Data Structures and Algorithms Winter term 2006/2007 Final Exam

Transcription

1 Page 0 German University in Cairo Winter 2006/2007 Computer Science Department Prof. Dr. Slim Abdennadher Data Structures and Algorithms Winter term 2006/2007 Final Exam Bar Code Instructions: Read carefully before proceeding. 1) Duration of the exam: 3 hours (180 minutes). 2) (Non-programmable) Calculators are allowed. 3) No books or other aids are permitted for this test. 4) This exam booklet contains 15 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 three 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! Don t write anything below;-) Exercise Marks Final Marks

2 Data Structures and Algorithms, Final Exam, Winter 2006/2007 Page 1 Exercise 1 (6 Marks) a) True or False? Justify your answer. Searching for a valuexin a binary search tree is always faster than searching for the same value in a binary tree. False: A binary search tree could be a linear tree with search in O(n) which is the same as search in a binary tree. b) Give one reason why it is helpful for a programmer to think in terms of Abstract Data Types (ADT). It is better to think in terms of ADT because this will eliminate the overhead of how a certain Data structure is implemented, we do not care about its implementation, but instead we only care about the Interface that we are going to use. c) A List ADT can be implemented using either an array or a linked-list. Give one advantage for each implementation. An advantage of using a linked-list is that, unlike an array, it is not restricted to a size that cannot be changed after its creation, which means that a link-list is more useful when the size of the data is not known at the time of creation of the data structure. An advantage of using an array is having a random access time of O(1), which is useful when frequent random access to individual elements using an index is needed.

3 Data Structures and Algorithms, Final Exam, Winter 2006/2007 Page 2 Exercise 2 (10 Marks) Consider the following problem. You are given an array ofnnumbers. The array contains only0 s and1 s, and it is sorted in increasing order. For example, the array might bea = [0, 0, 0, 0, 0, 0, 1, 1, 1]. a) Write an algorithm that prints the indexiat which the numbers switch from0 s to1 s, i.e.a[i] = 0 and A[i + 1] = 1. In the example above, it should print"5", as A[5] = 0 and A[6] = 1. public static void print(int[]a) for(int i=0;i<a.length;i++) if(a[i] == 1) System.out.println(i-1); break; b) What is the time complexity of your implementation in parta)? O(n) c) If the time complexity of your algorithm is O(n) or worse, describe in one short sentence a modification to your algorithm that will perform in better that O(n). We can use the concept of binary search so that we check the middle of the array, if it is a zero and the next value is 1 we print the current index, if it is a zero and the next value is zero too then we binary search the right half of the array, otherwise we binary search the left half.

4 Data Structures and Algorithms, Final Exam, Winter 2006/2007 Page 3 Exercise 3 (10 Marks) Write a methodislegal(strings) that takes as argument a Stringsand checks if it is a legal postfix expression. For example: is legal. * 5 4 is not legal / + is not legal. You are required to use stacks to implement your method. public static boolean islegal(string s) Stack operands = new Stack(); for(int i=0; i < s.length(); i++) if(character.isdigit(s.charat(i))) operands.push(1); else if(operands.isempty()) return false; operands.pop(); if(operands.isempty()) return false; if(operands.isempty()) return false; operands.pop(); return operands.isempty();

5 Data Structures and Algorithms, Final Exam, Winter 2006/2007 Page 4 Exercise 4 (6 Marks) a) What does the following code fragment print? int [] a = new int[3]; IntQueueq=new IntQueue(); for (int b : a) q.enqueue(b); while(!q.isempty()) for (int i = 2; i > q.peek();i--) q.enqueue(q.dequeue()); System.out.println(q.dequeue()); b) What does the following code fragment print? IntQueueq=new IntQueue(); q.enqueue(0); q.enqueue(0); q.enqueue(0); for (int i = 0; i < 8; i++) int c = q.dequeue(); int b = q.dequeue(); int a = q.dequeue(); System.out.println(a+ " " + b + " " + c); c++; b += c / 2; a += b / 2; c = c % 2; b = b % 2; a = a % 2; q.enqueue(c); q.enqueue(b); q.enqueue(a);

6 Data Structures and Algorithms, Final Exam, Winter 2006/2007 Page 5 Exercise 5 (10 Marks) a) Why is a Binary Search Tree called Binary? Why is it convenient for searching? A tree where each node has no more than 2 children is called binary. The search tree is convenient because at each node you can select which of the two branches to search and completely eliminate the other - the tree is organized so that one side is always greater, the other less, than the current node. If the tree is full and balanced, then you effectively halve the search area with each step, eliminating one or the other subtree. b) Given the following inputs sequence Construct the corresponding binary search tree by inserting the elements of the sequence starting from left to right Redraw the tree when Node 63 is deleted from the tree

7 Data Structures and Algorithms, Final Exam, Winter 2006/2007 Page 6 Exercise 6 Suppose that linked lists of integers are made from objects belonging to the class (12 Marks) class Node int item; Node next; The linked list is defined as follows class LinkedList public Node first; public LinkedList() first = null; a) Write a methodvoid organize(inti) that will modifythis list, with the elements starting at indexibeing split in half. The first half should be placed at the beginning of the list and the second half at the end of the list. For example, given the list[1, 2, 3, 4, 5, 6, 7] andi = 3, the list should be modified to[4, 5, 1, 2, 3, 6, 7]. Do not assume the existence of any methods. public void organize(int i) // solutionfor 0 < i < size - 1 Node current = first; int size; for(size = 0; current!= null; size++) current = current.next; Node a = first,b=first; for(intj=1; j < i; j++) a = a.next; for(intj=1; j < i + ((size-i) / 2); j++) b = b.next; Node oldfirst = first; first = a.next; a.next = b.next; b.next = oldfirst; b) Re-implement the method organize given that you now supposed to return a new linked list with the required organization and that you are allowed to use the methods insertfirst, insertlast, removefirst, andremovelast of the classlinkedlist. public static LinkedList organize(linkedlist l, int i) LinkedList temp = new LinkedList(); int size = 0; while(!l.isempty())

