NAME: c. (true or false) The median is always stored at the root of a binary search tree.

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1 EE 322C Spring 2009 (Chase) Exam 2: READ THIS FIRST. Please use the back side of each page for scratch paper. For some of the questions you may need to think quite a bit before you write down an answer. If you write while you think, use the back side of the page and copy your final (neatly written) answer to the front side. I will not give credit for illegible answers (and I reserve sole judgment for what is legible ). The exam is out of 100 points, not all questions are equally hard or take equally long to complete. Some hard questions are not worth very much. Some easy questions are worth a lot. Good luck! 1. (12 pts) True/false questions. NAME: a. (true or false): In java, any subtype Object can be inserted into a HashSet or HashMap (HashSet and HashMap are the two standard hash-table data structures in the java standard library). b. (true or false): In java, any subtype Object can be inserted into a TreeSet or TreeMap (TreeSet and TreeMap are the two standard balanced, binary search tree data structures in the java standard library). c. (true or false) The median is always stored at the root of a binary search tree. d. (true or false) The median is always stored at the root of a red-black tree. e. (true or false) Setting local variables to null before returning from a function can reduce the time spent performing garbage collection. f. (true or false) In Java, it is normal for programs to run slightly faster immediately following a garbage collection and to gradually slow down as more garbage is created.

2 2. (15 pts) Garbage collection problems: a. (6 pts) Write a program that calls new an infinite number of times, but never runs out of memory. b. (3 pts) What sort of garbage collector would be better for your program in part (a), a copy collector or a mark and sweep collector? c. (6 pts) Write a program that calls new as many times as possible but never creates any garbage. Obviously, this program will eventually run out of memory.

3 Reference / cheat sheet for Question 3. ArrayList has the following methods (there are others, but you shouldn t need to use any more than these. class ArrayList implements List { public boolean add(object x) adds the element at the end public boolean remove(object x) finds and removes x from the list public boolean contains(object x) returns true if x is in the list Object get(int k) returns the object stored at position k in the list Object set(int k, Object v) stores v at position k in the list Iterator iterator() returns an iterator for this list Int size() returns the number of elements in the list

4 3. (20 pts) Write the contains (10 pts) and remove (10 pts) methods for the following hash table. Your solution must be consistent with the add method shown. class HT { int M = 100; // table size, fixed at 100 for this example Object[] table = new Object[M]; int length = 0; // number of elements in the table public void add(object x) { if (x == null) { return; // cannot be null if (contains(x)) { return; // no duplicates allowed in this table int k = x.hashcode(); Object entry = table[k]; // entry is null, is a single element in the table, or is an ArrayList of elements in the table if (entry == null) { table[k] = x; if (entry instanceof ArrayList) { ArrayList list = (ArrayList) entry; list.add(x); else { ArrayList list = new ArrayList(); list.add(entry); list.add(x); table[k] = list; length += 1; public boolean contains(object x) { // return true if x is in the table, false otherwise public void remove(object x) { // ON THE NEXT PAGE, PLEASE

5 public void remove(object x) { // if x is in the table, remove it. Do nothing otherwise if (! contains(x)) { return; 4. (9 pts) In the Java standard library, the Iterator interface defines a method called remove. While we didn t talk about this much in class, it is (an optional) part of the interface. What remove() is supposed to do (when it s implemented, which is optional), is remove the element that was most recently returned by next(). If you were to implement remove() what would you expect the worst case time complexity to be for one call to remove() for each of the following data structures (assume there are N elements in the data structure). a. A LinkedList (assume doubly-linked, if that matters) b. A Binary Search Tree (assume red-black, if that matters) 5. c. A Hash Table (assume the table size is M and chaining is used, if that matters)

6 (20 pts) Consider the following attempt to sort the values in an array using a tree. void sort(int[] x) { TreeType tree = new TreeType(); // creates an empty tree for (int k = 0; k < x.length; k += 1) { tree.insert(new Integer(k)); Iterator p = tree.iterator(); int k = 0; while (p.hasnext()) { Integer v = p.next(); x[k] = v.intvalue(); k = k + 1; a. Will this algorithm correctly sort the values in x? If the answer depends on whether an ordinary binary search tree is used, or if a red-black tree is used, be sure to indicate that in your response. b. If TreeType is an ordinary binary search tree, what is the total worst case time complexity for this function? c. What is an example sequence of array values that would produce this worst case time complexity when TreeType is an ordinary BST. (there may be many different correct answers, I just want one example). d. If TreeType is a red-black tree, what is the worst case time complexity for this function? e. What is an example set of array values (in sequence) that would produce this worst case time complexity when TreeType is a red-black tree. (there may be many different correct answers, I just want one example).

7 6. (24 pts) Draw a red black tree containing the values { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. Indicate red nodes by writing the value followed by the letter R (e.g., 5R, 10R, etc.) Indicate black nodes by writing the value followed by the letter B (e.g., 15B, 20B). a. (8 pts) Draw the tree with the smallest possible height. (4 pts) What is the black height of the tree you have drawn for part a? b. (8 pts) Draw the tree (using the same values as above) with the largest possible height. (note the answer to part b could be the same as the answer to part a depending on the properties of red/black trees). (4 pts) What is the black height of the tree you have drawn for part b?\