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Q1:In a double-entry accounting system, every business transaction has to be entered as two separate transactions, in different two accounts, once as a debit and once as a credit. For example, if a business borrows $10,000 from a bank, the business should enter a debit of $10,000 to its Cash account and a credit of $10,000 to its notes Payable account. In order to be in balance, every debit in such a system must have a matching credit. Describe an efficient algorithm to test if a double-entry accounting system is in balance. What is the running time of your method in terms of n, the number of business transactions?See Answer
Q2:We saw in the lectures how to convert a 2-SAT instance into a digraph. Which arcs are created in the digraph from a clause (a V b) in the 2- SAT instance? Select all that apply.See Answer
Q3:Ques 1: Find the asymptotic uppler and lower bound of T(n) = 3T(n/2) + n/(log(n)) and T(n) = 3T (n-1). Use recursion tree or master theorem Show stepsSee Answer
Q4:Ques 2: find the asymptotic upper and lower bound of T(n) = 3T (n-1)? also need to use recursion tree or master theorem with steps.See Answer
Q5:1. The citizens of the Scandinavian town of Garnsholm are trying to solve a puzzle: is it possible to take a wundertur, that is, a cycle ride through Garnsholm that crosses each of its seven bridges exactly once? (Note that a wundertur does not have to start and end at the same point). North Island B South Island M (a) [3 marks]. Turn this map of Garnsholm into a graph, and find its degree sequence. Is this graph simple? (b) [4 marks]. Explain why there can be no wundertur through Garnsholm. (You can assume any results we proved in the lectures, but you should state what they are.) (c) [3 marks]. The citizens of South Island are so annoyed by your argument that they decide to demolish one of the three bridges connecting them to the rest of Garnsholm. Indicate on your graph which of the bridges they could choose to remove if they want to make it possible to take a wundertur. Give a brief justification.See Answer
Q6:2. A simple connected graph G has ten vertices of degree 7 and one of degree 6, and no other vertices. (a) [3 marks]. How many edges does it have? Be sure to state clearly any results that you use to get your answer. (b) [2 marks]. If this graph had a planar drawing, how many faces would it have? Be sure to state clearly any results that you use to get your answer. (c) [4 marks]. Explain why it follows that G cannot be planar.See Answer
Q7:3. Consider the following network of computers: E 3 The numbers labelling each edge indicate the number of Macquariecoin it costs to send a message along that network connection. We wish to minimise the costs of sending messages by finding paths of minimal length in this weighted graph. Use Dijkstra's algorithm to construct a shortest-path spanning tree rooted at computer C. In your an- swer, you should: (a) [3 marks]. Draw the graph and illustrate the spanning tree you construct within it. (b) [5 marks]. Show your working by displaying, at each step of execution, the contents of the fringe list, clearly identifying the edge which will be added to the tree next. (c) [3 marks]. List the length of each shortest path starting from the chosen root computer C to each of the other computers in the network.See Answer
Q8:1. Consider a Halloween party. Consider the Graph G with vertices denoting people at the party and an edge between 2 vertices if their costumes are related in some way. G contains 2k such vertices and k such edges, where k is a positive integer. (a) Prove that if G has no isolated vertices then it has exactly k connected components. (b) Prove that if G has no isolated vertices then it has no vertices of degree >= 2.See Answer
Q9:2. Consider a graph G that has k vertices are k-2 connected components, for k >= 4. What is the maximum possible number of edges in G? Prove your answer.See Answer
Q10:3. Let n >= 2 and p>= 1 be two positive integers. Let G be a graph with n vertices such that each vertex has p or more incident edges. Prove that if p > ((n-2)/2) then G is connected.See Answer
Q11:[10 pts] Suppose John's biking environment consists of n ≥ 3 landmarks, which are linked by bike route in a cyclical manner. That is, there is a bike route between landmark 1 and 2, between landmark 2 and 3, and so on until we link landmark n back to landmark 1. In the center of these is a mountain which has a bike route to every single landmark. Besides these, there are no other bike routes in the biking environment. You can think of the landmarks and the single mountain as nodes, and the bike routes as edges, which altogether form a graph G. A path is a sequence of bike routes. (a) [6 pts] Find the number of paths of length 2 in the graph in terms of n. Justify your answer. (b) [6 pts] Find the number of cycles of length 3 in the graph in terms of n. Justify your answer. (c) [6 pts] Find the number of cycles in the graph in terms of n. Justify your answer.See Answer
Q12:[10 pts] This problem will use the concept of a graph's degree sequence. This is a list of the degrees of all the vertices in the graph, in descending order of degree. For example, the graph a C (†) has degree sequence (4,3,3,2,2,0) because there is one node with degree 4 (c), two nodes with degree 3 (b and d), two nodes with degree 2 (a and e), and one node with degree 0 (f). For each of the following, either list the set of edges of a tree with vertex set {a, b, c, d, e, f} that has the stated degree sequence, or show that no such tree exists. There's no need to draw out the tree here, but it may help you to do so on paper. (a) [4 pts] (3,3,3,1, 1, 1) (b) [4 pts] (3,3,1, 1, 1, 1) (c) [4 pts] (4,3, 1, 1, 1, 1)See Answer
