1-4 CHARACTERS CODE(if you want your final grade posted)

 
CIS 680: Final Exam
 
1:50 minutes
Open Notes, Open Books
The exam consists of five problems
Answers not appearing in designated spaces WILL NOT be graded

• Problem #1 (10 points)

  Add the keys 6.1, 6.2, 6.3, 6.4, and 6.5 to the following B-tree of degree 3, and show all the intermediate trees and the final tree.

  

• Problem #2 (10 points)
  1. Describe an algorithm to determine whether a directed graph has exactly one topological order.

  2. What is the time complexity of your algorithm?
• Problem #3 (10 points) Show the linked lists of the union-find algorithm, at all the instances of applying Kruskal’s algorithm on the following graph.

  

• Problem #4 (10 points) The 3-vertex cover problem for graphs asks to assign a vertex cover of cardinality three to the given graphs. Assume an undirected graph whose edges are (1,2), (1,4), (1,5), (2,3), (2,4), (3,4), (4,5).
  1. Provide a spanning tree of a solution space, for the 3-vertex cover problem on the given graph.

  2. What vertex cover the backtracking algorithm provides for the above tree?

  3. What vertex cover the FIFO branch-and-bound algorithm provides on the above tree?
• Problem #5 (10 points) Consider the following recursive function.

  function value(i) { 
     if (i >= n) return 0; 
     return 1 + max( value(2*i+1), value(2*i+2), 
                     value(4*i+3) + value(4*i+4) + value(4*i+5), 
                     value(4*i+5) + value(4*i+6)                 ); 
  } 
  

  Translate the code into an iterative bottom-up dynamic programming algorithm.