v22.0310-003: Homework 2

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.

• SOLUTION (posted 10/14/99): Solutions for Questions 1-3 in postscript format. Java Program for Question 4 by Archi.
• Question 1 Hint
• The readString Function on different Operating Systems.
• Here is a sample input file called "cities" for your node.java program: it is a list of 20 cities.

1. The following recurrence actually arise in the analysis of a famous fast algorithm for multiplying binary numbers: T(x) = 3 T (x/2) + x if x > 1; otherwise T(x) = 0. Prove that T(x) = Theta(x^c) where c=lg 3 (approximately 1.585). HINT: since we already tell you the correct answer, you can just use induction directly. Remember that a Theta bound should be broken up into two parts.
2. Use real induction to give an exact solution for the following recurrence on real numbers: T(x) = 2 T (x/2) + x if x > 1; otherwise T(x) = 1. Note that only difference from our lecture example is that the initial condition.
3. Problem 3.8 (Chapter 3, page 72) of the text book. It says ``briefly explain'', which we will interprete to mean that you show the implicit constants in the big-Oh, big-Omega or big-Theta.

   [Note: if a pair of functions are big-Omega of each other, then you must show a big-Omega result to get full credit.]

4. In the file node.java we have designed a nodeInterface interface. You must give a complete implementation of this interface by defining a class called node. We have started this implementation process by providing you with a skeleton class. Please fill in all the remaining details. You are not allowed to modify any details we have specified. We have also specified the functionality we expect in your main method.

   We have written two methods for you, both having to do with DataInputStream and/or FileInputStream. These are two important topics that will help you in doing other problems -- so you should read this code carefully and understand what its doing.