Algorithms and Analysis Assignment 2: Algorithm Design & Complexity Analysis
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 Date Week 12, 11:59pm, May 27, 2022 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 is a binary tree, or more precisely, its root R. We denote by RLeft and RRight the left and the right children of R in the tree. Algorithm mystery(R : root of a binary tree) if R =∅ then return 0; else return 1+mystery(RLeft)+mystery(RRight); end if a) [2 marks] What does the algorithm compute? Justify your answer. b) [1 mark] What is the algorithmic paradigm that the algorithm belongs to? c) [2 marks] Assume that the tree is a perfect binary tree of height h (a binary tree is called perfect if every non-leaf node has precisely two children and all leaf-nodes are at the same level). Write the recurrence relation for C(h), the number of addi- tions required by mystery(). Convention: a single-node tree has height 0. d) [2 marks] Solve the above recurrence relation by the backward substitution method to obtain an explicit formula for C(h) in h for the perfect binary tree of height h. e) [1 mark] Write the complexity class that C(h) belongs to using the Big-Θ notation. Problem 2 (8 marks, 1.5 pages). The Australian Government Department of Health maintains a list of n people who have been double vaccinated against Covid, called list_double. At the end of 2022, an aggregated list of m people who have been vac- cinated the third time in 2022 with booster shots is created, called list_triple. Note that m ≤ n. People in each list are identified by their unique ID numbers (e.g., passport numbers) and are entered into the lists in chronological order. It is required to design an algorithm that takes as input the two lists and returns a new list of people who are double vaccinated but haven’t received their third shots. A reminder will be sent to all people in this list to take their booster shots. a) [2 marks] Design (describe + complexity analysis) a brute-force algorithm that per- forms the aforementioned task [1 mark]. Analyse the time complexity of the algo- rithm using the big-O notation [1 mark]. Pseudocode is NOT required. b) [3 marks] Design (describe + complexity analysis) a transform-and-conquer algo- rithm with time complexity O(n logm) that performs the aforementioned task using at most a constant amount of extra space (apart from the input/output). c) [3 marks] Design (describe + pseudocode + complexity analysis) an algorithm with (average-case) time complexity O(n) that performs the aforementioned task [2 marks]. There is NO restriction on the space complexity. 3 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 4 8 6 3 6 1 4 1 1 4 2 2 73Source 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 4. 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 b(a,4). Draw the heap after b(a,4) has been removed and the heap has been heapified (fixed), assuming that ∞≥∞. No intermediate steps are required. b(a, 4) c(a, 8) d(a, 6) e(−,∞) f(−,∞) g(−,∞) h(−,∞) Figure 2: The min-heap (priority queue) after a(a,0) has been removed. b) [2 marks] Draw the heap(s) after the neighbours of b have been updated and the heap 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., b must be updated before c. No intermediate steps are required. Follow the dis- cussion on Ed Forum for how to update a node in a heap. 4 S: vertices whose shortest paths have been known Priority queue of remaining vertices 1 a(a,0) b(a,4), c(a,8),d(a,6), e(−,∞), f (−,∞), g(−,∞),h(−,∞) 2 a(a,0),b(a,4) 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 → b | 4. Shortest Paths Distances a a → a 0 b a → b 4 c d e f g h Table 2: Complete this table for Part d). 5 2 Part II: Advanced Problem 4 (7 marks, 2 pages). Each Bitcoin transaction, in its simplest form, has one input coin and several output coins (see Fig. 3). The input coin is genuine in the sense that it is spent by the transaction. This creates the issue of traceability for Bitcoin, that is, the entire history of each coin can be traced, e.g., how it was created, split, spent, and by which users, etc., which can sometimes be too revealing and undesirable for the users. To make the cryptocurrency untraceable, it has been proposed that instead of using only genuine inputs, the cryptocurrency wallet should also include other fake inputs so that an observer won’t know which input is the genuine one for that transaction. We will ignore the fact how this can be done technically and only focus on the mixing part where genuine and fake inputs are mixed in the transactions to provide untraceability. Assume that each coin can be used as a genuine input for exactly one transaction and that the number of coins is the same as the number of transactions in the system. Figure 3: Example of input and output coins in a Bitcoin transaction. Figure 4: Examples of mixing genuine and fake inputs for transactions to provide untraceability. The mixing strategy in a) fails because an observer can determine a unique mapping M that matches genuine coins with transactions: M[1] = 2, M[2] = 3, M[3]= 4, and M[4]= 1. The mixing in b) is not the best in disguising the actual mapping, but still confuses an observer as there are two possibilities to match the coins with the transactions. The mixing strategy sometimes fails because not all mixings are done in a proper way. For example, in Fig. 4 a), an observer can determine which coin is the genuine input of which transaction for ALL the coins. We refer to such a mixing as a bad mixing strategy. 6 Note that a mixing strategy can be described by a bipartite graph with the left-side vertices corresponding to the coins and the right-side vertices corresponding to the trans- actions, and there is an edge between a coin Ci and a transaction T j if Ci is included in T j as an input (genuine or fake). Design an algorithm of time complexity O(n+m), where n is the number of coins (or equivalently, the number of transactions) and m is the num- ber of edges in the bipartite graph, that determines if a particular mixing is bad, i.e., a unique mapping M that maps ALL coins to their corresponding transactions could be found. The algorithm must output M if the mixing is bad. a) [3 marks] Describe the algorithm in