CIS 680: Mid Term Exam
 
Mo, October 25, 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) What are the time and space complexities of the following code segments.

  (a)

  for i := 1 to n do 
     for j := 1 to n2 do 
       for k := 1 to n3do 
         a[i,j,k] := 0 
   

  (b)

  function comp(n) 
  { 
    if n < 2 then return 1 
    return comp( n / 2 ) 
  } 
  

• Problem #2 (10 points) Assume n is an integer value divisible by 6 and larger than 1000.

  (a)

  Solve the following recurrence equation exactly.

  T(n) = (b)

  Does the solution to the following recurrence equation equal to, smaller than, or greater than the solution to the equation of (a).

  T(n) = ___________

• Problem #3 (10 points) Which of the following statements is correct?
  1. Each AVL tree is also a balanced tree

  2. Each heap tree is also a height-biased leftist tree

  3. The ratio between the longest and shortest paths from a root to a leaf in a red-black tree is at most 2.

  4. Each height-biased leftist tree is also a binary search tree.

  5. An AVL tree with more than 3 keys can’t be a binary search tree.
• Problem #4 (10 points)
  1. Let the height of a tree be the number of nodes in the longest path from the root to the leaves. Provide a recursive algorithm that when given a binary tree determines the height of the tree.

  2. What is the time and space complexities of your algorithm?

  3. What is the time and space complexities of your algorithm, if only AVL trees are allowed?

  4. What is the time and space complexities of your algorithm, if only height-biased leftist trees are involved?
• 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 a 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 4

  Insertion of the key 16.

• Problem #7 (10 points)

  Consider the following red-black tree (the red nodes are double circled). What kind of discrepancy (Lb0, RRb, etc.) each of the following operations create. (Note: The different operations are independent. Each of them starts from the above tree.)

  

  1. Addition of the key 11.
  2. Addition of the key 13.
  3. Deletion of the key 20.
  4. Deletion of the key 12.
  5. Deletion of the subtree rooted at 4.