8 Data Structures and Algorithms, Final Exam, Winter 2006/2007 Page 7 size++; temp.insertlast(l.deletefirst().item); for(intj=0; j < i; j++) l.insertlast(temp.deletefirst().item); LinkedList t = new LinkedList(); for(intj=0; j < (size - i) / 2; j++) t.insertlast(temp.deletefirst().item); for(intj=0; j < (size - i) / 2; j++) l.insertfirst(t.deletelast().item); while(!temp.isempty()) l.insertlast(temp.deletefirst().item); return l;

9 Data Structures and Algorithms, Final Exam, Winter 2006/2007 Page 8 Exercise 7 (10 Marks) A Quaternary Search Tree is search tree where each node has at most four children. This tree will be used to store information aboutbooks that are located on a block of shelves in a bookstore. A book has to be to the right,left,above, orbelow any other book. Given that a methodint locate(bookx, Book y) will return one of the values1, 2, 3, 4 meaning that bookyis located to theright,left,above, below compared to bookx. Assuming that the time complexity of thelocate method is O(1) and that books are stored in the Quaternary Search Tree using the same concept that is used to store integer values in a binary search tree. Answer the following questions: a) What is the worst case time complexity of searching for a book in a Quaternary Search Tree? O(n) b) What is the worst case time complexity of searching for a book in a Quaternary Search Tree with an ideal topology, which is defined similarly as the ideal topology of binary search trees presented in class? O(log n) c) Assuming that the root of the tree is at level 0 and its children are at level 1, what is the maximum number of books that can be stored at level 3? 64

10 Data Structures and Algorithms, Final Exam, Winter 2006/2007 Page 9 Exercise 8 (8 Marks) a) For a binary search tree (BST) with n items, what is the worse-case running time to find the maximum difference between any two elements? Justify your answer. Does the running time depend on whether the binary search tree is balanced? Say why or why not. To find the maximum difference, sufficient to find the minimum element in the BST, the maximum element in the BST and compute their difference. Running time depends on the worst-case depth of the minimum and maximum elements in the BST. Worst case running time for an unbalanced BST is O(n) Worst case running time for a balanced BST is O(log(n)) b) Add an instance method to thebinarysearchtree class that swaps all nodes to the right of the root with all nodes to the left of the root. Does the resulting tree satifies a similar property like the binary search tree presented in the lecture. public void swap() if(root!= null) Node temp = root.left; root.left = root.right; root.right = temp;

11 Data Structures and Algorithms, Final Exam, Winter 2006/2007 Page 10 Exercise 9 Write a static recursive method (8 Marks) public static boolean sameshape(tree x, Tree y) that takes two tree references as inputs and determines whether they have the same tree shape and pointer structure. public static boolean sameshape(tree t1, Tree t2) return(sameshape(t1.root,t2.root)); public static boolean sameshape(node x, Node y) if (x == null && y == null)returntrue; // both null if (x == null y == null)returnfalse; // exactlyone null return sameshape(x.left, y.left)&& sameshape(x.right, y.right);

12 Data Structures and Algorithms, Final Exam, Winter 2006/2007 Page 11 Exercise 10 (8 Marks) In an attempt to describe a family tree we would like to modify the tree implementation presented in class in order to make nodes represent persons. Each person should be allowed to have an arbitrary (not predefined) number of children and also have a spouse (another person to which the person is married). a) What are the instance variables needed to implement the family tree described above? We need two more instance variable: A reference to the spouse A reference to an array (or some other data structure) of children class Node Node spouse; Node [] children; b) Implement a boolean method that checks if every node in the tree and its spouse have the same children. public boolean check(node c) if(c!= null) Node s = c.spouse; if(s!=null) if(c.children!= s.children) return false; for(inti=0; i < c.children.length;i++) if(check(c.children[i]) == false) return false; return true; c) Implement a boolean method that checks if two given nodes are siblings, i.e. have the same parent. public Node find(node p, Node c) if(p!= null) if(p.children!= null) for(inti=0; i < p.children.length;i++) if(c == p.children[i]) return p; for(inti=0; i < p.children.length;i++) Node x = find(p.children[i], c); if(x!= null) returnx;

13 Data Structures and Algorithms, Final Exam, Winter 2006/2007 Page 12 return null; public boolean siblings(node c1, Node c2) Node p1 = find(root, c1); Node p2 = find(root, c2); if(p1 == p2) return true; else return false;

14 Data Structures and Algorithms, Final Exam, Winter 2006/2007 Page 13 Exercise 11 (10 Marks) Consider inserting the keys 104, 221, 316, 444, 6042, 281, 174, 880, 442, 253 into a hash table of sizeb = 12 and using the hash function h(x)=sum-of-digitis(x)%b. Use a strategy of your choice to resolve further collisions. The function sum-of-digits(n) will sum up the digits of n. For example, if n = 221, then sum-of-digits(221) = 2+2+1=5. a) What is the strategy that you will use for resolving collisions? Explain the strategy in few words. Linear Probing b) Insert the keys in the given order in the hash table below

15 Data Structures and Algorithms, Final Exam, Winter 2006/2007 Page 14 Extra Sheet

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