THIS IS THE QUESTION
Algorithms and Analysis COSC 2123/1285 Assignment 2: Algorithm Design & Complexity Analysis Assessment Type Individual Assignment. Submit online via Canvas → Assign- ments → Assignment 2. Clarifications/updates/FAQs can be found in Ed Forum → Assignment 2: General Discussions. Due Dates Deadline 1 Week 11, October 4, 23:59, for Problems 1-3, Deadline 2 Week 12, October 16, 23:59, for Problems 4-5 Marks 40 IMPORTANT NOTES • If you are asked to design an algorithm, you need to describe it in plain En- glish first, say a paragraph, and then provide an unambiguous pseudo code, unless specified otherwise. The description must include enough details to under- stand how the algorithm runs and what the complexity is roughly. All algorithm descriptions and pseudo codes required in this assignment are at most half a page in length. Worst-case complexity is assumed unless specified otherwise. • Standard array operations such as sorting, linear search, binary search, sum, max/min elements, as well as algorithms discussed in the pre-recorded lectures can be used straight away (but make sure to include the input and output if you are using them as a library). However, if some modification is needed, you have to provide a full description. If you are not clear whether certain algorithms/opera- tions are standard or not, post it to Ed Discussion Forum or drop Hoang a Team message. • Marks are given based on correctness, conciseness (with page limits), and clar- ity of your answers. If the marker thinks that the answer is completely not under- standable, a zero mark might be given. If correct, ambiguous solutions may still receive a deduction of 0.5 mark for the lack of clarity. • Page limits apply to ALL problems in this assignment. Over-length answers may attract mark deduction (0.5 per question). We do this to (1) make sure you develop a concise solution and (2) to keep the reading/marking time under control. Please do NOT include the problem statements in your submission because this may increase Turnitin’s similarity scores significantly. • This is an individual assignment. While you are encouraged to seek clarifications for questions on Ed Forum, please do NOT discuss solutions or post hints leading to solutions. • In the submission (your PDF file), you will be required to certify that the submitted solution represents your own work only by agreeing to the following statement: I certify that this is all my own original work. If I took any parts from elsewhere, then they were non-essential parts of the assignment, and they are clearly attributed in my submission. I will show that I agree to this honour code by typing “Yes": 2 1 Part I: Fundamental Problem 1 (8 marks, 1 page). Consider the algorithm mystery() whose input consists of an array A of n integers, two nonnegative integers `,u satisfying 0≤ `≤ u ≤ n−1, and an integer k. We assume that n is a power of 2. Algorithm mystery(A[0, . . . , (n−1)],`,u,k) if `== u then if A[`]== k then return 1; else return 0; end if else m = b(`+u−1)/2c; return mystery(A,`,m,k)+mystery(A,m+1,u,k); end if a) [2 marks] What does mystery(A[0..(n−1)],0,n−1,k) compute (0.5 mark)? Justify your answer (1.5 marks). b) [1 mark] What is the algorithmic paradigm that the algorithm belongs to? c) [2 marks] Write the recurrence relation for C(n), the number of additions required by mystery(A,0,n−1,k). d) [2 marks] Solve the above recurrence relation by the backward substitution method to obtain an explicit formula for C(n) in n. e) [1 mark] Write the complexity class that C(n) belongs to using the Big-Θ notation. 3 Problem 2 (8 marks, 1.5 pages). Let A be an array of n integers. a) [2 marks] Describe a brute-force algorithm that finds the minimum difference be- tween two distinct elements of the array, where the difference between a and b is defined to be |a− b| [1 mark]. Analyse the time complexity (worst-case) of the algorithm using the big-O notation [1 mark]. Pseudocode/example demonstration are NOT required. Example: A = [3,−6,1,−3,20,6,−9,−15], output is 2 = 3-1. b) [2 marks] Design a transform-and-conquer algorithm that finds the minimum dif- ference between two distinct elements of the array with worst-case time complexity O(n log(n)): description [1 mark], complexity analysis [1 mark]. Pseudocode/ex- ample demonstration are NOT required. If your algorithm only has average-case complexity O(n log(n)) then a 0.5 mark deduction applies. c) [4 marks] Given that A is already sorted in a non-decreasing order, design an al- gorithm with worst-case time complexity O(n) that outputs the absolute values of the elements of A in an increasing order with no duplications: description and pseudocode [2 marks], complexity analysis [1 mark], example demonstration on the provided A [1 mark]. If your algorithm only has average-case complexity O(n) then 2 marks will be deducted. Example: for A = [3,−6,1,−3,20,6,−9,−15], the output is B = [1,3,6,9,15,20]. 