v22.0310-003: Homework 3

Due date: Wed Oct 20.

This homework has a programming part and a written part. The programming part should be submitted as usual by email. The written part must be handed in either after the lectures on Wednesday, OR to the TA during his office hours on Wednesay. [Please do not just stuff in our mail boxes without asking first]

Look here for additional information after the original posting of assignment:
  1. Problem 3.4 (page 71) in Text.
    [parts (a), (b) and (c)]
  2. Problem 3.10 (page 73) in Text.
    [relating n! and n log n]
  3. Let us insert the following keys (in this order) into an initially empty binary search tree:
    9, 7, 2, 8, 12, 5, 1, 16, 10, 13. 11
    (a) Please show the binary search tree after each insertion.
    (b) Let T be the final tree in (a). List the keys in a pre-order traversal of T.
    (c) Repeat (b) but for an in-order traversal.
    (d) Repeat (b) but for a post-order traversal.
    (e) Draw the tree after deleting node 12 from T.
  4. Suppose A is an array initially containing the keys in the previous question (9, 7, 2, ..., 11). Apply the buildheap() method (page 156, Figure 5.24) to array A.
    (a) Show the MaxHeap after each application of the siftdown method. NOTE: draw the MaxHeap as a complete binary tree, not as an array.
    (b) Suppose we apply the method removemax() to the result of (a). Show the resulting MaxHeap.
  5. Programming Question. Download and read the following "abstract" java program, myNode.java. It is "abstract" because we omit the various bodies for methods and putting dummy bodies where we could not. [But you can call "javac" on it without any error.] We want you to implement the merge sort algorithm (discussed in the first lecture of class, but see p.242, section 8.5).

    To write your own myNode.java program, first remove all occurences of the key word "abstract" from the sample program, then fill in the appropriate code. Remove also any dummy code. The main method has already been written for you. Here is a sample input file, input.0.


  6. RECOMMENDED EXTRA PROBLEMS: Do not hand in, as they will not be graded. But we will give solutions and discuss this. Please try them yourself first!
    • (a) 2n = O(n)
    • (b) n! = O(2n) is false. To show this, show that for any constant C> 0, if n is large enough, then
      n! > C 2n
    • (c) 3n = O(2n) is false.
    • (d) Consider the following property: at each node u of a binary tree we store a key. Moreover,
      u_L.Key < u.Key < u_R.Key
      Is this a binary search tree? Give counter example.
    • (e) In chapter 5.3, Schaffer said that if we eliminate pointers from leaves in full binary trees then the overhead fraction is p/(p+d). If the binary tree is not necessarily full, we can still achieve this overhead fraction. How? Explain your reasoning.