CIS 680: Mid Term Exam
 
Mo, May 3, 1999, 50 minutes
Open Notes, Open Books
The exam consists of seven problems
Answers not appearing in designated spaces WILL NOT be graded

• Problem #1 (10 points)
  1. What are the time and space complexities of the following code segment.

               read( n ) 
               for i = 1 to n do 
                 for j = 1 to i do   write( i, j ) 
     

  2. Solve the following reccurence equation exactly.

     T(n) =

• Problem #2 (10 points)
  1. Provide a recursive algorithm that given a binary tree determines the number of leaves in the tree.

  2. What is the time and space complexities of your algorithm?
• Problem #3 (10 points) Let the height of a tree be the number of nodes in the longest path from the root to the leaves. For each of the following types of trees, determine the minimum number of nodes a tree of height 4 can have.
  1. Heaps
  2. Height-biased leftist trees
  3. Binary search trees
  4. AVL trees
  5. Red-black trees
• Problem #4 (10 points)

  Consider the linear time algorithm for initializing max heaps. Show all the intermediate, and final, trees the algorithm creates on the given tree.

  

  

  

• Problem #5 (10 points)

  Consider the priority queue represented by the given height-biased leftist tree. Show the modified tree under each of the following operations. (Note: The two operations are independent. Each of them starts from the above tree.)

  ________________________________________

  Insertion of the key 7.

  Deletion of the largest key.

• Problem #6 (10 points) Consider the following AVL tree.

  

  Show the modified tree under each of the following operations. (Note: The two operations are independent. Each of them starts from the above tree.)

  ________________________________________

  Deletion of the key 5

  Insertion of the key 10.

• Problem #7 (10 points)

  Consider the following red-black tree (the red nodes are double circled). What kind of discrepency (Lb0, RRb, etc.) each of the following operations create.

  

  1. Addission of the key 21.
  2. Addission of the key 25.
  3. Deletion of the key 18.
  4. Deletion of the key 2.
  5. Deletion of the subtree rooted at 12.