CSE 373 Data Structures 11au, Homework 2

Due at the BEGINNING of class, Friday, 10/14/2011

Here are some questions on complexity, algorithm analysis, and the basics of binary trees. You only need to turn in written solutions, although you will need to run some code for one of the problems.

Problems

  1. Prove (using Induction) that:

Hints: Start with N=1 as the base case, then show how  ends up being equal to .  More hints: You already know what the sum of  is, and you should use the induction hypothesis to come up with your answer.  Referring to the induction examples on pages 6 and 7 and the examples from the slides may be helpful.

  1. Order the functions given in Weiss question 2.1 on page 50 from slowest growth rate to fastest growth rate. IN ADDITION add these functions in: log N, log2 N.  If any of the functions grow at the same rate, be sure to indicate this.
  2. Weiss question 2.2 on p.50.  You do not need to prove an item is true (just saying true is enough for full credit), but you must give a counter example in order to demonstrate an item is false if you want full credit.  To give a counter example, give values for T1(N), T2(N) and f(N) (e.g. “The statement is false if T1(N)=100N, T2(N)=2N2 and f(N)=N3”) for which the statement is false.  Hints: Think about the definitions of big O and little o.  
  3. Weiss, question 2.7 on p.51 (You only need to do this question for the first FIVE program segments – you may ignore the last loop (6)). For part (b) and (c), please turn in a printout of your Java code, (no electronic submission required). Hints: you will want to use assorted large values of n to get meaningful experimental results. You may find the library function System.nanoTime() to be useful in timing code fragments.  A link to some Java code showing an example of timing can be found here.

Note that there are THREE parts to this question, do all 3.  a) calculate big-O, b) run the code *for several values of N* (4 or more) and time it, c) talk about what you see.  For part c, be sure to say something about what you saw in your run-times, are they what you expected based on your big-O calculations?  If not, any ideas why not?  Graphing the values you got from part b might be useful for your discussion. Remember that when giving the big-O running time we always want the tightest bound we can get.

  1. Show that the function 50n + 16n3 + 2012 is O(n3). (You will need to use the definition of O(f(n)) to do this.  In other words, find values for c and n0 such that the definition of big-O holds true as we did with the examples in lecture.
  2. (Unbalanced binary search trees)
    1. Draw a picture of the integer-valued BST that results when these values are inserted in this order: 18, 21, 12, 1, 20, 7, 8, 19, 52, 89, 63.
    2. Which nodes are the leaves of this tree? Which node is the root?
    3. What is the depth of the node containing 8? What is the height of the node containing 52?
    4. Write down the order in which the node values are reached by (i) a preorder, (ii) an inorder, and (iii) a postorder traversal of the tree.