Data Structures and Algorithms Winter term 2016

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1 Page 0 German University in Cairo December 26, 2016 Media Engineering and Technology Prof. Dr. Slim Abdennadher Dr. Wael Abouelsaadat Data Structures and Algorithms Winter term 2016 Final Exam Bar Code Instructions: Read carefully before proceeding. 1) Please tick your major 2) Duration of the exam: 3 hours (180 minutes). 3) No books or other aids are permitted for this test. Major MET IET Mechatronics BI 4) This exam booklet contains 12 pages, including this one. Three extra sheets of scratch paper are attached and have to be kept attached. Note that if one or more pages are missing, you will lose their points. Thus, you must check that your exam booklet is complete. 5) Write your solutions in the space provided. If you need more space, write on the back of the sheet containing the problem or on the four extra sheets and make an arrow indicating that. Scratch sheets will not be graded unless an arrow on the problem page indicates that the solution extends to the scratch sheets. 6) When you are told that time is up, stop working on the test. Good Luck!

2 Data Structures and Algorithms, Final Exam, December 26, 2016 Page 1 Exercise 1 (3+3+3=9 Marks) a) For each of the following applications, explain which data structure from the given list you would choose and give your reasons for doing so. There may not be a single ârightâ answer to some of the questions. Your choices will be evaluated on whether they are appropriate for the given application and the quality of the (brief) reason(s) that you give in support of your choice. The reasons should be related to the order of magnitude of the operations to be performed. Choose from the following list of data structures: Sorted or Unsorted Arrays Doubly Linked list Binary tree, not a binary search tree Balanced binary search tree Hashtables 1. List of names of all GUC students. Search operations tend to be random and are much more frequent than additions and deletions. The application also needs to be able to list alphabetically all names in the directory starting at a particular name and ending at another name, e.g., list all names from Mina to Sarah, inclusive. Sorted arrays, as we want to list a certain range so we will search for the first element in the range which will cost O(logn) then we will list the elements till the last element in the range that will cost O(m), so the complexity will be O(log(n) + m) 2. Parts inventory in an automobile supply shop. There are hundreds of thousands of different parts in the inventory and, given a part number, information about the part with that number needs to be accessed quickly. Hashtables as searching using an ideal hash function will be O(1) 3. The history list recording sites visited by the user of a web browser. As new sites are visited they are added to the list. The list also supports the operation of going back to the web page that was previously visited before the current page and going forward to the next page visited. DoublyLinkedList going back and forth will be of O(1)

3 Data Structures and Algorithms, Final Exam, December 26, 2016 Page 2 Exercise 2 (5+5=10 Marks) a) List the order in which the given keys would be contained in a Priority Queue ADT implemented using a sorted sequence, where keys are ordered in increasing order of their priority value. For this problem the priority value of a key is defined as its minimum absolute value difference from the various powers of 2. For example, the priority value of the key 12 is 4, as it is 4 away from both 8, and 16. The priority value of the key 66 is 2, as it is 2 away from 64. A table of the powers of 2 is below, for your reference: k k Fill in the sequence diagram below with your answer, given the following list of keys in random order: Keys PQ b) A complete binary trees can be stored in a array. 1. Propose a way to store complete binary tree in an array. Show your approach using the following tree Level order traversal 2. What are the formulas for finding the indices of the parent, left child, and right child of a node in a complete binary tree when that tree is stored in an array, and the node in question is at index i? Parent = i 1/2 Left = i Right = i 2 + 2

4 Data Structures and Algorithms, Final Exam, December 26, 2016 Page 3 Exercise 3 Assume we have the following implementation of a single-ended singly linked list: (10 Marks) public class SSL { private Link head ; public SSL( Link head ) { this.head = head ; write an evenodd method in the SSL class above that changes the linked list to contain every other link of the original linked list, starting with the first node. Your method must also return a linked list that contains every other link of the original linked list, starting with the second node. Your method should not create any new Node object. If an SLL list contains zero elements or only one element, a call to evenodd should return null. For example, if an SSL list initially contains the elements [5, 2, 3, 1, 4], then a call to evenodd should return an SSL list with the elements [2, 1] and and after the call, the original SSL list should contain the elements [5, 3, 4]. Complete the following method: public SSL evenodd() { public MyLinkedList evenodd() { if (head == null head.next == null) return null; ListNode curr = head; ListNode ret = head.next; MyLinkedList retlist = new MyLinkedList(ret); while (curr!= null && ret!= null) { curr.next = ret.next; curr = curr.next; if (curr!= null) { ret.next = curr.next; ret = curr.next; return retlist;

5 Data Structures and Algorithms, Final Exam, December 26, 2016 Page 4 Exercise 4 (4+4+6=14 Marks) a) Given the following orders of traversal of a binary tree, recreate the tree: Postorder: G,D,B,H,I,E,F,C,A Inorder: D,G,B,A,H,E,I,C,F A B C / D E F \ G H I b) Draw a diagram of the binary search tree (BST) that results when the following numbers are added to the tree in the given order without any additional balancing or rearranging of the nodes in the tree \ \ / c) Given the following expression: x + (a + b) * r + z. 1. Construct an expression tree for the above expression. In an expression tree, each operator and operand is represented by a node, and each operator node has children that represent its operands. + + z x * + r a b 2. Write down the order in which the nodes of the above tree are encountered by a preorder traversal of the tree. Include all of the nodes, both operators and operands. + + x +abrz

