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Unit 7: Trees - Assignment Total points for Assignment: 35 points. Assignments must be submitted as a Microsoft Word document and uploaded to the Dropbox for Unit 7. All Assignments are due by Tuesday at 11:59 PM ET of the assigned Unit. NOTE: Assignment problems should not be posted to the Discussion threads. Questions on the Assignment problems should be addressed to the instructor by sending an email or by attending office hours. You must show your work on all problems. If a problem is worth 2 points and you only show the answer, then you will receive only 1 point credit. If you use a calculator or online website, give the source and tell me exactly what you provided as input. For example, if you used Excel to compute 16 * 16, state “I typed =16*16 into Excel and got 256. You may type your answer right into this document. Part I. Basic Computations 1. Using the following tree, name two vertices that are considered the following. Explain in your own words how you know: (a) parentchild (2 points). Answer: 202 - 401 Explanation: the parent of a vertex is the vertex connected to it on the path to the root, 202 is the parent and 401 is the child (b) sibling nodes (2 points): Answer: 301, 302, 303 Explanation: If two vertices are children of the same parent, then these two vertices are called siblings, 301, 302, 303 have the same parent that is 201 (c) leaf nodes (2 points) Answer: 301, 302, 303, 401 Explanation: the leaves are all terminal vertices Copyright 2010 – Kaplan University – All Rights Reserved 1|Page 2. Determine if each of the following graphs is considered a tree. Explain why or why not, using what you learned in this unit. a. (2 points) Answer: Not a tree Explanation: A tree is a connected graph with no cycles, and there is cycles in this graph b. (2 points) Answer: Yes Explanation: A tree is a connected graph with no cycles, and there is no cycles in this graph c. (1 point) Answer: No Explanation: A tree is a connected graph with no cycles, and there a cycle in this graph 3. Determine and sketch two different spanning trees for this graph: a. (1 point) Copyright 2010 – Kaplan University – All Rights Reserved b. (1 point) 2|Page 4. Consider this graph: 10 A 5 5 C E a. Determine the total weight for this graph. Show your work. (1 point) B 10 Answer: 122 Explanation: 5 + 10 +5 +10 +17 +15 + 5 +4+4+8+5+6+13+7+8 D 5 8 17 13 15 F 7 4 6 J G 4 5 H 8 b. Draw one spanning tree for this weighted graph and determine its weight. Show your work. (1 point) [Hint: One way to do this is to make a copy of the graph using cut/paste, then show your spanning tree.] Answer: I Total weight: Copyright 2010 – Kaplan University – All Rights Reserved 3|Page Part II. Case Study The Case of the Yip-it-tee DooDahs This week, the cast and crew of the “Patty Madeye Mysteries” will be filming on-location at the Yip-it-tee Amusement Park. The episode will be centered around a gang of purse-snatchers (they call them selves the “Doodahs”) who prey on vistors at the Yip-it-tee Amusement Park. While customers are having fun on the water rides, the purse-snatchers help themselves to any valuables that are left behind. The park owner has provided a detailed map of the amusement park. • • • • • • The Entrance Gate is identified by the little house (location H) Numbered locations identify the 7 rides in the park (see legend at the bottom) Food Stands are identified by the “hamburger” icons (A, C, D, F) Restrooms are identified as locations G, B, and E The Storage and Maintenance Facility is identified as location S. The distance between each of the 16 locations is measured in meters and listed in green letters. Copyright 2010 – Kaplan University – All Rights Reserved 4|Page Task #1: (4 points) The first scene of the episode will be focused on ride #6 (Wippee Water Slide). Your first task is to sketch a binary spanning tree (that would be a spanning tree in which each parent has no more than 2 children) using the locations as vertices and the paths as edges. Use location 6 (Wippee Water Slide) as the root of the tree. [Hint: Remember that all 16 locations must be included somewhere in your tree.] Answer: Task #2. (4 points) Your next task is to describe for Patty and the filming crew all the available paths from the Entrance (Location H). Sketch a spanning tree (it does not have to be a binary tree) using Location H as the root. Answer: Task #3. (4 points) Determine whether it is possible to visit each of the 16 locations on this map without retracing your steps on any given path. If so, describe the path below. If not, explain how you know that such a path is not possible. Answer: Task #4. (8 points) The crew will need to lay cable along the paths of the park so that they can film in each of the 16 locations. Your next task is to use a minimum spanning tree to determine how much cable the film crew will need to use. Sketch the tree and compute the total weight of the tree, using the distances indicated on the graph. [Hint: One way to do this is to print the graph, then use a highlighter to draw your spanning tree, then take a picture of your paper using your cell phone. Paste the picture into this document] Answer: Copyright 2010 – Kaplan University – All Rights Reserved 5|Page Unit 8: Algorithms - Assignment Total points for Assignment: 35 points. Assignments must be submitted as a Microsoft Word document and uploaded to the Dropbox for Unit 8. All Assignments are due by Tuesday at 11:59 PM ET of the assigned Unit. NOTE: Assignment problems should not be posted to the Discussion