Data Structures and Algorithms Winter Semester

Transcription

1 Page 0 German University in Cairo October 24, 2018 Media Engineering and Technology Faculty Prof. Dr. Slim Abdennadher Dr. Wael Abouelsadaat Data Structures and Algorithms Winter Semester Midterm Exam Bar Code Instructions: Read carefully before proceeding. 1) Please tick your major Major IET/MET Mechatronics BI 2) Duration of the exam: 2 hours (120 minutes). 3) No books or other aids are permitted for this test. 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! Don t write anything below ;-) Exercise Possible Marks Final Marks

2 Data Structures and Algorithms, Midterm Exam, October 24, 2018 Page 1 Exercise 1 (4+4+8=16 Marks) a) Consider a queue on which an intermixed sequence of enqueue and dequeue operations are performed. The enqueue operations put the numbers from zero until nine (in ascending) order on the queue and the dequeue operations print out the result of removing a number from the back of the queue. Which of the following print sequence(s) could not occur? Justify why? b) Convert the following expression from Postfix notation: H A B C + D * + F * G * E * * J + Note A,B,C,D,E,F,G,H and J are the operands, and +, * are the operators. to Infix notation (full-bracketed expression): ((H ((((A + ((B + C) D)) F ) G) E)) + J) to Prefix notation: + * H * * * + A * + B C D F G E J c) In lectures, we discussed an algorithm to transform an expression from Infix notation into an expression in Postfix notation. Assume we have a String representing a sequence of digits and the following binary operators +,-,*,/. You are asked to write an algorithm (in English) that converts any expression from Prefix to Infix notation. Use the most appropriate data structure in such a way your algorithm will have a time complexity O(n). The expression should be broken into String items. Each item is either an operator or an operand. We enqueue these items into a queue of Objects. We start inspecting the items from the front of the queue and follow the following algorithm: If we find an operator followed by two operands, we dequeue the three items then enqueue back a String containing the application of the operator on the operands in infix notation (for example if we found * followed by 2 and 3, then we enqueue 2*3 ) If we find an operator followed by an operand then another operator, we dequeue the first opertor and the operand and enqueue them back at the rear of the queue. If we find two consecutive operators, we dequeue the first one and enqueue it back at the rear of the queue. If we find an operand, we dequeue it and enqueue it back at the rear of the queue. We continue executing the above steps till the queue contains only one item which is the result. d) Trace your algorithm with the following example: * - A B / C D The following is the state of the queue after each step: front * - A B / C D rear front - A B / C D * rear front / C D * A-B rear front A-B*C/D rear

3 Data Structures and Algorithms, Midterm Exam, October 24, 2018 Page 2 Exercise 2 (8 Marks) A sorting algorithm is stable if it preserves the order of elements with equal keys. For example, given the following array: 32, 12a, 4, 12b, 39, 19 where 12a and 12b represent records that both have a key of 12, a stable sorting algorithm would produce the following sorted array: 4, 12a, 12b, 19, 32, 39 Note that 12a comes before 12b, just as it did in the original, unsorted array. State whether the following sorting algorithms are stable or not. If an algorithm is stable, justify your answer. If an algorithm is unstable, then construct an example of an input array containing two elements with equal keys whose order is reversed by this sorting algorithm. a) Bubble Sort Stable. Bubble sort swaps two adjacent elements only if the first one is strictly greater than the second one. In case of equal keys, the first one will never be swapped with the second one so their order will be preserved. b) Selection Sort Unstable. Consider a given array with the following keys: 15a b 1 7. After the first pass, the 1 will be swapped with the 15a putting it after the second 15b. Since the selection sort only searches for the minimum each pass, 15b will be considered as minimum before 15a. Hence, it will be placed before 15a which results in not keeping the original order of the equal keys. c) Insertion Sort Stable. The loop that inserts each key in its correct position relative to the previous keys stops once it find a key with less or equal value. Which means that a key wont be inserted before another key with equal value that appeared before him hence, the order of the equal keys will be preserved.

4 Data Structures and Algorithms, Midterm Exam, October 24, 2018 Page 3 Exercise 3 (8 Marks) In the lectures, we presented a priority queue with an insertion method of O(n) and a removal method of O(1). Assume we want to implement a stack using a priority queue. a) What will be the keys of the values that we want to push in the stack? How will you implement that? The time stamp of each value should be the key of this value. This can be done using an integer that will be attached to the key. For each insertion, that counter will be decreased so the values that are pushed later will have lower key. Therefore, they will have a higher priority. b) Explain in English how will the push method be implemented and what is its corresponding time complexity? Each time an element is pushed into the stack, we decrement the counter representing the time stamp, attach it to the element and call the add method of the priority queue. The time complexity will be O(n). c) Explain in English how will the pop method be implemented and what is its corresponding time complexity? We will just call the remove() method of the priority queue. The time complexity will be O(1).