4 Problem 3. [10 marks, 1.5 pages] (Dijkstra’s algorithm + min-heap) Given a graph as in Fig. 1, we are interested in finding the shortest paths from the source a to all other vertices using the Dijkstra’s algorithm and a min-heap as a priority queue. Note that a min-heap is the same as a max-heap, except that the key stored at a parent node is required to be smaller than or equal to the keys stored at its two child nodes. In the context of the Dijkstra’s algorithm, a node in the min-heap tree has the format v(pv,dv), where dv is the length of the current shortest path from the source to v and pv is the second to last node along that part (right before v). For example, b(a,4) is one such node. We treat dv as the key of Node v in the heap, where v ∈ {a,b, c,d, e, f , g,h}. a b c d e f g h 7 1 5 6 2 15 3 49 7 3 9 1 3 Source 10 Figure 1: An input graph for the Dijkstra’s algorithm. Edge weights are given as integers next to the edges. For example, the weight of the edge (a,b) is 7. a) [1 mark] The min-heap after a(a,0) is removed is given in Fig. 2. The next node to be removed from the heap is c(a,1). Draw the heap after c(a,1) has been removed and the tree has been heapified, assuming that ∞≥∞ (note: no need to swap if both parent and children are ∞). No intermediate steps are required. c(a, 1) b(a, 7) h(−,∞) e(−,∞) f(−,∞) g(−,∞)d(a, 5) Figure 2: The min-heap (priority queue) after a(a,0) has been removed. b) [2 marks] Draw the heap(s) after each neighbour of c has been updated and the tree has been heapified (see the pseudocode in the lecture Slide 30, Week 9). If there are multiple updates then draw multiple heaps, each of which is obtained after one update. Note that neighbours are updated in the alphabetical order, e.g., d must be updated before e. No intermediate steps, i.e., swaps, are required. Follow the discussion on Ed Forum for how to update a node in a heap. 5 S: vertices whose shortest paths have been known Priority queue of remaining vertices 1 a(a,0) b(a,7), c(a,1),d(a,5), e(−,∞), f (−,∞), g(−,∞),h(−,∞) 2 a(a,0), c(a,1) 3 4 5 6 7 8 Table 1: Complete this table for Part c). c) [5 marks] Complete Table 1 with correct answers. You are required to follow strictly the steps in the Dijkstra’s algorithm taught in the lecture of Week 9. d) [2 marks] Fill Table 2 with the shortest paths AND the corresponding distances from a to ALL other vertices in the format a →?→?→ v | dv, for instance, a → c | 1. Shortest Paths Distances a a → a 0 b c a → c 1 d e f g h Table 2: Complete this table for Part d). 6 2 Submission The final submission (via Canvas) will consist of: • Your solutions to all questions in a PDF file of font size 12pt and your agreement to the honour code (see the first page). You may also submit the code in Problem 5. Late Submission Penalty: Late submissions will incur a 10% penalty on the total marks of the corresponding assessment task per day or part of day late, i.e, 4 marks per day. Submissions that are late by 5 days or more are not accepted and will be awarded zero, unless Special Consideration has been granted. Granted Special Considerations with new due date set after the results have been released (typically 2 weeks after the deadline) will automatically result in an equivalent assessment in the form of a practical test, assessing the same knowledge and skills of the assignment (location and time to be arranged by the coordinator). Please ensure your submission is correct and up-to-date, re-submissions after the due date and time will be considered as late sub- missions. The core teaching servers and Canvas can be slow, so please do double check ensure you have your assignments done and submitted a little before the submission deadline to avoid submitting late. Assessment declaration: By submitting this assessment, you agree to the assess- ment declaration - https://www.rmit.edu.au/students/student-essentials/ assessment-and- exams/assessment/assessment-declaration 3 Plagiarism Policy University Policy on Academic Honesty and Plagiarism: You are reminded that all sub- mitted work in this subject is to be the work of you alone. It should not be shared with other students. Multiple automated similarity checking software will be used to compare submissions. It is University policy that cheating by students in any form is not permit- ted, and that work submitted for assessment purposes must be the independent work of the student(s) concerned. Plagiarism of any form will result in zero marks being given for this assessment, and can result in disciplinary action. For more details, please see the policy at https://www.rmit.edu.au/students/student-essentials/assessment-and-results/ academic-integrity. 4 Getting Help There are multiple venues to get help. We will hold separate Q&A sessions exclusively for Assignment 2. We encourage you to check and participate in the Ed Discussion Fo- rum, on which we have a pinned discussion thread for this assignment. Although we encourage participation in the forums, please refrain from posting solutions or sugges- tions leading to solutions. 7 https://www.rmit.edu.au/students/student-essentials/assessment-and-results/academic-integrity https://www.rmit.edu.au/students/student-essentials/assessment-and-results/academic-integrity Part I: Fundamental Submission Plagiarism Policy Getting Help