threads. Questions on the Assignment problems should be addressed to the instructor by sending an email or by attending office hours. You must show your work on all problems. If a problem is worth 2 points and you only show the answer, then you will receive only 1 point credit. If you use a calculator or online website, give the source and tell me exactly what you provided as input. For example, if you used Excel to compute 16 * 16, state “I typed =16*16 into Excel and got 256. You may type your answer right into this document. Part I. Basic Computations m 1. (5 points) Consider the following pseudo code for determining any sum in this form:  (5k 2 + 2k − 10) k =6 ComputeSum: INPUT m sum = 0 for k = 6 to m do sum = sum + 5*k^2 + 2*k -10 next k OUTPUT sum a. Are the inputs and outputs for this algorithm well defined? Explain. b. Identify an iterative operation in this algorithm. Copyright 2010 – Kaplan University – All Rights Reserved 1|Page 2. ( 5 points) Consider this algorithm used to sort an array in this form: S(n) = 5 8 6 3 5 1 6 19 7 4 BUBBLESORT: INPUT S, n (an array called S, with n elements) FOR i = 1 to n – 1 FOR j = 1 to n – 1 if S(j) > S(J+1) then exchange the contents of S(j) with S(j+1) NEXT NEXT OUTPUT S Using the following grid, show a trace for this algorithm using the following values: Copyright 2010 – Kaplan University – All Rights Reserved S = {25,1, 26,9, 2) n=5 2|Page Part II. Case Study Season Finale: Unlocking the Treasure The directors and producers of the “Patty Madeye Mysteries” have decided that the season finale for the show will be filmed in a secret location. They have identifed a remote island and visited it to determine what facilities and features are available on the island. The interns that were sent to scout out the island only had time to draw this map of the island and collect some other information. In order to open the lock to the treasure chest, you will need to solve a puzzle using discrete math. Shown below are the paths and distances that connect the 7 locations on the map. Copyright 2010 – Kaplan University – All Rights Reserved From To C D Distance (in meters) 80 D E 25 E G 80 D G 80 D A 100 A F 80 A G 80 B A 25 C B 25 C E 100 B E 100 G F 25 B F 80 A C 80 3|Page Clue #1: (3 points) One of the things that the film crew will need to know about is the condition of light during the day at the island. The only marking on the island was this binary representation of the longitude. Convert this longitude to decimal so that the crew can estimate time of sunrise and sunset. Show your work. Longitude: 10010100 Answer: Clue #2. (3 points) The cast and crew will be staying in the 8 tents shown in the Tent Village. Two of the tents will be set aside for to be used for “Hair and Makeup” (one for the males, one for the females). How many different ways can you select these special tents out of the 8 available? Answer: Clue #3. (3 points) Since the natives on this remote island only speak Spanish, it is important to know how many people in the cast & crew of 56 people. The producers have provided the following information and asked you to prepare a Venn Diagram depicting the various sets and subsets of this group. F = {members of the film crew} S = {members of the cast & crew who speak Spanish} M = {male members of the cast & crew} n(F) = 10 n(S) = 14 n(M) = 15 n(F and S) = 4 n(F and M) = 5 n(M and S and F) = 3 n(M or S or F) = 28 How many members of the cast & crew are male and speak Spanish, but are NOT on the Film Crew? Answer: Copyright 2010 – Kaplan University – All Rights Reserved 4|Page Clue #4. (3 points) A strange children’s game was found on the island with the following markings: 53 56 59 62 … Determine the 20th element of the sequence using the general term for the sequence. Answer Clue #5. (3 points) This clue requires solving a logic problem: p: q: r: s: cast member is male cast member speaks Spanish cast member has short hair it is NOT raining If a cast member is male and has short hair, does not speak Spanish, and it is raining, determine the truth value for the following expression: (~ p  q )  (r  ~ s )  ( s  ~ q) Answer: Clue #6. (3 points) To solve this clue, you must determine if there exists an Euler path or Euler circuit between the 7 locations on the island. If so, describe it. If not, explain how you know that such a path or circuit does not exist. Answer: Copyright 2010 – Kaplan University – All Rights Reserved 5|Page Clue #7. (2 points) Determine a Minimum Spanning Tree for the graph described by the paths on the island, using the distances between the locations as weights. What is the weight of the resulting spanning tree? Answer: Copyright 2010 – Kaplan University – All Rights Reserved 6|Page Clue #8. (5 points) Record your clues here, then use the following algorithm to determine the 5 digit combination to unlock the treasure. Clue #1: Longitude expressed in decimal: DIGIT1 DIGIT2 Clue #2: Number of combinations: Clue #3: n(M + S +~F) Clue #4: a20 = Clue #5: Truth Value Clue #6: Path, Circuit, or None? Clue #7: Weight of spanning tree: DIGIT3 DIGIT4 DIGIT5 To unlock the treasure, you must determine the combination for the lock: DIGIT1 = 2 FOR i = 1 to Clue#3 DIGIT1 = DIGIT1 + 2 NEXT DIGIT2 = (Clue#1 / 4) – Clue#2 IF Clue#4 = 71 THEN DIGIT3 = 4 ELSE DIGIT3 = 7 IF Clue#5 = True AND Clue #7 < 275 THEN DIGIT4 = 2 ELSE DIGIT4 = 1 IF Path exists in Clue #6 THEN DIGIT5 = 0 ELSE IF Circuit exists in Clue #6 THEN DIGIT5 = 2 ELSE DIGIT5 = 8 Copyright 2010 – Kaplan University – All Rights Reserved 7|Page
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