6 Data Structures and Algorithms, Final Exam, December 26, 2016 Page 5 3. Write down the order in which the nodes of the above tree are encountered by a postorder traversal of the tree. Include all of the nodes, both operators and operands. xab + r +z+

7 Data Structures and Algorithms, Final Exam, December 26, 2016 Page 6 Exercise 5 Assume that we modified the Node class in a binary search tree as follows: (12 Marks) class BSTNode { public int data; public BSTNode left; public BSTNode right; public BSTNode parent; // data stored in this node // left subtree; null if none // right subtree; null if none // this nodeâs parent; null if // none (i.e., null if this is // the root node) In this representation, each node contains a reference to its parent in the tree, if there is one, in addition to its left and right subtrees. Complete the definition of the following Java method so that, given a reference to some node p in a binary search tree, the method returns a reference to the node that precedes it in an in-order traversal of the tree. The result should be null if there is no node preceding p. Note: For full credits, the method should be implemented iteratively without traversing the whole tree. In case, your method will traverse the whole tree, you will receive half the grades. For example, for the following tree / 24 / 20 if p is the node with data 7, the method should return the node with data 2. if p is the node with data 15, the method should return the node with data 14. if p is the node with data 20, the method should return the node with data 19. if p is the node with data 0, the method should return null. public BSTNode previousnode(bstnode p) {

8 Data Structures and Algorithms, Final Exam, December 26, 2016 Page 7 public BSTNode previousnode(bstnode p) { // Strategy: if p has a left child, then the previous node // is the largest one on the left. If not, then it is the // closest ancestor node whose item is less than p.item BSTNode q; if (p.left!= null) { q = p.left; while (q.right!= null) { q = q.right; else { q = p.parent; while (q!= null && q.item.compareto(p.item) > 0) { q = q.parent; return q; Another way to search for the nearest ancestor that contains a smaller value is to ascend the tree as long as the current node is the left child of its parent. q = p; while (q.parent!= null && q.parent.left == q) { q = q.parent; return q.parent;

9 Data Structures and Algorithms, Final Exam, December 26, 2016 Page 8 Exercise 6 (8 Marks) Assume we have a hash function for strings that returns the ordinal value of the first character in the string modulus size of the table, where the ordinal value is defined as follows ord(a) = 0... and ord(z) = 25. Draw a hash table of size 10 where the keys are strings. Process the following strings and insert them into the table in the order given. we, eat, ham, and, jam, and, spam, a, lot, in, camelot a) using linear probing b) using quadratic probing c) using chained hashtable 0 and 1 and 2 we 3 a 4 eat 5 lot 6 in 7 ham 8 spam 9 jam 0 and 1 and 2 we 3 in 4 eat 5 lot 6 a 7 ham 8 spam 9 jam 0 and and a 1 lot 2 we camelot 3 4 eat ham 8 spam in 9 jam

10 Data Structures and Algorithms, Final Exam, December 26, 2016 Page 9 Exercise 7 ( =15 Marks) In this question, we would like to implement an ADT for sorted sets using binary search trees whose nodes are defined as follows: class Node { public Comparable item; // item referred to by this node public Node left; // left subtree; null if none public Node right; // right subtree; null if none... Assume the following interface for the BST class: class BST { public Node root; public void insert(comparable item); public Node search(comparable item); public Node delete(comparable item); Implement the class OrderedSet with the corresponding instance variables and the following constructor and methods: a) void add(comparable item) that adds item to the set if not already present b) boolean contains(item) that returns true if the set contains item; false if not c) int size() that returns the number of items in the set d) void remove(comparable item) that removes item from the set if present e) LinkedList tolist() that returns a linked list with the items of the set sorted in increasing order. You may define additional helper methods. Hints: Use a LinkedList to accumulate the result, and recall that these lists include an insertlast(â) method to add new items to the end of a linked list. Complete the following class: public class OrderedSet {

11 Data Structures and Algorithms, Final Exam, December 26, 2016 Page 10 Solution Exercise 7 class BasicOrderedTreeSet{ private BSTNode root; // root of the tree or null if the // set (tree) is empty /** Return the contents of this set as an ordered list */ public List tolist() { ArrayList values = new ArrayList(); addsubtree(root, values); return values; // add contents of subtree with root r to values in order private void addsubtree(bstnode r, List values) { if (r == null) { return; addsubtree(r.left, values); values.add(r.item); addsubtree(r.right, values);

12 Data Structures and Algorithms, Final Exam, December 26, 2016 Page 11 Scratch paper

13 Data Structures and Algorithms, Final Exam, December 26, 2016 Page 12 Scratch paper

14 Data Structures and Algorithms, Final Exam, December 26, 2016 Page 13 Scratch paper