2005 S

1. (15%) Consider the graph in Figure 1. Find a minimum-cost spanning tree by Prim's algorithm and by Kruskal's algorithm.

2. (10%) In Figure 2 use Dikstra's algorithm to find the shortest paths from node a to other vertices.

3. (5%) The external path length of a rooted tree is the sum over all leaves of the lengths of the from the root to each leaf. What is the external path length of a full binary tree with n levels?

4. (10%) Given tight big-oh and big-omega upper and lower bounds on the function T(n) defined by:
 * T(1)=1
 * T(n)=4T(n/2)+for n=2, 4, 8...

Your bounds need apply only when n is power of 2, i.e., you may assume that for all m, T(m)=T, where is the smallest power of 2 greater than or equal to m. Show your reasoning.

5. (10%) Let