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Q1: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
Q2: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
Q3:1. 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
Q4:2. 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
Q5:3. 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
Q6:2. 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
Q7:[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
Q8:[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
Q9:[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
Q10: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
Q11: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
Q12:5. Each game of bridge involves two teams of two partners each. In a bridge club, a game can not be played if any two of the four people have previously been partners. At the start of the evening at this bridge club, there are 14 members present who play games until each has played exactly four times. After this, they are able to play six more games (total 12 more partnerships). Just as they are ready to wrap-up for the night, a 15th member arrives. Prove that the arrival of this new member allows at least one more game to be played.See Answer
Q13:Ꮖ 4. The following network N has source x and sink y with arc capacities as shown. An initial flow of this network is given in parentheses. Starting from this flow, use the labelling algorithm to find a maximum flow and a minimum cut in N. In each iteration of the algorithm, you are required to label all vertices that can be labelled. Show every stage of the algorithm, state explicitly the value of the maximum flow found, and give explicitly the minimum cut found and its capacity. [8 marks] 1See Answer
Q14:3. Prove that any connected graph with minimum degree no less than 3 contains at least three distinct cycles. [5 marks]See Answer
Q15:Q1 Because of the incredible popularity of his class Math for Computer Science, TA Mike decides to give up on regular office hours. Instead, he arranges for each student to join some study groups. Each group must choose a representative to talk to the staff, but there is a staff rule that a student can only represent one group. The problem is to find a representative from each group while obeying the staff rule. a) Explain how to model the delegate selection problem as a bipartite matching problem. (This is a{modeling problem); we aren't looking for a description of an algorithm to solve the problem.) b) The staff's records show that each student is a member of at most 4 groups, and all the groups have 4 or more members. Is that enough to guarantee there is a proper delegate selection? Explain.See Answer
Q16:Q2 6.042 at MIT is often taught using recitations. Suppose it happened that 8 recitations were needed, with two or three staff members running each recitation. The assignment of staff to recitation sections, using their secret code names, is as follows: \item R1: Maverick, Goose, Iceman R2: Maverick, Stinger, Viper R3: Goose, Merlin R4: Slider, Stinger, Cougar R5: Slider, Jester, Viper R6: Jester, Merlin R7: Jester, Stinger R8: Goose, Merlin, Viper Two recitations can not be held in the same 90-minute time slot if some staff member is assigned to both recitations. The problem is to determine the minimum number of time slots required to complete all the recitations. a) Recast (translate) this problem as a question about coloring the vertices of a particular graph. b) Draw the graph and explain what the vertices, edges, and colors represent. (One free flow-chart building application that will work to make such an image is https://draw.io c) Show a coloring of this graph using the fewest possible colors; prove why no fewer colors will work. d) What is the resulting schedule of recitations implied by the coloring?See Answer
Q17:How many strongly connected components does the graph below have?See Answer
Q18: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
Q19: 3. (10 pts.) You are in charge of the United States Mint. The money-printing machine has developed a strange bug: it will only print a bill if you give it one first. If you give it a d-dollar bill, it is only willing to print bills of value d² mod 400 and d² + 1 mod 400. For example, if you give it a $6 bill, it is willing to print $36 and $37 bills, and if you then give it a $36-dollar bill, it is willing to print $96 and $97. You start out with only a $1 bill to give the machine. Every time the machine prints a bill, you are allowed togive that bill back to the machine, and it will print new bills according to the rule described above. You want to know if there is a sequence of actions that will allow you to print a $20 bill, starting from your $1 bill.Model this task as a graph problem: give a precise definition of the graph (what are the vertices and edges)involved and state the specific question about this graph that needs to be answered. Give an algorithm to solve the stated problem and give the running time of your algorithm.See Answer
Q20: ots.) Consider the following directed graph. 1. What are the sources and sinks of the graph? 2. Give one linearization of this graph. 3. How many linearization does this graph have?See Answer

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