Computer Science 245
Spring 2009
Homework 8:  B-Trees and Connected Components
Due Monday, April 20th 3:30 p.m.

  1. Consider the following graph:


    Graph


    1. Show the Vertex / Distance / Path table after Dijstra's algorithm is run on this graph, starting at vertex 0.
    2. Show thet Vertex / Distance / Path table after Prim's algorithm is run on this graph, starting at vertex 0.


  2. Consider a B-Tree with maximum degree k (that is, all interior nodes have between (k+1)/2 and k children -- a 2-3 tree is a B-Tree with maximim degree 3). 
    1. What is the largest number of keys that can be stored in a  B-Tree of height h with maximum degree k?
    2. What is the smallest number of keys that can be stored in an B-Tree of height h with maximum degree k?

    Show your work!

    (Hint:  You may find the following formula helpful:)


     n
     ∑      xi = (x(n+1) - 1) / (x-1)   
    i = 0     


  3. Consider the following directed graph:


    Graph



    Run the connected component algoirhtm on this graph.   Show all discovery and finish times for all vertices, as well as the Depth-First Forrest of the final pass of the algorithm.