consider the directed weighted graph G with vertex and edge sets below:

  V = {A,B,C,D,E,F,G}

  E = {(A,B,10),
       (A,D,20),
       (B,C,2),
       (B,D,6),
       (B,F,6),
       (B,G,8),
       (C,E,8),
       (D,F,5),
       (D,G,1)
       (G,E,1)
       (G,F,3)}

1. Suppose we are considering adding edges to or removing edges from G.
   1. For the given size of V, what is the maximum possible size of E (assuming no self-loops)?
   2. For the given size of V, what is the minimum possible size of E?

2. Assume that an integer takes one cell and a pointer takes two cells, and it is not necessary to store the actual label on each vertex,
   1. How much space would it take to store G using
      1. an adjacency list?
      2. an adjacency matrix?
   2. Develop a general formula in terms of |V| and |E| for the space required for each representation under these assumptions. Then evaluate the space tradeoffs when
      1. |E| = |V|
      2. |E| = |V|²

   For questions 3-7, assume that "for each adjacent vertex" in any relevant algorithm takes the vertices in alphabetical order.

3. Show the order in which the vertices will be visited in a breadth-first search from A.

4. Show the order in which the vertices will be visited in a depth-first search from A.

5. Find a topological sort of the vertices using
   1. a post-visit depth-first search.
   2. the queue-based method.

6. Use Dijkstra's algorithm to find the shortest route and distance from A to every other vertex.

7. Viewing this as an undirected graph, find a minimum spanning tree using
   1. Prim's algorithm
   2. Kruskal's algorithm