Module Contact: Dr Tony Bagnall, CMP Copyright of the University of East Anglia Version 1

Transcription

1 UNIVERSITY OF EAST ANGLIA School of Computing Sciences Main Series UG Examination DATA STRUCTURES AND ALGORITHMS CMP-5014Y Time allowed: 2 hours Section A (Attempt all questions: 75 marks) Section B (Attempt 1 question: 45 marks) Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. CMP-5014Y Module Contact: Dr Tony Bagnall, CMP Copyright of the University of East Anglia Version 1

2 Page 2 SECTION A 1. (a) Outline a general strategy for describing the complexity of an algorithm. (b) The PrefixAverages problem is specified as follows: Problem: PrefixAverages Given an array of n numbers, X, to output an array of numbers, A, such that A[i] is the average of X[1] to X[i]. Two algorithms for solving this problem are given below: Algorithm: PrefixAverages_1 for i := 1 to n do sum := 0 for j := 1 to i do sum := sum + X[j] endfor // j for loop A[i] := sum / i endfor // i for loop return A Algorithm: PrefixAverages_2 sum := 0 for i := 1 to n do sum := sum + X[i] A[i] := sum / i endfor // i for loop return A Analyze both algorithms and hence determine which is more efficient. [11 marks] 2. (a) Give a Java interface specifying a stack abstract data type. Assume stack elements are of a Comparable type. [5 marks] CMP-5014Y Version 1

3 Page 3 (b) Describe an array implementation for the stack interface. Your answer should include descriptions of how each of the methods specified in your answer to part 2(a) is implemented. You may assume that a method, doublearraysize(), is available for doubling the length of the underlying array when necessary. [10 marks] 3. (a) Describe the property a Binary Search tree (BST) must possess for it to be an AVL tree. (b) Draw diagrams to illustrate the effect in a binary tree of (i) a left rotation, (ii) a right rotation. [5 marks] (c) Consider the BST below. Draw the binary search tree that results when 39 is deleted from this tree. Explain what you have done. Is the resulting tree an AVL tree? Justify your answer When a set of keys is represented by a trie structure, to what do individual keys correspond in the trie structure? [2 marks] Given the following set of keys: S = {145,1452,1459,147,351,3514,35146,853,8538} (a) draw the trie corresponding to S; (b) represent the trie from (a) using a two-dimensional array. [7 marks] CMP-5014Y Version 1 TURN OVER

4 Page 4 5. (a) Define what is meant by a B-tree of order m. [5 marks] (b) Draw diagrams to illustrate the insertion of the following keys, in the order given below, into a 2-3 tree. 25,90,23,37,42,61,34,69,82,45,19,87,20. [10 marks] CMP-5014Y Version 1

5 Page 5 SECTION B 6. (a) Describe the insertion sort algorithm both informally and formally in pseudo-code. (b) Perform a best-case and worst-case analysis of insertion sort. (c) Describe the merge sort algorithm both informally and formally in pseudo-code. (d) Perform a worst-case analysis of merge sort. [12 marks] (e) Discuss the relative merits of insertion sort and merge sort, indicating when you might choose to use each. 7. (a) Define what is meant by a complete binary tree and describe how a one-dimensional array may be used to represent a complete binary tree. (b) Define what is meant by a (max) heap. (c) Describe a linear-time algorithm for constructing a heap. Draw diagrams to illustrate your answer by creating a heap from the following sequence of integers: [15 marks] 37,45,11,31,77,54,59,63,39,48,67,86,43. (d) Give the one-dimensional array corresponding to the final tree you obtained in part 7(c). (e) Describe the main ideas underlying the deletemax() method for a heap. Illustrate the operations of this method on the array you obtained in part (d). What is the worst-case run-time complexity of deletemax(). Justify your answer. (f) Suppose n elements are held in a max heap, h. Give an algorithm to output these elements in decreasing order. Determine the worst-case run-time complexity of your algorithm. [7 marks] END OF PAPER CMP-5014Y Version 1

6 Data Structures and Algorithms Exam Feedback This is the first year this has been a two hour exam, and the new structure possibly made it a little too long. I took this into account when marking and the overall marks distribution is in line with those in previous years. However, for next year I think we will restructure to be 4 from 6, as with programming 2. As usual, the distribution was bimodal. The median mark was strangely 50%. The top mark an impressive 95.8% and 11 students got over 80%. 24% of the cohort got a first class mark. However, 31% failed the exam and 22 students failed both programming 2 and data structures. Better coursework marks mean that many of these will pass the module overall, but it is still a concern. There were some huge discrepancies between coursework and exam marks Part A Question 1: Complexity: generally quite well done (mean 59%) but many of you still struggle with forming the run time complexity function and solving a summation. I think it is just lack of practice, so we will do more on this next year. Question 2: Lists. The wording and structure of this question possibly confused you, first asking for an interface, then a description. It was not completely clear whether you were required to write code, pseudo code or a description, so I took this into account when marking. Nevertheless, the mean mark was just 49%. Question 3: Binary Search Trees. The common mistake here was confusing the operations of deleting an element into a binary search tree and rebalancing a balanced binary search tree. To delete an element, the in order successor must take its place (in the example, 44). The left field of the parent of the in order successor (node 55) is set to point at the right child of the in order successor (node 48). Most people just reconstructed the tree in an ad hoc way or using rotations. I think this is partly due to the question structure (i.e. talk about rotations in part b made people think rotations were asked for in part c. We will make the distinction clearer on the module next year. Question 4: Tries: This was the best answered question (65% average mark), mistakes were mostly on forming the array representation of the trie. We will do more lab exercises for this in the future. Question 5: B-Trees. Most confusion was around the algorithm for inserting elements. The most common violation was of the requirement that all leaf nodes are at the same level. Again, more practice examples may have helped. Part B The split between the two choices was around 50/50. Question 6: Sorting. Marks were bought down by a failure in the analysis. The analysis of merge sort is fairly standard and we covered identical analysis twice in the lectures. Some people confused insertion sort with selection sort, and there were some bizarre solutions involving bubble sort.

7 Question 7: Heaps: generally better answered than the sorting question. One common mistake was thinking that a complete binary tree is in fact a heap, so just entering elements into an array solves part c). This is incorrect, a heap is a complete binary tree with the extra condition that all parents are bigger than (for a max heap) than off spring. So it is not a choice of heap, max heap or min heap. It is there are two types of heap: max and min. Another more common error was getting the heapify algorithm wrong, or rather using the obvious n long n algorithm of inserting them from the top rather than the order n algorithm of staring at level l-1 and sifting down. Part e) was actually really easy, if you understood it was just asking for heapsort.