## Description

1. Execute Primâ€™s minimum spanning tree algorithm by hand on the graph below showing

how the data structures evolve specifically indicating when the distance from a fringe

vertex to the tree is updated. Clearly indicate which edges become part of the minimum

spanning tree and in which order. Start at vertex A.

2. Execute Kruskalâ€™s algorithm on the weighted tree shown below. Assume that edges of

equal weight will be in the priority queue in alphabetical order. Clearly show what

happens each time an edge is removed from the priority queue and how the dynamic

equivalence relation changes on each step and show the final minimum spanning tree that

is generated.

3. Give an example of a weighted graph for which the minimum spanning tree is unique.

Indicate what the minimum spanning tree is for that graph. Give another example of a

weighted graph that has more than one minimum spanning tree. Show two different

minimum spanning trees for that graph. What determines whether a graph has more than

one minimum spanning tree?

4. Given the following adjacency lists (with edge weights in parentheses) for a directed

graph:

A: B(5), C(3), D(1)

B: C(1), D(3)

C: B(3), D(7), E(1)

D: A(6), C(3)

E: F(5)

F: D(3), A(4)

Execute Dijkstraâ€™s shortest-path algorithm by hand on this graph, showing how the data

structures evolve, with A as the starting vertex. Clearly indicate which edges become part

of the shortest path and in which order.