UNIVERSITY OF MANITOBA. SIGNATURE: (in ink) (I understand that cheating is a serious offense)

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1 DATE: Nov. 21, 2013 NAME: (Print in ink) STUDENT NUMBER: SIGNATURE: (in ink) (I understand that cheating is a serious offense) INSTRUCTIONS TO STUDENTS: This is a 90 minute exam. Cearly show all necessary work. Texts, notes, cell phones, electronic translators, and other electronic devices are not permitted. This exam has a title page and 11 pages, including 2 blank pages for rough work. Check that you have all the pages. You may remove blank pages, but be careful not to loosen the staple. If you need more scrap paper, use the backs of pages. Solutions may be continued either on the backs of sheets or on the blank pages, but clearly indicate where a question is to be continued or the remainder will not be marked. To show that you read these instructions and for 2 bonus marks, circle the date on this page. Question Points Score Total: 100

2 PAGE: 1 of 11 [20] 1. Read each statement carefully and indicate, in the left margin beside each, whether it is true (T) or false (F). Statements that are absurd, uninterpretable, or which fail to be true in every case, are considered false. (a) The sum of the indegrees of the vertices of a digraph is always equal to the sum of its outdegrees. (b) A simple graph with 10 vertices cannot have 50 edges. (c) Every graph is isomorphic to itself. (d) If x, y are vertices in a digraph D in which there is no x y path or y x path then D is not connected. (e) Every tree is a forest. (f) A depth first search (DFS) algorithm that begins immediately with vertex v first examines a vertex, u, whose distance from v is maximum. (g) The number of distinct functions from a 7-set to a 4-set is 7 4. (h) If every edge of a connected graph has a different weight, then the graph has exactly one minimal spanning tree. (i) If A, B, C sets and A, C are disjoint, then A B C = A + B + C A B B C. (j) Warshall s algorithm finds the shortest path between vertices. 2. Short answers [3] (a) How many one-to-one functions are there from a five-element set to an eight-element set?

3 PAGE: 2 of 11 [3] (b) How many triples (a, b, c) exist, where a, b, c {1, 2, 3,..., 10} and a < b < c? [4] (c) In a class of 25 engineering students, 9 received a grade of A, 7 received a B, 5 received a C and the rest received a D (nobody fails). How many different ways could these grades be assigned to the 25 students? [4] (d) For n 4, how many subgraphs of K n are isomorphic to C 4 (the cycle of length 4)?

4 PAGE: 3 of 11 [6] 3. The first of the following three graphs is the Peterson Graph (seen in class). For each of the other two either label them to show that they are isomorphic to the first, or explain why they are not isomorphic to it [6] 4. Use inclusion/exclusion to count the number of possible 7-letter passwords that consist of letters (A,B,..., Z), numerical symbols (0,1,..., 9) and special symbols (restricted to #, $, %, &, *) subject to the restriction that one of each of the three types must appear in the password. (HINT: There are three different ways the condition can fail. Altogether, how many strings fail the condition?)

5 PAGE: 4 of 11 [11] 5. For the following graph, G: (a) Find the adjacency matrix A = A G (AzaToth, Wikimedia Commons) (b) Calculate A 4 (c) How many 3 5 paths are there in G? (d) How many distinct open paths of length 4?

6 PAGE: 5 of 11 [4] 6. (a) Construct a tree from the following ordered list of words using algorithm BUILDTREE (which used lexicographical ordering) as seen in class: Grumpy, Happy, Cupid, Dopey, Sleepy, Doc, Bashful, Sneezy, Dasher, Dancer, Prancer, Vixen, Comet, Donner, Blitzen [9] (b) Traverse the tree constructed above in each of the following ways, writing down the words in order processed: Preorder Inorder Postorder

7 PAGE: 6 of For the following (weighted) digraph D: 3 b e 3 a 2 3 d g c [9] (a) Use Dijkstras algorithm to find the shortest a f path in D. Use a table to keep track of the values of d(v) in each iteration. f

8 PAGE: 7 of 11 [9] (b) Apply Warshall s algorithm to D, showing the steps of evolution of the matrix P (until you are sure it has stopped changing).

9 PAGE: 8 of 11 [12] 8. (a) Suppose T is any tree with at least one edge. If a single edge is deleted from T, what kind of graph results? Be as specific as possible. (b) Let us call the result of part (a) the Tree Pruning Lemma. Use complete induction on the order of a tree to prove the result seen in class that the size of a tree is one less than its order. (Begin by introducing a proposition, P n, for each n 0.)

10 PAGE: 9 of 11 BONUS: Do ONLY ONE of the following questions, for extra credit. If you attempt more than one, indicate clearly which you wish marked. There may be more work than you can complete, but any significant progress on a bonus question will receive some credit; marks (up to about 10%) will be assigned according to the quality and completeness of your response, taking into account the difficulty of the question. A: Evaluate the expression S(n) = i+j+k=n n! for n = 0, 1, 2. Guess a general i!j!k! formula for the value of S(n). Prove this formula to be correct, using any method. B: Write an AL algorithm that takes as input an indexed list of numbers a 1,..., a n and the number n, and puts the numbers into increasing order. The only list-editing command available (i.e., you are not allowed to directly assign values to the a i s.) is flip(k), which reverses the order of the first k numbers in the list. for example, if the current list is 3, 5, 2, 7, 1, 4, then the command flip(4) returns the list 7, 2, 5, 3, 1, 4 (before, a 3 = 2; after, a 3 = 5). C: State the rule of inclusion/exclusion for the most general case: n sets, where n can be any positive integer. Use this rule to derive a formula for the number of derangements of an n-set. (A derangements of a set is a permutation of the set in which no element is returned to its original position. For example, 312 is a derangement of 123 but 321 is not.)

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