U08606 Discrete Mathematics

Semester 2 2013/14Assignment 2Please provide the following informationSubmissionPlease complete this cover sheet and attach it to the front of your assignment solutions. You can hand in your work at the start of the lecture, or post your assignment in the coursework box labelled U08606 in the R building,Wheatley campus on or before the due date. Coursework submitted late without a valid justification will not be accepted unless you have made an application for an extension due to mitigating circumstances.AssessmentThis assignment carries 25% of the total marks for the module. All questions should be attempted and marks will be allocated as indicated. Full details of your working should be handed in for assessment.Important noticeYou are bound by University regulations which require that coursework submitted for assessment purposes is genuinely your own and is not borrowed or stolen from any source, including fellow students. Ask yourpractical tutor or attend a surgery if you need advice.1U08606 DISCRETE MATHEMATICS – Coursework 2There are seven questions on this assignmentQuestion 1 (5 marks)The functions f :R ?R and g :R R ? are given by1 ( ) 3x f x ? ? and gx x () 3 2 ? ? .Determine the functions ?gf x ? ?( ) and ? f ? g x ?( ).Question 2 (5 marks)Of a company’s personnel, 9 work in manufacturing, 6 in marketing and 3 in accounting. Aproject group of 6 is to be formed to plan the launch of a new product.In how many ways can the project group be formed if(a) the project group includes two members from each department?(b) manufacturing is to have at least two representatives?Question 3 (5 marks)Consider the following graphs G1 and G2.1 12 2 3 4 53 45 6 6G1 G2For each of the graphs G1 and G2 , state whether or not the graph is(a) Eulerian and (b) Hamiltonian.If the graph is Eulerian give an Eulerian trail and if the graph is Hamiltonian give aHamiltonian cycle.2Question 4 (5 marks)The following table gives the distances (in miles) between six towns A, B, C, D, E and F.A B C D E FA – 7 26 28 37 16B 7 – 18 23 28 20C 26 18 – 8 11 30D 28 23 8 – 13 42E 37 28 11 13 – 29F 16 20 30 42 29 –(a) Use the minimal spanning tree algorithm to find a network of minimal total lengthlinking all six towns. What is the total length of the network?(b) Use the nearest neighbour algorithm (starting at town C) to find a circular routevisiting all six towns. What is the total length of the route?Question 5 (5 marks)Given the following table of modules and their prerequisites, use the Topological SortAlgorithm to find a total ordering in which the modules can be taken sequentially. Showyour working.Module PrerequisitemodulesAdvanced Mathematical Methods BBasic Mathematical Methods noneCore Electronics BDigital Imaging E, FElectronics CFourier Analysis A, C3Question 6 (10 marks)Let C denote the set of lower case characters of the English alphabet, and let S denote theset of strings of such characters. N is the set of natural numbers.The following basic primitive functions for manipulating elements of S are given:CHAR : S ? C where CHAR(s) is the first character of the non-empty string s,REST : S ? S where REST(s) is the string obtained from the non-empty string s byremoving its first character,ADD : C ? S ? S where ADD(c, s) is the string obtained by adding the character c to thefront of the string s,LEN : S ? N where LEN(s) is the number of characters in s,REV : S ? S where REV(s) is the string obtained from the non-empty string s byreversing the characters in s.(a) A function F : S ? S is given byF(s) = ADD(CHAR(REV(s)), REV(REST(REV(s)))).Evaluate F(s) when s = “apples”.Describe in words the effect of F on any string s containing at least two characters.(b) The function PEN(s) removes the penultimate (last but one) character from an inputstring s of length at least two, leaving the rest of the string intact. For example,PEN(“robes”) = “robs”. Express the function PEN(s) in terms of the basic primitivefunctions above.(c) Trace the values of u, v, s, LEN(v) and LEN(s) in the following algorithm when s =“potato”.beginInput swhile LEN(v) < LEN(s) dobeginu := CHAR(s);s := REST(s);v := ADD(u,v);endOutput vv := REV(v);endDescribe in words the effect of the algorithm on an arbitrary string s. What happens ifthe input string is of odd length?4Question 7 (10 marks)The following algorithm, bubblesort, sorts a list of n integers into increasing order bysuccessively comparing adjacent elements and interchanging them if they are in the wrongorder.Input x1, x2, x3, …, xn ; beginfor i := 1 to (n – 1) dofor j := 1 to (n – i) doif xj > xj+1thenbegintemp:= xj ;xj := xj+1 ;xj+1:= temp ; endelse {do-nothing};endOutput x1, x2, x3, …, xn ;(a) When n = 4, trace the values of i, j, x1, x2, x3 and x4 for the input values x1 = 3, x2 = 2,x3 = 7, x4 =1.(b) A time complexity function T(n) can be found for bubblesort by counting thenumber of times the comparison j x > j 1 x ? is made for a general positive input n.Show that T(n) is ? ?. 2 O n(c) Another sorting algorithm requires comparisons to sort a list of n integersinto ascending order. Is this algorithm more efficient than bubblesort? Brieflyjustify your answer.2 3 log n n