5 Data Structures and Algorithms, Midterm Exam, October 24, 2018 Page 4 Exercise 4 Your task is to write a method called scramble that takes as argument a queue of integers (16 Marks) public static void scramble(queue q) The function should reverse the order of SOME of the elements in the queue, and maintain the order of others, according to the following pattern: The first element stays on the front of the queue. Then the next two elements are reversed. Then the next three elements are placed on the queue in their original order. Then the next four elements are reversed. Then the next five elements are place on the queue in their original order. etc. For full credits, only one single stack can be used as an additional data structure. If you use more than one stack then you will loose some marks. For example, given the following queue, front back after calling the method scramble, we get the following result: front back Please note that any leftover numbers should be handled as if their block was complete. (See the way 15 and 16 were treated in our example above.) Stack ADT public class Stack public Stack(int s); public void push(int k); public int pop(); public int size(); public boolean isempty(); public boolean isfull(); Queue ADT public class Queue public Queue(int s); public void enqueue(int k); public int dequeue(); public int size(); public boolean isempty(); public boolean isfull();

6 Data Structures and Algorithms, Midterm Exam, October 24, 2018 Page 5 public static void scramble (QueueObj q) StackObj temp = new StackObj(q.size()); boolean reverse= false; int iteration=0; int i; for( i = 0 ; i +iteration+1< q.size(); i=i+iteration) iteration++; if(reverse) for(int k = 0; k< iteration; k++ ) temp.push(q.dequeue()); while(!temp.isempty()) q.enqueue(temp.pop()); else for(int k = 0 ; k < iteration; k++) q.enqueue(q.dequeue()); reverse=!reverse; i=q.size()-i; if(reverse) for(int k = 0; k< i; k++ ) temp.push(q.dequeue()); while(!temp.isempty()) q.enqueue(temp.pop()); else for(int k = 0 ; k < i; k++) q.enqueue(q.dequeue());

7 Data Structures and Algorithms, Midterm Exam, October 24, 2018 Page 6 Exercise 5 We want to implement a data type MiddleStack which will support the following operations: (15 Marks) a) void push(int k) which adds an element to the top of stack in O(1) time complexity b) int pop() which removes an element from top of stack in O(1) time complexity. c) int findmiddle() which will return middle element of the stack in O(1) time complexity. d) int deletemiddle() which will delete and return the middle element in O(n) time complexity. You should decide about the most appropriate data structure to be used. You should implement the class MiddleStack including the attributes and at least one constructor. public classs MiddleStack public class MiddleStack int[] elements; int top; public MiddleStack(int n) elements= new int [n]; top=-1; public void push(int k) elements[++top]=k; public int pop() return elements[top--]; public int findmiddle() return elements[top/2]; public void deletemiddle() for(int i = top/2;i<top;i++) elements[i]= elements[i+1]; top--;

8 Data Structures and Algorithms, Midterm Exam, October 24, 2018 Page 7 Exercise 6 Given the following mystery method: (10 Marks) static void mystery(int arr[], int l, int h) if (l >= h) return; if (arr[l] > arr[h]) int t = arr[l]; arr[l] = arr[h]; arr[h] = t; if (h - l + 1 > 2) int t = (h - l + 1) / 3; mystery(arr, l, h - t); mystery(arr, l + t, h); mystery(arr, l, h - t); and given the following call: int arr[] = 2, 4, 5, 3, 1 ; mystery(arr, 0, arr.length - 1); a) Assume that the display method prints out the elements of the array, write out the elements of arr after the execution of the first if statement and after the termination of every recursive call. Hint: Do not try to trace all recursive calls. Understand the functionality of the program by reading the code carefully. static void mystery(int arr[], int l, int h) if (l >= h) return; if (arr[l] > arr[h]) int t = arr[l]; arr[l] = arr[h]; arr[h] = t; display(arr); -----> arr = if (h - l + 1 > 2) int t = (h - l + 1) / 3; mystery(arr, l, h - t); display(arr); -----> arr = mystery(arr, l + t, h); display(arr); -----> arr = mystery(arr, l, h - t); display(arr); -----> arr =

9 Data Structures and Algorithms, Midterm Exam, October 24, 2018 Page 8 First display: Second display: Third display: Fourth display: b) Explain in English what the mystery method does for any array and how it works? The method sorts any given array by first swapping the first and last elements if applicable. Then sorts the first 2/3rd of the array. After that it sorts the second 2/3rd of the array. Finally it sorts the first 2/3rd of the array again in case it has been messed up by the sorting of the second 2/3 of the array.

10 Data Structures and Algorithms, Midterm Exam, October 24, 2018 Page 9 Scratch paper

11 Data Structures and Algorithms, Midterm Exam, October 24, 2018 Page 10 Scratch paper

12 Data Structures and Algorithms, Midterm Exam, October 24, 2018 Page 11 Scratch paper