Q13:[10 pts] Imagine a country that has n N cities that are linked by highways, where a highway (which can be traversed in either direction) links exactly two cities and the only way to enter or exit a city is via highway(s). The president of this country hates wastefulness, so she ensured that for any pair of cities in the country, there exists only one sequence of highway(s) linking the two cities; in other words, the graph formed by the cities (vertices) and the highways (edges) is a tree. Further, suppose that there exists at least one city in this country such that there exist d EN distinct highways via which you can enter or exit the city. Prove that there are at least d cities such that: for each of the d cities, there is only one highway to enter or exit the city.See Answer
Q14:Let G be a graph with n ≥ 3 vertices that has a clique of size n-2 but no cliques of size n 1. Prove that G has two distinct independent sets of size 2.See Answer
Q15:Let p> 2. Consider the graph G, whose set of 2p vertices is [0..(2p-1)] and whose set of 2p + 1 edges is {0-1, 1-2,.... (p-1) p.p (p+1), (2p 2)-(2p-1), (2p-1)-0} U {0-p} For illustration, here is how G3 looks like: 3 Count the number of distinct spanning trees of Gp. Express your answer in terms of p. Justify your answer.See Answer
Q16:Let G (V, E) be a connected graph that has two distinct spanning trees. Prove that |E| > = IVI-1.See Answer
Q17:Prove that there exists a graph G with 5 nodes such that both G and G (the complement of G) have chromatic number > 3. Separately, prove that the cycle graph C5, is isomorphic to its complement.See Answer
Q18:Question 3 (25 marks) This question focuses on Part 3 (selection) and Part 4 (repetition). As inspiration for this question, you might like to revisit Activity 4.8 of Part 4, which uses a list of specific characters. The teacher wishes to see if her pupils can make sense of words in which certain letters have been removed. She requires a when[space]key_pressed script that will take a word from the user and display a new word that is the same as the original except that each vowel has been removed. For clarification, the vowels should be considered as: a, e, i, o and u. The user should input the word as a single string and the new word that is displayed should be a single string For example, if the user enters 'ratse then 'rs' is displayed. One way to create the new word to be displayed is to start with the new word as an empty string and then work through the letters in the original word from first to last, creating the new word incrementally. If the current letter is not a vowel then we add it to the new word, otherwise we ignore it. This might be visualised as in Figure 2. Original word New word so far: Test number 1 2 New word so far: Y 3 a a. Create and write down an algorithm to solve this problem. You might like to use the idea above, or an alternative of your own. New word so far: Y (10 marks) b. Create a when[space]key_pressed script to implement your algorithm. Depending on your algorithm and the way in which you choose to implement it, you may also decide to have a when_green_flag_clicked script. Take a screenshot of your script(s) and paste it into your TMA document (11 marks) c. Copy the following table into your TMA document and add to it two tests you would perform to check whether the completed program fulfils the specification. (Several tests might be appropriate; however, you are only required to add two.) Test purpose New word so far: First and last letters are vowels New word so far: 'rs Inputs Word apple Expected results New word displayed ppl New word so far: 'Y/nThe teacher wishes to see if her pupils can make sense of words in which certain letters have been removed. She requires a when[space]key_pressed script that will take a word from the user and display a new word that is the same as the original except that each vowel has been removed. For clarification, the vowels should be considered as: a, e, i, o and u. The user should input the word as a single string and the new word that is displayed should be a single string. For example, if the user enters 'raise' then 'rs' is displayed. One way to create the new word to be displayed is to start with the new word as an empty string and then work through the letters in the original word from first to last, creating the new word incrementally. If the current letter is not a vowel then we add it to the new word, otherwise we ignore it. This might be visualised as in Figure 2. Original word New word so far: "* r New word so far: 'r' New word so far: 'r' i New word so far: 'r' S New word so far: 'rs' CD New word so far: 'rs'See Answer
Q19:Questions: Part 1: Choose one of the following questions: 1. In the lost cow problem: what's the best you can do if there are three optional paths from the origin, instead of just two? 2. What's the competitive ratio of FIFO (first in first out - if eviction is needed, evict the page that entered the cache first)? How about LIFO (last in first out)? What can you say about the competitive ratio of MTF if all swaps cost 1, including those involving the searched entry? 3.See Answer
Q20:Part 2: Choose one of the following questions: 1. Argue why linked list update in the cost model where swaps involving the searched item cost 0 is not a metrical task system. Model the problem as a request-answer game. 2. Prove that if a MTS algorithm ais C-competitive against sequences of elementary cost vectors, then it is C-competitive against any sequence of (non-negative) cost vectors. 3. Show that layered graph traversal where the input satisfies that for all t IX: ≤ 2 has a deterministic 9-competitive online algorithm. You may assume that the minimum non-zero distance is 1.See Answer

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